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by Copyright Ryan Paul Russell 2004 The Dissertation Committee for Ryan Paul Russell certifies that this is the approved version of the following dissertation: Global Search and Optimization for Free-Return Earth-Mars Cyclers Committee: Cesar A. Ocampo, Supervisor Robert H. Bishop Wallace T. Fowler David G. Hull Timothy P. Crain Global Search and Optimization for Free-Return Earth-Mars Cyclers by Ryan Paul Russell, B.S., M.S. Dissertation Presented to the Faculty of the Graduate School of The University of Texas at Austin in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy The University of Texas at Austin August, 2004 Acknowledgements I would like to give thanks to the God of the Christian faith, from whom all blessings flow; to my wonderful bride, Jen, for loving, encouraging, and supporting me despite my faults; to my parents, for all of their sacrifices and for raising me to chase my dreams; to my sister, for walking through childhood with me and blazing many common paths; to my family, for giving me a tremendous foundation for life and for their unconditional support; and finally, to my friends and all others who have supported, encouraged, or simply tolerated me in this profound yet obscure endeavor. My time at the University of Texas has been rich with experiences, both personal and academic. I would like to extend a formal thank you to each member of the current dissertation committee. In particular, I thank Dr. Fowler for serving as my advisor for a Masters degree. His passion for teaching and public service is contagious. Also, I thank Dr. Hull for sharing his expertise in optimal control, and answering many questions along the way. Of course, I am very thankful for my advisor, Cesar Ocampo, for his guidance and extraordinary enthusiasm for astrodynamics. I would also like to acknowledge Troy McConaghy, a peer from Purdue who is also working on cyclers, for his willingness to collaborate. Lastly, I would like to acknowledge the sources of funding for this research and graduate school in general. These include the United States Department of Defense, NASA's Goddard Space Flight Center, the American Astronautical Society, the University of Texas Graduate Studies Office, and the Department of Aerospace Engineering and Engineering Mechanics. iv Global Search and Optimization for Free-Return Earth-Mars Cyclers Publication No._____________ Ryan Paul Russell, Ph.D. The University of Texas at Austin, 2004 Supervisor: Cesar A. Ocampo A planetary cycler trajectory is a periodic orbit that shuttles a spaceship indefinitely between two or more planets, ideally using no powered maneuvers. Recently, the cycler concept has been revived as an alternative to the more traditional human-crewed Mars missions. This dissertation investigates a class of idealized Earth-Mars cyclers that are composed of Earth to Earth free-returns trajectories patched together with gravityassisted flybys. A systematic method is presented to identify all feasible free-return trajectories following an arbitrary gravity-assisted flyby. The multiple-revolution Lambert's Problem is solved in the context of half-rev, full-rev, and generic returns. The solutions are expressed geometrically, and the resulting velocity diagram is a mission-planning tool with applications including but not limited to Earth-Mars cyclers. Two different global search methods are then developed and applied, taking advantage of all three types of free-return solutions. The first method results in twentyfour ballistic cyclers with periods of two to four synodic periods, ninety-two ballistic v cyclers with periods of five or six synodic periods, and hundreds of near-ballistic cyclers. Most of the solutions are previously undocumented. The second and more generalized method only searches for the more practical cyclers with repeat times of three-synodic periods or less. This global approach uses combinatorial analysis and minimax optimization to identify 203 promising ballistic or near-ballistic mostly new cyclers. Finally, the feasibility of accurate ephemeris versions of the promising idealized cyclers is demonstrated. An efficient optimization method that utilizes analytic gradients is developed for long duration, ballistic, patched-conic trajectories with multiple flybys. The approach is applied at every step of a continuation method that transitions the simple model solutions to accurate ephemeris solutions. Hundreds of ballistic launch opportunities for accurate ephemeris cyclers are documented. Remarkably, twenty parent cyclers are found to have an average total maneuver requirement over all twenty-one launch windows of less than 100 m/s for seven-cycle propagations. In summary, the Earth-Mars cycler problem is fully addressed from the problem definition stage all the way to solutions in a reasonably accurate ephemeris model for a broad class of cyclers. The most promising solutions are viable options for sustaining a Earth-Mars transportation system. vi Table of Contents List of Figures ................................................................................................................. x List of Tables ................................................................................................................. xii Nomenclature ............................................................................................................... xiv 1 Introduction............................................................................................................. 1 1.1 1.2 1.3 1.4 1.5 Cycler Definitions and Solar System Models ........................................................................... 1 Motivation................................................................................................................................. 3 Previous Research on Cyclers................................................................................................... 4 Contributions ............................................................................................................................ 7 Dissertation Organization ......................................................................................................... 9 2 Free-Return Trajectories ..................................................................................... 10 2.1 2.2 2.3 2.4 2.4.1 2.4.2 2.5 2.6 2.7 2.7.1 2.7.2 2.7.3 2.7.4 2.7.5 2.8 Chapter Summary ................................................................................................................... 10 Introduction............................................................................................................................. 10 Lambert's Problem.................................................................................................................. 12 Generalized Full-Revolution Return Solutions ....................................................................... 18 Expressions Based on Orbital Periods................................................................................ 19 Additional Insight from Lambert's Equation ..................................................................... 19 Generalized Half-Revolution Return Solutions ...................................................................... 25 Generic Free-Return Solutions................................................................................................ 30 Flyby Velocity Diagrams ........................................................................................................ 31 Full-revolution Velocity Diagrams .................................................................................... 32 Half-revolution Velocity Diagrams.................................................................................... 35 Generic Free-Return Velocity Diagrams............................................................................ 41 Flyby Diagrams Containing All Free-Return Solutions ..................................................... 48 Numerical Considerations and Velocity Diagram Generation ........................................... 52 Chapter Conclusions ............................................................................................................... 54 3 Idealized Free-Return Cyclers Composed of Transfers, 2 Transfers, and 3.1 3.2 3.3 3.4 Chapter Summary ................................................................................................................... 56 Introduction............................................................................................................................. 57 Simplified Solar System.......................................................................................................... 59 Free-return Trajectories........................................................................................................... 60 One or More Identical Generic Returns..................................................................... 56 vii 3.4.1 3.4.2 3.4.3 3.5 3.5.1 3.5.2 3.6 3.7 3.8 3.9 2 Full-Revolution Returns................................................................................................ 61 1 Half-Revolution Returns ............................................................................................... 62 One or More Identical Generic Free Returns ..................................................................... 63 Cyclers With One Generic Return .......................................................................................... 64 Preliminaries ...................................................................................................................... 64 Turn Angle Optimization ................................................................................................... 69 Cyclers with Multiple Identical Generic Returns.................................................................... 76 Algorithm Overview ............................................................................................................... 78 Results..................................................................................................................................... 79 Chapter Conclusions ............................................................................................................... 93 4 Idealized Free-Return Cyclers Composed of n Transfers and One or More 4.1 4.2 4.3 4.4 4.5 4.6 4.6.1 4.6.2 4.6.3 4.6.4 4.7 4.7.1 4.7.2 4.7.3 4.8 4.9 Chapter Summary ................................................................................................................... 95 Introduction............................................................................................................................. 96 Motivation and A Sampling of Expected Results ................................................................... 98 Idealized Free-Return Cycler Definition............................................................................... 101 Problem Setup....................................................................................................................... 101 Combinatorics ....................................................................................................................... 107 General Counting ............................................................................................................. 108 n-Tuples ........................................................................................................................... 109 Beaded Necklace Problem ............................................................................................... 110 Partitions of an integer ..................................................................................................... 111 Solution Method.................................................................................................................... 112 Main Algorithm and Explanation..................................................................................... 113 Post-processing ................................................................................................................ 123 Using full-revolution returns for the Earth-Mars transfers............................................... 124 Results................................................................................................................................... 125 Chapter Conclusions ............................................................................................................. 135 Identical or Different Generic Returns....................................................................... 95 5 Finding Cyclers In an Accurate Solar System ................................................. 137 5.1 5.2 5.3 5.4 Chapter Summary ................................................................................................................. 137 Introduction........................................................................................................................... 138 Problem Definition and Assumptions ................................................................................... 142 Solution Method.................................................................................................................... 146 viii 5.4.1 5.4.2 5.4.3 5.4.4 5.4.5 5.4.6 Continuation Method........................................................................................................ 146 Non-Analytic Solutions.................................................................................................... 147 Gradient Method .............................................................................................................. 148 Multiple Shooting Method ............................................................................................... 149 Ballistic Flybys with Powered Maneuvers at the Sphere of Influence ............................. 152 Analytic Gradient Calculations ........................................................................................ 156 Partials of the final state with respect to the initial state......................................... 161 Partials of the final state with respect to the initial time ......................................... 163 Partials of the powered v constraint ..................................................................... 164 5.4.6.1 5.4.6.2 5.4.6.3 5.4.7 5.4.8 5.5 5.6 Constraint infeasibilities and post-processing considerations .......................................... 165 Algorithm Description...................................................................................................... 168 Results................................................................................................................................... 172 Chapter Conclusions ............................................................................................................. 183 6 Conclusions.......................................................................................................... 185 6.1 6.2 6.3 6.4 6.5 Dissertation Summary........................................................................................................... 185 Global Claims ....................................................................................................................... 188 General Conclusions ............................................................................................................. 189 Most Promising Cyclers........................................................................................................ 190 Final Remarks ....................................................................................................................... 190 Appendix A: Dynamics of a Translating, Rotating, and Pulsating (TRP) Reference Frame ........................................................................................................................... 192 Appendix B: Combinatorics and Generating Algorithms ..................................... 196 n-Tuples algorithm .............................................................................................................................. 196 Beaded Necklace Problem................................................................................................................... 197 Partitions of an integer ........................................................................................................................ 199 Appendix C: Cycler Trajectories Using a Ephemeris Model ................................ 201 References.................................................................................................................... 246 Vita ............................................................................................................................... 253 ix List of Figures Figure 2.1: Procedure to find vacant foci................................................................................................... 13 Figure 2.2: The four possible arcs of Lambert's Equation for a given N .................................................. 15 Figure 2.3: Example multiple revolution Lambert solution (r1=r2=1 AU, =2N + 4 /7) ..................... 17 Figure 2.4: Limit r1 r2. Lambert transfers using . ............................................................................ 24 Figure 2.5: Limit r1 r2. Lambert transfers using ................................................................................ 24 Figure 2.6: Limit r1 r2. All Lambert transfers...................................................................................... 25 Figure 2.7: Half-rev solutions, |r1|=|r2|...................................................................................................... 27 Figure 2.8: Zoom view of the N=4 curve .................................................................................................. 28 Figure 2.9: Half-rev solutions, |r1| |r2|..................................................................................................... 29 Figure 2.10: Example generic return solutions ......................................................................................... 31 Figure 2.11: Full-rev velocity diagram, N=7, M=4 ................................................................................. 34 Figure 2.12: Full-rev velocity diagram, N=111, M=4 ......................................................................... 34 Figure 2.13: Outbound velocity components for half-rev free-returns...................................................... 36 Figure 2.14: Half-rev velocity diagram, M=3.5, N=2 slow ..................................................................... 38 Figure 2.15: Half-rev velocity diagram, M=3.5 ....................................................................................... 40 Figure 2.16: N=1 solutions. TOF=06 flyby body periods ..................................................................... 43 Figure 2.17: All Solutions N=115. TOF=06 .................................................................................... 43 Figure 2.18: vs. v for generic returns to a body in circular orbit. N=07. ........................................ 44 Figure 2.19: vs. v for generic returns to a body in circular orbit. N=915. ...................................... 45 Figure 2.20: TOF vs. v for generic returns to a body in circular orbit. N=07..................................... 46 Figure 2.21: TOF vs. v for generic returns to a body in circular orbit. N=815................................... 47 Figure 2.22: Free-return solutions for |v|=0.1838 AU/TU....................................................................... 49 Figure 3.1: Three types of free-returns...................................................................................................... 61 Figure 3.2: Flyby velocity diagram including potential 1 half-rev and 2 full-rev free-return transfers.62 Figure 3.3: Conditions required to re-initiate generic return trajectory..................................................... 65 Figure 3.4: Velocity diagram of a gravity-assisted flyby that re-initiates a generic return ...................... 66 Figure 3.5: Gravity-assisted flyby velocity diagrams with 2 full-rev returns ......................................... 67 Figure 3.6: Gravity-assisted velocity diagram with 1 half-rev return ..................................................... 68 Figure 3.7: Velocity diagram of a 1 half-rev return with ve fixed........................................................... 69 Figure 3.8: Turn angle optimization for a re-initiation that requires five flybys....................................... 73 Figure 3.9: Algorithm summary................................................................................................................ 79 Figure 3.10: Top-down views of Cycler 4.3.1.-5 and Cycler 2.5.1.+0 ..................................................... 85 x Figure 3.11: Three Dimensional view of Cycler 3.1.2.+1 ........................................................................ 87 Figure 3.12: Number of cyclers found vs. period...................................................................................... 88 Figure 4.1: Example v diagram with free-return solutions for |v|=0.1838 AU/TU.............................. 102 Figure 4.2: Velocity diagram for the discussed example cycler.............................................................. 122 Figure 5.1: Trajectory leg diagram.......................................................................................................... 151 Figure 5.2: Powered v required at the SOI of a flyby. ........................................................................... 155 Figure 5.3: Path from simple to real model............................................................................................. 169 Figure 5.4: General algorithm ................................................................................................................. 170 Figure 5.5: Total v for optimized ephemeris cyclers. Part I.................................................................. 174 Figure 5.6: Total v for optimized ephemeris cyclers. Part II ................................................................ 175 Figure 5.7: Ballistic Aldrin cycler 6.399G1(#1), launch: Aug 6, 2003 ................................................... 181 Figure 5.8: Aldrin cycler from Figure 5.7 plotted in TRP frame ............................................................ 181 Figure 5.9: High energy cycler 8.049gGf2(#188), launch: July 26, 2042, vtot=420 m/s, avg. transit 95 days............................................................................................................................................ 182 Figure 5.10: High energy cycler from Figure 5.9 plotted in TRP frame ................................................. 182 Figure A1: Translating, Rotating, and Pulsating Reference Frame......................................................... 192 Figure A2: Example Trajectory in the TRP Frame ................................................................................. 195 xi List of Tables Table 2.1: Properties of Lambert solutions in the limit as r1 r2 [2N+, 2(N+1) -] .................. 21 Table 2.2: N=1 generic return solutions.................................................................................................... 42 Table 2.3: Details of generic return solutions in Figure 2.22.................................................................... 51 Table 3.1: Possible combinations for free-returns for a cycler with h=3 and s=2..................................... 58 Table 3.2: Optimal number of flybys necessary to re-initiate a generic return ......................................... 72 Table 3.3: Coordinates for heliocentric flyby velocities ............................................................................ 75 Table 3.4: Two, three, and four-synodic period ballistic or near-ballistic cyclers ARMIN=0.9, and TRMIN=0.85 ....................................................................................................................................... 83 Table 3.5: Cycler 2.5.1.+0 ........................................................................................................................ 84 Table 3.6: Cycler 3.1.2.+1 ........................................................................................................................ 84 Table 3.7: Cycler 4.3.1.-5 ......................................................................................................................... 84 Table 3.8: Cycler 4.5.2.-2 ......................................................................................................................... 84 Table 3.9: Five-synodic period ballistic (or near) cyclers ARMIN=0.9, and TRMIN=0.85............................ 90 Table 3.10: Six-synodic period ballistic (or near) cyclers. Part I ARMIN=0.9, and TRMIN=0.85 ................. 91 Table 3.11: Six-synodic period ballistic (or near) cyclers. Part II ARMIN=0.9, and TRMIN=0.85............... 92 Table 4.1 Examples of improved solutions to cyclers documented in Chapter 3..................................... 100 Table 4.2: Inbound v for free-returns a ................................................................................................... 103 Table 4.3: Criteria for a sequence of free-returns to be a ballistic cycler................................................. 104 Table 4.4: Total TOF allotted for generic returns, = 2.135384 yra ....................................................... 106 Table 4.5: Generic return solutions for |v|=5.5 km/s (0.1838 AU/TU) ................................................. 115 Table 4.6: Good n-tuples......................................................................................................................... 116 Table 4.7: Original list of ballistic cyclers .............................................................................................. 124 Table 4.8: Rearranged list of ballistic cyclers ........................................................................................ 124 Table 4.9: Ballistic two-synodic period cyclers ....................................................................................... 127 Table 4.10: Ballistic three-synodic period cyclers with generic transit legs. Part I. ............................... 128 Table 4.11: Ballistic three-synodic period cyclers with generic transit legs. Part II. ............................. 129 Table 4.12: Ballistic three-synodic period cyclers with full-rev transit legs. .......................................... 132 Table 4.13: Near-ballistic three-synodic period cyclers........................................................................... 134 Table 4.14: Summary of results ............................................................................................................... 134 Table 5.1: Earth-Mars resonances........................................................................................................... 143 Table 5.2: Integer multiples of circular-coplanar cycler repeat times ..................................................... 144 Table 5.3: Summary of parameters and constraints a ............................................................................... 154 xii Table 5.4: Mean Elements at J20005 ........................................................................................................ 169 Table 5.5: Characteristics of smallest v solutions for selected parent cyclers ....................................... 178 Table 5.6: Solutions sorted by average v over all the launch dates....................................................... 180 Table 6.1: Summary of global claims ...................................................................................................... 188 Table B1: The partitions of the integers 1 through 11a ........................................................................... 200 xiii Nomenclature Symbol , FR GR a E F fj h hj i Description latitude and longitude in reference to the velocity diagrams. latitude of the full revolution circle on the velocity diagram latitude of v+ and v- for a generic return semi-major axis eccentric anomaly focus of a transfer ellipse number of flybys required to re-initiate the jth generic return total number of half-years allotted for full or half-revolution free-returns during one cycler period, h 0 total number of half-years allotted for full or half-revolution returns associated with the jth generic return, 0 hj h the ith solution from a multiple revolution Lambert's problem. The absolute value of i indicates the number of complete revolutions made by the free-return transfer. The sign of i indicates a fast or slow solution. J M N p r s sj t T v W x , , S objective function number of complete revolutions made by the celestial body number of complete revolutions made by the spacecraft The period of the cycler is p synodic periods. p 1 position vector total number of identical generic returns during one cycler period, s 1 jth generic return time period velocity vector weighted tuning parameter for constraint violations cartesian state vector classic orbital element set intermediate variables for Lambert's Equation unconventional orbital element set flight path angle xiv , , r1, r2, c MAX MIN MINIMAX-j input geometry for Lambert's equation gravitational parameter of the primary synodic period referencing angle for generic returns heliocentric turning angle geocentric turning angle the minimized maximum of MINIMAX-j for j=1s minimum turning angle required to achieve a full-revolution free-return following a generic free-return. the minimized maximum turn angle required to re-initiate the jth generic return Subscript + 0 avail B e F f H l ns p plan r req s v Description before a flyby after a flyby initial relative to celestial body before or after flyby available celestial body Earth full-revolution return final half-revolution return long period solution number of solutions Periapse planet radial component required short period solution vacant focus transverse component xv 1 Introduction 1.1 CYCLER DEFINITIONS AND SOLAR SYSTEM MODELS A cycler orbit can be generally defined as a perfectly repeatable round-trip trajectory that shuttles between any two or more celestial bodies. The cycler concept is an alternative approach to developing and maintaining a human presence on Mars. 1,2 Rather than accelerating, decelerating, and possibly discarding the habitation module for each leg of an interplanetary flight, a cycler system provides a reusable vehicle that once placed in orbit can shuttle crews and cargo between planets using very little fuel. A cycler is considered to be ballistic if once set in motion, the only maneuvers required are gravity-assisted flybys with physically realizable altitudes. Typically, the main disadvantages of cyclers are the lengthy durations between successive interplanetary legs and the large hyperbolic velocities encountered at the planets. In order for a cycler trajectory to be exactly periodic, the relative geometry of all the associated bodies must also be periodic. For this reason, true cyclers exist only in simple solar systems with perfectly repeatable geometries, and the period of a cycler must be an integer multiple of the synodic period of the associated bodies. For nonsimple solar systems, approximated cyclers may be defined to exist as any sequence of planetary encounters with a total period that is near an approximate repeat time for the inertial positions of the planets. As a result, solutions must be generated for several decades, making the rather large assumption that the basic pattern is repeatable. In 1 general, it is desirable to propagate the trajectories for as long as possible. The obvious extension is to use several cycles of a true cycler solution from the simple case as an initial estimate for a solution in the more complicated model. This is the subject of the final main chapter of this dissertation. When referring to a simple model, the simplest possible is chosen: the circularcoplanar solar system. Although previous studies 3,4 have shown limited success in finding true cyclers in a model that includes eccentricity and inclination, the additional complexity and lack of symmetry compels most researchers to use the circular-coplanar model first. In this simple model, the Earth and Mars move in circular and coplanar orbits around the sun. When a spacecraft encounters the Earth, a zero-sphere-of-influence patched conic approximation is applied to simulate a gravity-assisted flyby. Any flyby altitude greater than 200 km is considered feasible. The simulated flyby instantaneously rotates the geocentric velocity of the spacecraft. In this model, Mars is not capable of providing a gravity-assist flyby. This is considered reasonable because the mass of Mars is approximately 1/9th that of the Earth. More importantly, however, this allows all Earth-Mars transits to exist on Earth-Earth free-return trajectories, where a free-return is defined to be any trajectory that leaves a celestial body and ballistically re-encounters the same body at a later time. Thus, an idealized free-return Earth-Mars cycler is defined to be a cycler in an ideal solar system that is composed of a sequence 2 of Earth-Earth free-return trajectories patched together with Earth gravity-assisted flybys. In this study, when looking for cyclers in a more realistic solar system, several of the prior assumptions remain valid. Namely, the spacecraft still moves in Keplerian two-body motion around an inertially fixed sun, and flybys are modeled with the same instantaneous and zero radius sphere-of-influence patched conic technique. However, flybys at Mars are included, and the positions of the planets are found using one of two methods. In the first method, the planets move in Keplerian motion based on respective sets of orbital elements. The elements can be chosen to reflect the simple circularcoplanar model, the true mean motion of the planets, or any model in between. The mean elements of the planets at J2000 are found in Ref. 5. The second method locates the planets using JPL's DE405 Ephemerides.6 The gravitational parameters of the sun and planets are taken from DE405, and the mean radius values used for Earth and Mars are 6378 km and 3397 km respectively. 1.2 MOTIVATION The main purpose of this dissertation is to identify and catalogue useful EarthMars cycler orbits. The goal is to use Earth powered gravity-assisted flybys to connect generic, half-revolution, and full-revolution free-return orbits in a logical manner such that the patched trajectories are periodic, require realistic flybys, and encounter Mars in a simplified solar system. Finally, approximate cycler orbits in a more realistic solar system are sought based on the idealized solutions in the circular-coplanar model. 3 It is left up to future researchers to interpret the feasibility of a complete interplanetary transportation system that utilizes the cycler concept. Rather, the ultimate goal of the present study is to document useful and efficient techniques to search for the highly constrained cycler trajectories. Furthermore, it is desirable to catalogue as many useful cycler trajectories as possible, both in the simple model and more accurate models. 1.3 PREVIOUS RESEARCH ON CYCLERS The concept of a cycler trajectory, one that shuttles between two or more celestial bodies, is not new. Several previous studies with favorable results have shown that many such trajectories exist, both ballistic and powered. Hollister first introduced the concept of patching Earth free-returns to form cyclers in the 1960's with several studies on Earth-Mars and Earth-Venus cyclers. The free-returns can be categorized into three mutually exclusive categories: 1) Halfrevolution (odd-n) transfers, 2) Full revolution (even-n) transfers, and 3) generic (non-n) transfers also referred to as symmetric returns. 7 , 8 Hollister8, Rall3 and Hollister, and Menning4 and Hollister used non-linear search methods to find feasible combinations of 1 transfers, 2 transfers, and a limited class of generic returns. Although these studies were successful in finding several ballistic cyclers, the limited computational power restricted the search to a very localized portion of the solution space and no Earth-Mars cyclers were found with repeat times of less than four synodic periods. 4 Cycler interest was revitalized in the mid 1980's and early 1990's following the discovery of Niehoff's VISIT9,10,11,12 cyclers and the Aldrin cycler1,13,14, named after its co-inventor, the well-known astronaut, Buzz Aldrin. Unlike most other cyclers, the VISIT class of cyclers have orbits that are intertially fixed and utilize resonance opportunities between the periods of the cycler, Earth, and Mars; and thus, require no flyby maneuvers at Earth or Mars. The Aldrin cycler is a traditional near-ballistic cycler with a short repeat time. Over the past two decades, Aldrin has promoted the cycler concept (interplanetary and lunar) to both technical and non-technical audiences around the country including Congress, the media, universities, and professional conferences. More recently, several studies have investigated further the idea of constructing Earth-Mars cyclers using Earth free-returns patched with flybys. 15 , 16 , 17 , 18 , 19 , 20 , 21 Reference 17 presents all cyclers consisting of a single generic return while Ref. 15 investigates several cyclers using two non-identical generic returns patched by an Earth flyby. Reference 16 shows solutions for two-synodic period cyclers with a single generic return patched with 1 half-year and 2 full-year transfers. Ref. 22 is a survey article discussing the benefits and drawbacks of cycler missions vs. traditional missions and the hybrid semi-cycler23 missions that use gravity assists at the Earth, but enter closed orbits around Mars. Although ballistic cyclers are typically preferable to those requiring maneuvers, several studies have also considered the design and analysis of powered cyclers.14,15,18,24 5 Periodic constraints associated with cyclers restrict true solutions to exist only in simplified solar system models. Therefore, the references mentioned above primarily focus on techniques to find solutions in a circular-coplanar model. Additionally, Refs. 3, 4, 8, 15, and 16 find corresponding ballistic solutions in an ephemeris model. Refs. 13, 15, and 18 include ephemeris model solutions that use impulsive or low-thrust maneuvers respectively when necessary. Ref 25 summarizes the current state of research on free-return cyclers and proposes a standard nomenclature for these complicated trajectories. The nomenclature is designed to encompass the most general cases for transfers with all three types of transfer angles including interplanetary legs and free-returns to planets in non-circular orbits. An Earth-Mars cycler is a patched trajectory that is periodic in reference frames that have repeatable geometry every synodic period. Although not identified in the literature as cyclers, some researchers have had success finding these orbits as periodic solutions to the circular restricted three-body problem (RTBP) when the mass of the secondary body (e.g. the mass of the Earth) is zero. This method is termed as Poincar e's method of constructing "second species"26,27,28 periodic solutions. It turns out that the cyclers documented in this dissertation could in fact be labeled as "second species" periodic orbits in the RTBP. 6 1.4 CONTRIBUTIONS In general, the current dissertation is a compilation of four stand-alone, but highly related papers dealing with free-returns and cyclers. The first three studies were presented at AAS/AIAA conferences,19,20,21 and expanded versions were subsequently published in AIAA journals (or have already been technically reviewed and are currently in-press).29,30,31 The results from the fourth study will also be submitted to an archival journal in the near future. In a collaborative effort with a similar research group at Purdue University, a fifth paper that outlines a formal nomenclature for cyclers is also currently in-press.25 In this dissertation, the four main topics will be unified into one broad treatise that sufficiently addresses its title: "Global Search and Optimization for Free-Return Earth-Mars Cyclers." The major contributions of each sub-study are discussed below. Reference 29 presents optimized solutions for all cyclers that use one or more identical generic returns patched by any combination of 1 half-year or 2 full-year transfers, including those discussed in Ref. 16 and Ref. 17. The original method identifies twenty-four ballistic cyclers with periods of two to four synodic periods, 92 ballistic cyclers with periods of five or six synodic periods, and hundreds of nearballistic cyclers, most of which are previously undocumented. Reference 30 outlines a systematic method to find all feasible n and generic returns following a ballistic flyby. The solutions for arbitrary cases are derived and presented in an original graphically based method. Additional insight is gained from an 7 original discussion on Lambert's problem in the context of n transfers. Based on the more generalized definitions for half and full-rev returns, several improved versions of the cyclers documented in Ref. 29 are presented. Depending on the methods discussed in Ref. 30, Ref. 31 uses combinatorics to exhaustively search the entire solution space of idealized Earth-Mars cyclers consisting of any combination of n and generic returns. The generalized global approach is original, yet it encompasses most of the previous works in this area, identifying all useful idealized Earth-Mars free-return cyclers. The results include 203 noteworthy cyclers with periods up to three synodic periods, including twenty-nine that use fullrevolution returns for the transit legs. Again, most of the solutions are previously undocumented. The final main chapter addresses the obvious next step of searching for cyclers using a more accurate solar system model. An original, efficient method that utilizes analytic gradients is developed to optimize long-duration, multiple-flyby, patchedconic, ballistic trajectories with potential applications beyond Earth-Mars cyclers. The method is applied to minimize powered requirements at every step of a continuation method that transitions the idealized solutions to accurate ephemeris solutions. The algorithm is run for twenty-one launch windows for each of the cyclers presented in Ref. 31. Over 4000 cases are examined, resulting in hundreds of ballistic launch opportunities for seven-cycle propagations of accurate ephemeris cyclers. A total of twenty different parent cyclers are found to have an average total maneuver requirement 8 over all twenty-one launch windows of less than 100 m/s. In general, the study demonstrates the broad feasibility for an entire class of accurate ephemeris cyclers. 1.5 DISSERTATION ORGANIZATION Before discussing applications regarding cyclers, Chapter 2 gives necessary background information on the three types of free-return trajectories and a method to calculate feasible solutions following a flyby with a given v. Chapter 3 outlines a method to find and optimize a class of free-return cyclers limited to half-year halfrevolution returns, full-year full-revolution returns, and one or more identical generic returns. These solutions include cyclers with total periods of up to six synodic periods. Chapter 4 outlines a completely different approach that solves the more general problem. Because the solutions only include periods of one, two, and three synodic periods, the results encompass some but not all of the results presented in Chapters 3. Chapter 5 then gives a procedure to take the idealized solutions from Chapter 4 and find analogous solutions in a more accurate solar system. conclusions and discusses options for future work. Appendix A discusses the dynamics of a translating, rotating, and pulsating reference frame that is convenient for the analysis and visualization of interplanetary cycler trajectories. Appendix B provides algorithms for several combinatorics subproblems used in Chapter 4. Lastly, Appendix C documents several ballistic or nearballistic fully reproducible accurate ephemeris cycler trajectories. Chapter 6 draws general 9 2 Free-Return Trajectories 2.1 CHAPTER SUMMARY The purpose of this chapter is to systematically identify all feasible trajectories following a gravity-assisted flyby that immediately return to the flyby body with no intermediate maneuvers. Every class of possible transfer angles is considered including even-n, odd-n, and generic return orbits. Lambert's Problem is solved for a desired time of flight range allowing the possibility for multiple spacecraft and celestial body revolutions. The solutions are expressed geometrically, and the resulting velocity diagram is a mission-planning tool with potential applications that include cycler trajectories and planetary moon tours. The generalized free-return solutions may be used to construct loitering orbits about one celestial body or transfers between multiple bodies. 2.2 INTRODUCTION Free-return trajectories have been the subject of many studies in the wake and anticipation of taking humans back to the Moon and beyond. 32 , 33 , 34 Proven to be invaluable during the Apollo missions, free-returns are useful for human exploration because they return to the original body, by design or as an abort option, without any powered maneuvers. Free-returns are also useful on interplanetary missions or moon 10 tours when consecutive flybys of the same body provide the appropriate timing and gravity-assisted maneuvers necessary to reach the next destination. Half- and full-rev free-returns, orbits with a transfer angle that is an odd or even integer multiple of respectively, are subsets of the general free-return transfer. First termed by Hollister8, half- and full-rev returns were originally defined such that the spacecraft and the celestial body were limited to a 1 or 2 transfer respectively, and were used as stalling mechanisms in the construction of interplanetary cycler orbits. The half and full-rev returns are of particular interest because the solution space for the targeting problem significantly increases if the transfer angle is an integer multiple of . The extra degrees of freedom make it possible to find free-return solutions that are further constrained by the matching v conditions associated with gravity-assisted flybys. This concept has been applied to missions involving the Moon,35,36 Jupiter,37 Venus,4 and Mars.3,16,19 Free-returns with transfer angles that are non-integer multiples of are termed generic returns.19 Unlike the half or full-rev return, these non-resonant return orbits have no extra degree or degrees of freedom for a transfer with a given semi-major axis. However, because the multiple-revolution Lambert Problem has many solutions, there may exist several generic free-returns, each with a different semi-major axis, for any sufficiently large time of flight. Similar to Uphoff et al, this paper develops equations that govern even- and oddn free-returns.37 Additionally, a detailed analysis using Lambert's Equation provides 11 further insight into the behavior of the solutions. Furthermore, a numeric method is presented to find all the generic returns that are feasible following a flyby. The solutions for the n and generic returns are then combined onto one velocity diagram to provide a general design tool for multiple-flyby missions. The first section gives an overview of Lambert's multiple revolution problem with an emphasis on n transfers. The next few sections outline methods to obtain semi-major axis values for full-rev, half-rev, and generic free-returns respectively. In the full-rev case, two methods are presented: a simple derivation based on orbital periods and a second more complete discussion based on Lambert's Equation. In the half-rev and the generic case, Lambert's Equation is used exclusively. The following section discusses the terminal velocity vectors that initiate free-returns, and expresses the solutions on a common flyby velocity diagram. Finally, conclusions are drawn in a general sense and in the context of the following chapters. 2.3 LAMBERT'S PROBLEM This section gives an introduction to Lambert's problem in general38 and in the context of n transfers. The following sections will examine the application of Lambert's Equation to n and generic free-returns in more detail. Half and full-rev returns are specific cases of the classic Lambert targeting problem when the transfer angle is an integer multiple of and 2 respectively. It is well known that these cases lead to singularities when computing terminal velocity vectors using standard approaches. In order to understand the behavior of the solutions 12 at these singularities, it is useful to look at them in the limit as they approach half and full-rev returns. The sum of the two chords connecting any point on a given ellipse to its two respective foci is a constant. From this familiar "tac and string" property, a diagram similar to that in Figure 2.1a can be drawn to find all vacant foci locations for elliptic transfers from r1 to r2 given any sufficiently large value for semi-major axis. Figure 2.1: Procedure to find vacant foci 13 The spheres centered at the tips of r1 and r2 are the locus of all vacant foci for ellipses of semi-major axis a, that contain the position vectors r1 and r2, respectively. The intersection of the two spheres is illustrated with a dotted circle in parts a and b. Note, in part c, the intersection is a sphere rather than a circle. In the general case, if r1 is not parallel to r2, then the transfer plane is defined by the two position vectors. The intersection of this transfer plane and the dotted circle are the two points labeled as Fv1 and Fv2. These are the two vacant foci locations for elliptic transfers from r1 to r2 with semi-major axis a. However, if the transfer angle is (2N+1), as is true for a half-rev return, then the transfer plane is no longer defined by the position vectors, and any point on the dotted circle on Figure 2.1b is a valid location for the vacant focus. Due to the "tac and string" property, if the transfer angle is 2N and r1 r2, then there is just one point of intersection between the two spheres, and the only transfers possible are on a rectilinear ellipse. If the transfer angle is 2N and r1 = r2, as is true for a full-rev return, then the two spheres merge into one. Thus, from Figure 2.1c, any location on the sphere of radius 2a-r is a valid location for a vacant focus. In summary, there are only two vacant foci locations in the general case, one degree of freedom is required to specify the location for a half-rev return, and lastly, two degrees of freedom are required to specify the vacant focus location for a full-rev return. Figure 2.2a shows the four transfers on the two possible ellipses in the general case. Figure 2.2b and Figure 2.2c illustrate the transfers in the limit as they approach a half and full-rev return respectively. Note, the angle, 2N, is defined to always be 14 less than or equal to . Therefore, -2N is always defined to be greater than or equal to . Figure 2.2: The four possible arcs of Lambert's Equation for a given N Lambert's theorem states that the time of flight connecting any two points on an elliptic orbit is a function only of its semi-major axis, the chord length between the two points, and the sum of the respective distances from the focus to the two points. 15 Equation (2.1) summarizes the Lagrangian formulation38 generalized to include multiple revolutions of the primary as defined in Figure 2.2a. Equation (2.1) holds true for all four transfers shown in Figure 2.2. The quadrant ambiguities associated with the angles and uniquely characterize each of the four arcs. TOF = a 3 2 2 N + fast , slow - - sin( fast , slow ) + sin( ) (2.1) where, sin ( 0 2) = S ( 2a ) fast = 0 slow = 2 - 0 sin ( 0 2) = ( S -c ) ( 2a ) = 0 if transfer angle is =- 0 if transfer angle is S =1 2( r1 + r2 + c ) Using Eq. (2.1), Figure 2.3 plots the time of flight, TOF, vs. semi-major axis for N = 09. The input geometry for this plot is r1=r2=1 AU, =2N + 4 /7. The curves in Figure 2.3 represent direct solutions, or =2N+4/7 for this example. A similar plot is obtained for the corresponding retrograde solutions using =2N+10/7. Notice the time of flight is double valued for each N due to the quadrant ambiguity in 0. Each value of N has a fast and a slow transfer curve, corresponding to the lower and upper branches respectively. The solution associated with = 0 is termed "fast" while the solution associated with = 2-0 is termed "slow," indicating a slow solution has a longer time of flight than a fast solution given two transfers with a common semi-major axis. If r1 = r2, then trajectories associated with 16 the fast transfer curve initially approach perihelion, while trajectories associated with the slow transfer curve initially approach aphelion. Also, for a transfer with a given semi-major axis value, the time of flight for the solution on the lower curve is less than the solution on the upper curve. For these reasons, the terms fast and slow seem appropriate. The literature is not consistent in naming the upper and lower branches. Other names, such as short and long, are often used, but can be confusing because it is unclear if they refer to time of flight or orbital period. Figure 2.3: Example multiple revolution Lambert solution (r1=r2=1 AU, =2N + 4 /7) 17 In a typical application of Lambert's theorem, the time of flight is given and the corresponding values for semi-major axis are solved iteratively. For a given geometry, an arbitrary time of flight has an associated NMAX. For example, from Figure 2.3, a time of flight of 20 TU's has an NMAX of 3. Given r1, r2, , and TOF, an algorithm is developed to identify NMAX and systematically solve for each of the 2NMAX +1 corresponding values of semi-major axis. Because the algorithm is very similar to the procedure described by Prussing39, it is not explained in detail. It uses standard rootfinding methods to bracket solutions and solve the transcendental equation for semimajor axis. When calculating terminal velocity vectors, a singularity exists if is any integer multiple of because the transfer plane is undefined. Half- and full-rev returns are subsets of these cases respectively. The following sections will address further how to use Lambert's Equation to find solutions for full-revolution returns, half-revolution returns, and generic returns. 2.4 GENERALIZED FULL-REVOLUTION RETURN SOLUTIONS The generalized full-rev return is defined to be any trajectory that leaves a celestial body and returns directly to the same body after completing N revolutions of the primary while the celestial body completes M revolutions. 18 2.4.1 Expressions Based on Orbital Periods Because the times of flight for full-rev returns are simple integer multiples of a celestial body's period, the governing equation of full-rev returns, Eq. (2.2), is derived by setting the times of flight for both the spacecraft and the celestial body to be equal. M 2 aB 3 = N 2 aF 3 (2.2) Solving for aF, the expression for the semi-major axis of a full-rev transfer becomes aF = aB ( M N ) 23 (2.3) 2.4.2 Additional Insight from Lambert's Equation Equation (2.2) can also be derived directly from Lambert's Equation. The first step in deriving Eq. (2.1), shown in Eq. (2.4), is to subtract Kepler's Equation applied to r1 from Kepler's Equation applied to r2. The angles and are alternate parameters that can be shown to be functions of E1 and E2. TOF = a 3 2 E2 - E1 - e ( sin( E2 ) - sin( E1 ) ) (2.4) When r2 = r1, and consequently E2 = E1+2N, it becomes TOF = a 3 2 [ 2 N ] (2.5) Equation (2.5) is identical to Eq. (2.2) when a=aF and TOF is constrained to be M times the period of the celestial body. The general form of Lambert's Equation gives additional insight to the problem when observing the solutions in the limit as r1 r2. Examining Figure 2.2a, as r1 r2 19 and consequently 2N+ and 2(N+1) -, the attracting focus, the tip of the position vectors, and the two empty foci become collinear as seen in Figure 2.2c. The only possible ellipse with a position vector directly between foci F and Fv2 is a rectilinear ellipse with e=1. The only possible ellipse with foci F and Fv1 that contains r is one whose apoapse occurs at r, thus e=r/a-1. Both solutions are shown in Figure 2.2c. In the limit, the two position vectors are very close, but never exactly parallel, thus a transfer plane is defined, and the two vacant foci are unique, as illustrated in Figure 2.2a. The transfers approach those illustrated in Figure 2.2c, and properties of the solutions smoothly approach values as outlined in Table 2.1. However, when r1 = r2 exactly, the transfer plane is no longer defined, and the vacant focus can be anywhere on the sphere of possible locations, as seen in Figure 2.1c. Thus, the rectilinear ellipse and the non-rectilinear ellipse in Figure 2.2c represent just two points on the vacant foci sphere. Transfers 1 and 4 are specific cases of the solutions described by Eq. (2.2). However, transfers 2 and 3 are not described by Eq. (2.2) because the eccentric anomalies at the beginning and end of the transfers are different. Examining Figure 2.2a, as r1 r2 , it is clear that Eo Ef for transfers 1 and 4, but not for transfers 2 and 3. Expressions for these values, as well as other noteworthy properties for each of the four transfers, are given in Table 2.1. 20 Table 2.1: Properties of Lambert solutions in the limit as r1 r2 [2N+, 2(N+1) -] transfer angle transfer ellipse period time of flight e initial E final E transfer 1 2N+ T NT r/a-1 + transfer 2 2N+ T N T + t 1 E0 2 E0 transfer 3 2(N+1) T (N+1) T - t 1 2 E0 E0 transfer 4 2(N+1) T (N+1) T r/a-1 + - where E0 = - cos -1 ( r - a ) a , t = 2 - 4 sin -1 ( r (2a ) + 2 2ar - r 2 a , and T = 2 a 3 ) Every point on the vacant foci sphere has a direct and retrograde transfer, each with a time of flight equal to a normalized value of 2M TU. However, the point on the sphere that is exterior to and collinear with F and r has two additional transfers because its associated ellipse is rectilinear. Equation (2.2) covers all the solutions such that Eo = Ef, including two of the four possible transfers on the rectilinear ellipse. The remaining two solutions on the rectilinear ellipse are transfers 2 and 3 as shown in Figure 2.2c where Eo Ef. For a given value of aB, each set of M and N has an associated vacant foci sphere with radius 2aF - r. The two transfers on each ellipse associated with every point on the vacant foci sphere in addition to the two rectilinear solutions such that Eo Ef comprise the set of all M:N resonant transfers with a common semi-major axis. In the limiting case when r1=r2, then c=0, 0=0, and Eq. (2.1) simplifies to Equations(2.6), (2.7), (2.8), and (2.9) respectively for transfers 1, 2, 3, and 4 shown in Figure 2.2c. 21 TOF = a 3 2 2 N (2.6) TOF = a 3 2 2 ( N + 1) - 4 sin -1 TOF = a 3 2 2 N + 4 sin -1 ( r (2a ) + 2 2ar - r 2 a ) (2.7) (2.8) (2.9) ( r (2a ) - 2 2ar - r 2 a ) TOF = a 3 2 2 ( N + 1) Eqs. (2.6) and (2.9) are identical to Eq. (2.5) except the N is phased by one in Eq. (2.9) because the transfer angle is rather than . As an example, suppose it is desirable to traverse one revolution of the primary. If transfer 1 is used, the associated N is one because the transfer angle, , approaches 2+ and a full revolution is completed as r1 r2 . However, if transfer 4 is used, the associated N is zero because the transfer angle, , approaches 2 , and a full revolution is not reached as r1 r2. Figure 2.4 and Figure 2.5 plot the four time of flight vs. semi-major axis curves expressed in Eqs. (2.6)-(2.9) representing all four full-rev transfer arcs. The plots were generated using canonical variables with =1 AU3/TU2, r=1 AU, and N=05. Figure 2.6 is the two prior figures plotted on the same axes. Note that the transfer 1 curve is identical to the transfer 4 curve when N is phased by one, as evidenced from Eqs. (2.6) and (2.9). Also, the values of the TOF(amin) and TOF/a(amin) are equal for the transfer 2 and 3 curves when N is phased by one. This is easily shown using Eqs. (2.7) and (2.8) and recognizing that amin=S/2. Figure 2.6 contains all time of flight and semi-major axis information for transfers that make 2N revolutions of the primary. More information is necessary to 22 pick out the specific transfers that leave and return to a particular celestial body. Clearly, the trajectory of the celestial body must also lie on one of the curves in Figure 2.6. Since a rectilinear ellipse is not a physically realistic solution, the trajectory of the body can only be a transfer 1 or transfer 4 curve. A vertical dotted line is drawn at a=aB. An example value of aB=1 AU is used in the figures. The intersections of this vertical line with the transfer 1 and 4 curves locate the solutions that correspond to the trajectory of the celestial body. The horizontal dotted lines are then drawn at each of the intersections. Each integer multiple of the celestial body's period is associated with a horizontal dotted line, corresponding to values of M in Eq. (2.2). Finally, the intersections of the transfer lines with these horizontal lines represent specific semimajor axis values that will yield orbits that return to the body after N revolutions. For example, there are five values of semi-major axis that yield non-rectilinear orbits that return after the celestial body makes two complete revolutions of the primary. Examining Figure 2.6, the five values for semi-major axis are found by locating the intersections of the top horizontal dotted line with the transfer 1 and 4 curves. The values are identical to those obtained with Eq. (2.3). 23 Figure 2.4: Limit r1 r2. Lambert transfers using . Figure 2.5: Limit r1 r2. Lambert transfers using . 24 Figure 2.6: Limit r1 r2. All Lambert transfers. 2.5 GENERALIZED HALF-REVOLUTION RETURN SOLUTIONS The generalized half-rev return is any trajectory that leaves a celestial body and returns directly to the same body after completing N+1/2 revolutions of the primary. When N=0, the half-rev return is also known as a backflip36 trajectory. Examining Figure 2.1b, when the transfer angle is (2N+1), any point on the dotted circle is a valid location for the vacant focus. The alternate focus locations represent a rotation of the transfer orbit about the position vectors. Thus, the shape of the transfer ellipse is fixed, but a free parameter is required to specify the transfer plane. 25 A simple expression similar to Eq. (2.2) is not available for the half-rev return because the times of flight are not simple multiples of the orbital periods. Lambert's Equation is necessary for the analysis. In the limit as the transfer angle approaches 2N+, as seen in Figure 2.2b, the first and second Lambert transfers become mirror images of the third and fourth respectively. Similar to the full-rev discussion, the two vacant foci of the ellipses in Figure 2.2b correspond to specific opposing locations on the dotted circle in Figure 2.1b. TOF = a 3 2 2 N + 2 sin -1 ( (r1 + r2 ) 2a - ) ( 2a - r1 - r2 ) ( r1 + r2 ) a a (2.10) (2.11) TOF = a 3 2 2 ( N + 1) - 2 sin -1 ( (r1 + r2 ) 2a + ) ( 2a - r1 - r2 ) ( r1 + r2 ) Equation (2.1) simplifies to Eq. (2.10) for transfers 1 and 3, and simplifies to Eq. (2.11) for transfers 2 and 4. The solution is indifferent to using the or direction because 0 = -0 =0 in Eq. (2.1). Figure 2.7 shows the time of flight vs. semi-major axis plots for a sample half-rev geometry using =1 AU3/TU2, r1 =r2 =aB =1 AU, and N=05. In canonical terms, one revolution for the celestial body is 2 TU. Thus, the vertical axis is labeled in celestial body periods. In Figure 2.7, the semi-major axes that correspond to half-rev solutions are denoted by the intersections of the dotted lines and the transfer curves. Note that r1 =r2, aB =amin, and the horizontal lines occur exactly at half periods, indicating the celestial body is in a circular orbit. As an example, if a spacecraft leaves a celestial body at r1 and the time of flight is 4.5 celestial body periods, then there are nine semi-major axis values that yield transfers that will re26 encounter the body at r2. The two solutions from N=4 are more easily seen in Figure 2.8. For the special case of aB > amin and r1 =r2, the celestial body must be in an elliptic orbit with its line of apses perpendicular to the terminal position vectors. In this case, the vertical dotted line corresponding to aB should be moved accordingly in Figure 2.7 and the procedure to find the half-rev solutions is similar to that described below when r1 r2. Figure 2.7: Half-rev solutions, |r1|=|r2| 27 Figure 2.8: Zoom view of the N=4 curve Figure 2.9 represents the more general case where r1 r2. For this example, r2=0.45 AU and the other parameters remain unchanged. Before identifying solutions that return to the body, it is necessary to specify whether the body is on a fast or slow path solution. If the body is on a fast path, then the times of flight corresponding to half-rev returns are denoted by the intersections of the vertical dotted line, positioned at a=aB, with the fast transfer curves. These intersections are marked with horizontal dashed lines, each with an associated value of M that increases with TOF. If the body is on a slow path, the intersections of the solution curves with the horizontal dotted lines indicate the half-rev return times of flight. Notice that successive dashed and dotted 28 lines are respectively spaced by the period of the celestial body. The bottom dashed and dotted lines correspond to the M=0 fast and slow path solutions respectively. Figure 2.9: Half-rev solutions, |r1| |r2| For example, from Figure 2.9, suppose the celestial body with aB=1 is on a fast path transfer, then there are seven M=2 solutions that return to the body. The M=2 solutions are marked by the dashed line with TOF=2.18 x 2 TU. The two N=3 intersections occur at a=0.73 and a= 0.75 AU. The two N=2 intersections occur at a=0.86 and a= 1 AU. Note the latter corresponds to the path of the celestial body. The two N=1 intersections occur at a=1.11 and a= 1.60 AU. intersection occurs at a=1.76 AU. 29 Lastly, the one N=0 2.6 GENERIC FREE-RETURN SOLUTIONS Generic free-returns consist of all free returns that are not even- or odd-n transfers. Suppose a celestial body is located at r1 along its orbit, as seen in Figure 2.2a. For sufficiently large times of flight, multiple generic return solutions exist, including the path of the body itself. For a given 0 < < , the input geometry, or , r1, r2, and c, can be easily calculated for the Lambert problem that connects the body's initial and final positions. Using these values, a plot similar to Figure 2.9 can be generated for each transfer angle using Eq. (2.1). However, for the generic returns, the plot will have four distinct sets of solutions corresponding to transfers 1-4 as opposed to the two seen in Figure 2.9. An example plot generated with =/4, r1 = r2 = 1 AU, and =1 AU3/TU2 is given in Figure 2.10. The solutions using , or transfers 1 and 2, represent either direct or retrograde transfers, while solutions using , or transfers 3 and 4, represent the opposite, depending on the motion of the celestial body. A similar procedure as described above in the half revolution case is required to solve for all semi-major axis values that correspond to generic return solutions that initiate and terminate at a particular body. The process can be repeated for every value of 0 < < at any desired interval. Each plot contains all solutions for +2M and +2M where M is any integer. 30 Figure 2.10: Example generic return solutions 2.7 FLYBY VELOCITY DIAGRAMS The previous sections provide methods to obtain semi-major axis values for all three types of free returns: full-rev, half-rev, and generic. Based on these values, this section discusses the outbound velocity vectors required to initiate the free-returns, and expresses the solutions on three-dimensional velocity diagrams. 31 2.7.1 Full-revolution Velocity Diagrams If a spacecraft leaves a celestial body on an orbit with the semi-major axis aF, obtained from Eq. (2.3) or a plot similar to Figure 2.6, then it will return directly after it completes N revolutions. Equation (2.12) is the vis-viva equation solved for spacecraft velocity magnitude. Inserting Eq. (2.3) into Eq. (2.12) gives Eq. (2.13), and restates the constraint in terms of velocity magnitude. vF = 2 r - a F (2.12) aB vF = 2 r - ( N M ) 23 (2.13) Thus, if a spacecraft leaves a celestial body with the speed relative to the primary given by Equation (2.13), then it will return after it completes N revolutions. The constraints described in Eqs. (2.3) and (2.13) are identical to the constraint that the absent focus must lie on the sphere of intersection from Figure 2.1c. The scalar nature of Eq. (2.13) is significant because the direction of the velocity magnitude is unconstrained. This provides a convenient method to parameterize the two degrees of freedom associated with full-rev returns. Feasible values for N, M, r, , and aB are combinations such that vF from Eq. (2.13) is non-imaginary. For fixed values of M, r, , and aB, N can vary from 1 to Nmax. If a celestial body with semi-major axis aB is located a distance r from the primary, then for a given M and N, the locus of all spacecraft velocity vector tips that initiate full-rev returns is a sphere. The full-rev sphere is centered at the base of the body's velocity and has a radius of vF. If a spacecraft approaches a body with an 32 arbitrary hyperbolic speed and direction, the locus of all feasible points for the velocity after an unpowered gravity-assisted flyby is the surface of a sphere with radius v centered at the tip of the body's velocity. This is the common three-dimensional velocity diagram for a gravity-assisted maneuver. A sample diagram with these spheres is illustrated in Figure 2.11. Canonical units were used to generate the figure with =1 AU3/TU2, r = aB =1 AU, N = 7, M= 4, and v= 0.5 AU/TU. The right-handed coordinate system chosen for illustration and analysis of these diagrams is centered at the tip of vB with the z axis aligned with vB and the y axis opposed to the celestial body's angular momentum vector. The sphere centered at the base of vB is the full-rev sphere associated with Eq. (2.13). The sphere centered at the tip of vB is the v sphere. The circle marks the intersection of the two spheres. If v + is located anywhere on this circle, the spacecraft and body will re-encounter after completing seven and four revolutions respectively. Figure 2.12 illustrates all of the full-rev spheres associated with M=4. It is clear that nine of the full-rev spheres intersect with the v sphere to form full-rev circles. If v + is located on any of the full-rev circles illustrated in Figure 2.12, then the spacecraft and body will re-encounter after the body completes four revolutions. Nmax is eleven in this example. However, all eleven spheres do not necessarily intersect with the v sphere. Equation (2.14) is a logical expression that must be true for an intersection to exist for a given vB, vF, and v . {v F vB - v } and {vF vB + v } (2.14) 33 Figure 2.11: Full-rev velocity diagram, N=7, M=4 Figure 2.12: Full-rev velocity diagram, N=111, M=4 34 The equation describing the surface of the v and vF spheres are given by Eqs. (2.15) and (2.16). 2 x 2 + y 2 + z 2 = v (2.15) (2.16) 2 x 2 + y 2 + ( z + vB ) = vF 2 Eliminating x and y, Eq. (2.17) gives the z value for the full-rev circle of intersection. 2 2 2 z F = vF - v - vB ( ) 2v B (2.17) 2.7.2 Half-revolution Velocity Diagrams The f and g functions are typically used to solve for the terminal velocity vectors on a transfer with a known value for semi-major axis. However, singularities exist in these functions when the transfer angle is an integer multiple of . For full-rev returns, it was determined that the terminal velocity vectors can be located anywhere on a sphere with radius given in Eq. (2.13). For a half-rev return, the components of the outbound terminal velocity vector are derived by Battin40 and expressed in Eqs. (2.18) and (2.19). 2 vHr = [ 2 (r1 + r2 ) - 1 a ] (2.18) (2.19) 2 vH = 2 r2 (r 21 + r1r2 ) Figure 2.13 is a diagram of the velocity vector, vH, in the ecliptic plane that produces a half-rev return orbit. The x axis of the primed coordinate system is aligned 35 with rB while the unprimed z axis is aligned with vB. Because the transfer plane is not specified, vH can be rotated by any angle about rB and the resulting velocity vector will initiate a half-rev return. Notice the squares in the component equations indicate double valued solutions. If the solution is a fast transfer, then it initially moves toward periapse, thus the radial component of the velocity that initiates the return must be negative. The opposite is true for a slow transfer. Examining Figure 2.2b, it is clear that the two fast transfers, 1 and 3, have negative initial radial components for velocity. It is also clear that the two slow transfers, 2 and 4, initially move toward apoapse and have positive initial radial components of velocity. For the transverse component, the positive and negatives values produce half-rev returns because the transfer plane is undefined. The positive value, however, is used to reference the zero-point for the free angle that specifies the transfer plane. Figure 2.13: Outbound velocity components for half-rev free-returns 36 Examining Figure 2.13, the rotation of vH about rB forms a cone. The base of this cone, the half-rev circle, is the locus of all terminal velocity vector tips that lead to a half-rev return with a given semi-major axis. Uphoff and Crouch36 refer to this cone as the "equal gamma cone." Figure 2.14 is an example velocity diagram that contains a half-rev circle. Generated using information from Figure 2.7, it includes rB, vB, v-, the v sphere, and the half-rev circle that corresponds to the slow N=2 solution with a time of flight of 3.5 celestial body periods. The slow N=2 curve from Figure 2.7 has a semi-major axis of 1.18 AU when the time of flight is 3.5 periods. Inserting this into Eqs. (2.18) and (2.19) gives vHr=0.153 AU/TU and vH=1 AU/TU. Because a=1.18 AU is a slow solution, meaning it comes from the upper of the two N=2 curves, vHr must be positive. In general, the plane of any half-rev circle is perpendicular to rB. Particular to this example, the plane of the half-rev circle in Figure 2.14 is the y-z plane because the celestial body is in a circular orbit. It is clear from Figure 2.13, that the directions of vHr and vH are aligned with the x and z axes respectively when the flight path angle is zero. 37 Figure 2.14: Half-rev velocity diagram, M=3.5, N=2 slow The v sphere and the half-rev circle intersect at two locations, one below and one above the ecliptic plane, or the x-z plane. They are marked with an `x' and `o' respectively. These are the two feasible locations for v + following an unpowered flyby that will initiate this particular half-rev return. From Figure 2.13, in terms of the primed coordinates, the expressions describing the half rev circle are given by Eqs. (2.20) and (2.21). ( y ) + [ z + vB cos( )] 2 2 2 = vH (2.20) (2.21) x = vHr - vB sin( ) 38 The v sphere, expressed in the prime coordinates, is ( x ) + ( y ) + ( z ) 2 2 2 2 = v (2.22) Solving for x ,y, and z from Eqs. (2.20)-(2.22) gives the location of the two half-rev points of intersection. The coordinates, expressed in the un-primed variables, are given in Eqs. (2.23)-(2.25). x = [ vHr - vB sin( ) ] cos( ) + K sin( ) [ 2vB cos( ) ] 2 y = v - [ vHr - vB sin( ) ] - K 2 [ 2vB cos( ) ] 2 2 (2.23) (2.24) (2.25) z = [ vHr - vB sin( ) ] sin( ) - K ( 2vB ) where 2 2 2 2 K = 2vHr vB sin( ) + vB 1 - 2 sin 2 ( ) - vH - vHr + v 39 Figure 2.15: Half-rev velocity diagram, M=3.5 Continuing with the example, it is clear from Figure 2.7 that a time of flight of 3.5 periods yields seven corresponding solutions for semi-major axis. Thus, for this one time of flight, there are seven half-rev circles. All seven are illustrated in Figure 2.15. The circles form a tube-like structure with a radius, vH, and centerline, rB. Equation (2.19) indicates that vH is not a function of a or TOF. Therefore, the diameter of the tube is a constant for a specified , r1, and r2. For this example, it is clear that only three out of the seven circles intersect with the v =0.5 AU/TU sphere. intersection to exist, the y coordinate given in Eq. (2.24) must be non-imaginary. 40 For an 2.7.3 Generic Free-Return Velocity Diagrams The outbound terminal velocity vector for generic free-returns can be easily calculated using the f and g functions.38 As an example, the outbound velocity vectors are calculated for all generic free-returns with times of flight up to six celestial body periods for a body in a circular orbit. Note, because the body is in a circular orbit, it is not necessary to specify its departure location along the orbit. The multiple-revolution Lambert problem is solved at half-degree intervals and the coordinates for each resulting outbound velocity is recorded. Only direct (posigrade) solutions are included. All solutions corresponding to N=1, meaning the spacecraft makes between one and two revolutions of the primary are displayed in Figure 2.16. Note that all generic return solutions are in the ecliptic plane or x-z plane in the unprimed coordinate system defined in Figure 2.13. The arbitrary v sphere and the N=1 solutions intersect at the points denoted by the large dots. Because the problem is solved at discrete half-degree intervals, the apparent lines are not continuous. The solution properties of the intersection points are interpolated linearly between the two bordering points with the v sphere. Table 2.2 gives the interpolated results for the 11 intersection points for this example. The angular coordinate, , denotes its location in the ecliptic plane on the v sphere referenced to vB, with positive being aligned with rB. Figure 2.17 shows all possible generic return solutions for N values of 1...Nmax. The seventy-five intersection points for the example v sphere are also shown. The spacing between the points seen in Figure 2.17 is a function of the transfer angle 41 interval. The half-degree interval was chosen to give good resolution, yet still illustrate that the lines are composed of many points. In some regions of the velocity space, the point density is significantly less than others. This indicates regions of less stable solutions, meaning the departure directions at Earth are more sensitive to the time of flight in these regions. A more detailed analysis of the solution behavior in these regions is left to future work. When solving for the intersection points, a transfer angle interval of 1/24 degree is used to decrease the probability that any solutions are skipped. Remember that each of the small dots represents a unique transfer with an associated time of flight, , and semi-major axis. These values are also interpolated when solving for the intersection points. For each point, Figure 2.18 and Figure 2.19 plot as a function of v , while Figure 2.20 and Figure 2.21 plot TOF as a function of v. With this interpolation technique, all possible generic returns can be found for any value of v. Table 2.2: N=1 generic return solutions # 1 2 3 4 5 6 7 8 9 10 11 (deg) -139.5 -96.33 -81.04 -73.19 -68.37 -65.06 74.29 78.20 83.74 92.73 114.0 TOF (flyby body periods) 0.899 1.571 2.406 3.336 4.299 5.276 5.655 4.621 3.567 2.469 1.250 a (AU) 0.540 1.245 2.146 3.114 4.099 5.090 2.941 2.433 1.920 1.392 0.804 N (rev) +1 -1 -1 -1 -1 -1 +1 +1 +1 +1 +1 42 Figure 2.16: N=1 solutions. TOF=06 flyby body periods Figure 2.17: All Solutions N=115. TOF=06 43 Figure 2.18: vs. v for generic returns to a body in circular orbit. N=07. 44 Figure 2.19: vs. v for generic returns to a body in circular orbit. N=915. 45 Figure 2.20: TOF vs. v for generic returns to a body in circular orbit. N=07. 46 Figure 2.21: TOF vs. v for generic returns to a body in circular orbit. N=815. 47 2.7.4 Flyby Diagrams Containing All Free-Return Solutions While Figure 2.12, Figure 2.15, and Figure 2.17, illustrate a few of the full-rev, half-rev, and generic free-return solutions, it may be desirable for a given value of of v, to obtain all free-return solutions including every possible combination of multiple revolutions by the spacecraft and the body. Examining the figures with velocity diagrams, the radius of the v spheres used for these examples is half the magnitude of the celestial body's circular velocity. This value was exaggerated intentionally for graphical purposes. Typically it is more desirable to have significantly lower hyperbolic energy levels, thus the radius for a more realistic v sphere may only be one tenth to one fourth of the body's velocity. Clearly, a v sphere with a smaller radius will have fewer intersection locations for free-returns solutions. An example diagram with a more realistic v value is shown in Figure 2.22. This map contains all possible locations for v+ following a flyby such that the spacecraft will return ballistically for times of flight up to six celestial body periods. 48 Figure 2.22: Free-return solutions for |v|=0.1838 AU/TU Similar to the previous diagrams, Figure 2.22 was generated using a celestial body in a circular orbit. It is a projection view of the v sphere along the y axis, thus the full-rev circles appear as horizontal lines, and each set of the two half-rev points appear as an `x' underneath an `o.' The labels indicate time of flight in celestial body periods. The generic return dots are labeled directly while the half-rev points and full-rev circles are labeled along the top and to the right of the plot respectively. The locations and 49 properties of each of the generic returns interpolated using the data from Figure 2.18 Figure 2.21 are shown in Table 2.3. Note that one of the half-rev point sets and several of the full-rev circles have multiple labels. This is only true in the zero-point patched conic model. Using a more realistic model, a point on the velocity diagram can not yield a true free-return trajectory if there is an intermediate encounter. Of course, in the real solar system, the radius of a celestial body's sphere-of-influence is non-zero, thus it is impossible to perform a flyby with a turning angle of zero. Consequently, v must rotate during a flyby. Technically, two consecutive half-rev returns on the same set of half-rev points are only possible in the realistic system if they are patched with a feasible flyby that rotates v from an `x' to an `o' or vice versa. Consecutive full-rev returns on the same full-rev circle are only possible if they are patched with feasible flybys and v- v+. For the purposes of this study, the zero-point patched conic model assumptions hold. However, when transitioning solutions from this simple model to a more realistic one, the comments above should be considered. In this example, the celestial body is in a non-inclined, circular orbit. Thus, if a spacecraft leaves the body on an inclined circular orbit in the zero-point patched conic model, it will re-encounter the body every revolution. This is why the middle x/o set is labeled with each half-year. Of course, this set also lies on the full-rev circle that corresponds to the celestial body's semi-major axis. 50 For any value of v , a velocity diagram similar to Figure 2.22 can be generated. The equations for the full-rev circles and the half-rev points are valid to find freereturns to a celestial body in any elliptic orbit. Remember, however, that the data in Figure 2.18 - Figure 2.21 must be regenerated in order to find the generic returns to a body in a non-circular orbit. Table 2.3: Details of generic return solutions in Figure 2.22 # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 a b (deg) -158.6 -158.2 -157.0 -139.9 -131.0 -130.1 -120.4 -115.0 -111.5 -93.07 -92.58 -91.80 -90.40 -86.96 -72.55 -66.53 -55.51 -45.64 -23.39 -22.95 TOF (flyby body periods) 5.944 3.943 1.939 4.882 5.843 2.838 3.783 4.745 5.718 5.534 4.529 3.520 2.504 1.466 5.325 4.278 3.208 5.158 4.073 2.071 a (AU) 0.668 0.669 0.671 0.718 0.754 0.758 0.809 0.842 0.866 1.022 1.026 1.034 1.049 1.086 1.263 1.347 1.515 1.674 2.006 2.011 Na (rev) +8 +5 +2 +6 +7 +3 +4 +5 +6 -5 -4 -3 -2 -1 -4 -3 -2 -3 -2 -1 b # 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 (deg) 24.15 24.31 24.79 46.61 56.97 57.50 68.53 74.52 78.31 97.52 98.03 98.84 100.3 104.2 117.6 123.3 133.5 141.8 160.5 161.6 TOF (flyby body periods) 5.925 3.924 1.922 4.837 5.784 2.781 3.707 4.658 5.623 5.417 4.411 3.403 2.387 1.348 5.236 4.199 3.145 5.111 4.051 2.048 a (AU) 1.996 1.994 1.988 1.659 1.492 1.484 1.318 1.236 1.188 0.979 0.974 0.967 0.954 0.921 0.825 0.792 0.743 0.711 0.665 0.663 Na (rev) +2 +1 +0 +2 +3 +1 +2 +3 +4 +5 +4 +3 +2 -1 -6 -5 -4 -7 -6 -3 the `+' or `-' indicates a slow or fast solution respectively. the second of two solutions on the same lower branch for the associated TOF. 51 2.7.5 Numerical Considerations and Velocity Diagram Generation The solutions for the full-rev returns are analytic. Equation (2.17) gives the z coordinates for all full-rev circles as a function of vB, v, and vF, where all feasible values of vF are found from Eqs. (2.13) and (2.14). Thus, the full-rev circles on a velocity diagram are generated by plotting the circles described by Eq. (2.15) and feasible values of z. The solutions for the half-rev returns are both numerical and analytic. They are numerical in the sense that an iterative procedure is required to solve a Lambert problem to obtain the semi-major axis of the transfer orbit. describes this problem in great detail. A previous section A variety of techniques are available that efficiently find all feasible solutions with accuracies on the order of machine precision.39,41 Once the transfer semi-major axis values are known, solving for the halfrev returns with a given v is analytically found using Eqs. (2.18), (2.19), and (2.23)(2.25). Each valid semi-major axis leads to a pair of coordinates, and all non-imaginary pairs are the feasible half-rev solutions. These coordinates are plotted on the velocity diagrams as above and below plane `o's and `x's. Unlike the half- or full-rev return solutions, the generic return solutions are entirely numerical in origin. As explained in detail, a Lambert problem is solved to find all free-returns to a celestial body at discreet time intervals up to a desired maximum time of flight. The solutions are then separated into common bins according to number of revolutions and whether they are `fast' or `slow.' Example properties of these 52 families of solutions are given in Figure 2.18 - Figure 2.21. As the figures indicate, each family may contain several distinct sub-families of solutions. Each sub-family is sorted according to v and finally, the generic returns for a given v are found via interpolation. It is critical that the two neighboring interpolation points exist on the same sub-family of solutions, otherwise, the result is not valid. Thus, caution is advised when distinguishing between sub-families of solutions. In the limit as the time interval used to generate the original list of solutions goes to zero, the probability of skipping a solution and the error in the interpolation also approaches zero. The time interval used for this study is chosen to be sufficiently small such that decreasing the value does not increase the number of generic solutions found for arbitrary values of v. This interval is then further reduced to provide additional confidence that no solutions are skipped. The final value chosen, approximately a halfhour interval, provides over 180,000 solutions over a six year time period. Thus, in order to find all generic returns with a given v, an algorithm must sift through a nontrivial number of data points. The tips of the vout vectors of the final solutions are plotted on the velocity diagrams as generic return `dots.' The accuracy of the final linearly interpolated solutions can be greatly improved by using them as initial guesses in a one-dimensional solver that fixes v and iterates on TOF and in a Keplarian integration until the trajectory re-encounters Earth to a desired tolerance. Thus, similar to the half-and full-rev returns, the accuracy of these final generic return solutions is also on the order of machine precision. 53 2.8 CHAPTER CONCLUSIONS This chapter defines and gives solution characteristics for each type of free- return trajectory: the half-rev (odd-n) return, the full-rev (even-n) return, and the generic return. The semi-major axis values for the even-n free-return transfers, or fullrev returns, are derived both with a simple approach and taking the limit as r1 approaches r2 in Lambert's problem. Several interesting properties of the Lambert solutions are noted including the obviously impractical rectilinear ellipse solution. The semi-major axis values for the odd-n and the generic free-return transfers can only be found using Lambert's Equation because the times of flight are not integer multiples of orbital periods. In velocity space, all solutions for a given half-rev return are shown to lie on a set of constant-diameter circles forming a tube, while all solutions for a given full-rev return are shown to lie on multiple concentric spheres. The solutions for the generic returns are shown to lie on a series of non-uniform arcs in the ecliptic plane. Intersections of these lines and surfaces with the flyby sphere of constant v identifies the half-rev points, full-rev circles, and generic return dots of intersection that lead to potential free-return transfers following a flyby. The equations for the locations of the full-rev circles and sets of half-rev points are derived, and a numerical method is outlined to find the locations for all generic return dots. 54 When including the possibility of multiple body and spacecraft revolutions, the final result is the common three-dimensional v sphere with a surface marked by all feasible free-return solutions. The generation of such a diagram is straight-forward, based on the geometry of the celestial body's orbit and the hyperbolic energy of the spacecraft. These diagrams are useful mission design tools for any application requiring consecutive flybys of the same body. In particular, the two following chapters will use similar diagrams in an effort to construct Earth-Mars cyclers using patched Earth-Earth free-returns. 55 3 Idealized Free-Return Cyclers Composed of Transfers, 2 Transfers, and One or More Identical Generic Returns 3.1 CHAPTER SUMMARY In this chapter a procedure for constructing idealized Earth-Mars cycler orbits in a simple solar system is presented. Solutions from the multiple revolution Lambert problem are utilized to find free-return Mars trajectories. Multiple combinations of these generic return orbits are patched to sequences of 2 full- and 1 half-rev return orbits with Earth-generated gravity-assisted maneuvers. An algorithm is developed to find all useful combinations of these free-returns that have a combined period of any integer multiple of the synodic period. Given a sequence of free-returns, an analytic procedure is then developed to minimize the maximum of all the turning angles associated with the flybys necessary to maintain and re-initiate the cycler. The method identifies twenty-four ballistic cyclers with periods of two to four synodic periods, ninety-two ballistic cyclers with periods of five or six synodic periods, and hundreds of near-ballistic cyclers, where a ballistic cycler is defined to be one that requires no powered maneuvers to maintain and has realistic turning angles with respect to the surface of the Earth. These resulting orbits have diverse characteristics that could benefit a variety of potential missions. While the method finds several known idealized cyclers, most of the orbits presented are previously undocumented. 56 3.2 INTRODUCTION Recent work by McConaghy, Longuski, and Byrnes17 found several cycler trajectories using single generic return orbits and one cycler using two different generic returns. Byrnes, McConaghy, and Longuski16 showed that energy characteristics of a particular two-synodic period cycler could be considerably improved by including half and full-rev returns. The work presented in this chapter significantly expands on the methods used in these studies to form one non-iterative method that scans a much larger defined solution space to identify and classify cycler orbits. Namely, the new method finds and optimizes all idealized cyclers composed of 2 -full-rev returns, 1 half-rev return returns, and one or more identical generic returns. A few specifics about the simplified solar system model are discussed first, followed by a section with specific comments on the three types of free-returns used in this chapter. The following two sections discuss the construction of a cycler using one generic return and any combination of half or full-rev returns. The free parameters associated with each intermediate flyby are then discussed, and a method is outlined to minimize the maximum required turning angle. The next section presents the logic associated with including multiple generic returns, followed by a summary of the algorithm, and finally, the results and conclusions. The method reveals many ballistic and near-ballistic cyclers with a variety of time and energy characteristics. Included in the results are several previously documented near-ballistic cyclers. These are the 57 Aldrin cycler1,13,14, the single generic return cyclers presented by McConaghy et al17, and the two-synodic period cyclers presented by Byrnes et al.16 In the context of this chapter, the naming convention for the cycler orbits are of the form p.h.s.i, where the letters represent four numbers that uniquely identify a class of cyclers. For example, a cycler of the class 4.3.2.-5 has a period of 4 synodic periods, includes 3 half-years allotted for full or half-rev returns, and includes 2 generic returns that complete 5 revolutions. The negative sign indicates the solution for the generic return is from the lower solution curve of the two associated with the five revolution Lambert's problem. Table 3.1: Possible combinations for free-returns for a cycler with h=3 and s=2 Number 1 2 3 4 Sequential Combinations generic generic half-rev full-rev generic generic half-rev half-rev half-rev generic full-rev generic half-rev generic half-rev generic half-rev half-rev Cyclers associated with a particular p.h.s.i constitute a class of cyclers because there are an infinite number of patched trajectories that share these qualities. The order of free-returns and the distinction between full and half-rev is not specified in the naming convention. Table 3.1 shows the four different combinations possible for the order of the free-returns for the example given. Additionally, there are free parameters, as described in Chapter 2, associated with each of the flybys that patch the free-returns. 58 Therefore, a cycler of the class 4.3.2.-5 is uniquely described if the order of the freereturns and all associated free parameters are specified. An algorithm is developed to choose this order and these free parameters such that the required maximum turning angle of all the flybys is minimized given any feasible p.h.s.i class of cyclers. The resulting optimized patched trajectory is referenced as Cycler p.h.s.i. The goal then, is to apply the algorithm to the entire solution space of feasible ranges for the variables p, h, s, and i. The fixed characteristics of the optimized Cyclers are then analyzed and compared. Of particular interest are the resulting cyclers that are entirely ballistic. Once set in orbit, ballistic cyclers use physically feasible flybys and require no powered maneuvers to maintain. 3.3 SIMPLIFIED SOLAR SYSTEM The method discussed in this chapter assumes the solar system is simplified as stated in Section 1.1. In addition, Mars is assumed to be in an orbit with a period of 1.875 yrs. This period is chosen such that the absolute geometry repeats every 15 yrs. This value is used to be consistent with previous studies16,17 and for verification purposes. This value could be changed to 1.8801 yrs., the true period of Mars, and have little effect on the results of the chapter. Remember, as stated in Chapter 1, a simplified model is necessary because the relative geometry of Mars and Earth must repeat exactly for a perfectly periodic cycler to exist. Canonical units are used for calculations and defined such that sun=1 The derived time conversion is 1 TU= 58.1324409 days. 59 AU3/TU2. The astronomical unit definition, 1 AU =149597871 km, and Earth's gravitational parameter, earth = 3.00348960E-6 AU3/TU2, are taken from JPL's DE405 Ephemerides.6 In this circular-coplanar zero-point patched conic model, it is assumed that a very minor correction maneuver can be applied to the spacecraft at a great distance to target any altitude for an Earth flyby. 3.4 FREE-RETURN TRAJECTORIES Earth-Mars transfers and Earth loitering orbits can both be constructed using Earth free-return trajectories. A loitering orbit is defined to be a trajectory that leaves the Earth on a free-return transfer in order to wait for an appropriately timed flyby without performing a powered maneuver. The loitering orbits are important because the total period of a cycler is constrained to be an integer multiple of the synodic period. If a spacecraft encounters the Earth prior to the necessary planetary alignment, a freereturn may be used to re-encounter the Earth at a later time when the alignment is correct. If the altitudes of the required Earth flybys are physically possible, the loitering maneuvers require no fuel. Free-return trajectories are defined and discussed in detail in Chapter 2. The definitions and equations from Chapter 2 are general to allow for the departure and arrival celestial body to be in an arbitrary elliptic orbit. For the purposes of this chapter, the celestial body of interest is the Earth, and it is assumed to be in a circular orbit. Thus, appropriate simplifications can be made to the equations and solution methods 60 presented in Chapter 2. The three types of free-returns are the half-rev, full-rev, and generic returns. A review of these three types are illustrated below in Figure 3.1. Figure 3.1: Three types of free-returns 3.4.1 2 Full-Revolution Returns This chapter uses a class of full-revolution returns limited to full-year 2 transfers only. The more general M-year 2N transfers will be included in the search for cycler orbits in Chapter 4. Similar to Figure 2.11, Figure 3.2 illustrates an example flyby velocity diagram containing the full-year 2 full-rev sphere and its associated fullrev circle. The profile view is added for clarity. See Section 2.7.1 for a general description of velocity diagrams that include full-rev returns. 61 Figure 3.2: Flyby velocity diagram including potential 1 half-rev and 2 full-rev free-return transfers. 3.4.2 1 Half-Revolution Returns Similar to full-rev returns, this chapter uses a class of the half-rev returns limited to half-year 1 transfers only. This restricted class is also referred to as backflip trajectories.16,36 These transfers have the same semi-major axis as the departure planet, but have an arbitrary inclination. Figure 3.1 shows the spacecraft coming out of plane and re-encountering the Earth on the other side. The more general M half-year (2N+1) transfers will be included in the search for cycler orbits in Chapter 4. Similar to Figure 2.14, Figure 3.2 illustrates an example flyby velocity diagram containing the 1 half-year set of half-rev points. See Section 2.7.2 for a general description of velocity diagrams that include half-rev returns. In the real solar system, the radius of a planet's sphere-of-influence is, of course, non-zero. Thus it is not possible to perform a flyby with a turning angle of zero. 62 Consequently, the v vector must rotate during a flyby. Thus, if two consecutive halfrevs are desirable in the real solar system, since the turning angle required to transfer from the top `X' to the bottom or vice-versa is infeasible, the only option is to approximate two consecutive half-revs. On the velocity diagram, if a point on the fullrev circle near but not on the `X' is targeted, then the spacecraft will return after one year. After a half-year, the spacecraft will be near the Earth but will not encounter it, thus reducing the effect of the Earth's gravity. Note that a flyby can patch two consecutive full-rev returns in the realistic solar system if v- v+, both vectors are on the full-rev circle, and the turning angle is feasible. 3.4.3 One or More Identical Generic Free Returns A generic return, as seen in Figure 3.1, and discussed in detail in Sections 2.3, 2.6, and 2.7.3, is characterized by an arbitrary time of flight and a transfer angle that is a non integer multiple of . The transfer is therefore limited to be co-planar with the Earth. This transfer has also been called a symmetric return because the departure and arrival points are symmetric with respect to the line of apses if the planet is in a circular orbit.7 The s generic free-returns in a given cycler of the form p.h.s.i, by definition, have identical geometric properties and times of flight. Equation (3.1) is an expression for this time of flight in years. It constrains the total period of the cycler to be p synodic periods. This timing constraint ensures the repeatability of the relative geometry 63 between Earth and Mars and is the most fundamental condition associated with idealized cyclers. TOF = p-h 2 s (3.1) In order for a cycler to be ballistic, every flyby connecting two legs must have matching v's. From Figure 3.2, it is clear that the terminal v's from the initial generic return and all half and full-rev returns match. If additional generic returns are included, one method to ensure matching v's is to constrain all generic returns to be identical. However, for a cycler with multiple different generic returns, it is possible to search for departure and flyby times that give common terminal v values for each of the returns. This technique has been used successfully to find several cyclers with two generic returns17 and will be used more extensively in Chapter 4. The method discussed in this chapter avoids the search variables by simply considering only identical generic returns. 3.5 3.5.1 CYCLERS WITH ONE GENERIC RETURN Preliminaries This section discusses the construction of cyclers that include one generic return. Each cycle consists of a generic return followed by any combination of 1 half or 2 full-rev returns. In order to repeat a cycle, the generic return must be re-initiated following the last leg of the previous cycle. initiation are discussed below. The conditions required for this re- 64 Figure 3.3: Conditions required to re-initiate generic return trajectory Figure 3.3 shows an example of the geometry required to re-initiate a generic return trajectory using no intermediate returns. The heliocentric spacecraft velocities at Earth departure and arrival are v1+ and v2- respectively. Due to symmetry, the magnitudes of these vectors are equal, and the orientation of v2- with respect to ve2 is a mirror image of v1+ with respect to ve1. The velocity required to re-initiate an identical generic return is v2+. Thus, it is clear that the required heliocentric turn angle is 2. At t2, it is also clear that v- and v+ are coplanar with the Earth's orbit. These conditions apply for the re-initiation of any identical generic return trajectory. The mirror image symmetry can also be seen in Figure 3.4. 65 Figure 3.4: Velocity diagram of a gravity-assisted flyby that re-initiates a generic return If only one flyby is used, Figure 3.4 indicates that a planar v is required to reinitiate the generic return. Alternatively, if the timing is not constrained, one or more full-rev free-returns can be performed prior to the re-initiation of the generic return. This is possible because the geometry of the velocity diagram remains the same before and after a full-rev return. In order to enter a full-rev free-return orbit, the tip of the intermediate v must lie somewhere on the intersection of the two spheres on the fullrev circle. Figure 3.5a shows an example that uses an intermediate full-rev return requiring two flybys to re-initiate the generic return. The geocentric velocity at the arrival location on a generic return is v1-. The first flyby requires a turn angle of 1 in order to place the spacecraft on a full-rev free-return orbit. If no powered or gravityassisted maneuvers occur, in the simple model stated, the Earth and spacecraft will return to this location with the same heliocentric velocities each year. 66 As a consequence, the velocity diagram remains unchanged at the time of the second flyby, thus v1+ = v2-, and a turn angle of 2 is required to achieve v2+, thus re-initiating the generic return. Figure 3.5b is a similar example requiring three flybys and two In these examples, the exact positions of the intermediate full-rev return orbits. intermediate v's on the dotted full-rev circle are chosen arbitrarily. A following section will discuss a method to optimize the selection of these locations. Figure 3.5: Gravity-assisted flyby velocity diagrams with 2 full-rev returns 67 Figure 3.6: Gravity-assisted velocity diagram with 1 half-rev return Consider the example shown in Figure 3.5a, however choose the location of the intermediate v to be the half-rev `X' above the plane of the paper. This is illustrated in Figure 3.6a. Following the first flyby with a turn angle of 1, the spacecraft enters a half-rev return trajectory. Again, for the assumed simple model, if no powered or gravity-assisted maneuvers occur, the spacecraft will encounter the Earth after every integer multiple of a half-year. If the next gravity-assisted maneuver occurs after an even number of half-revs, then the velocity diagram after the transfer is identical to that of a full-rev return. However, if it occurs after an odd number of half-revs, then the velocity diagram changes. This is illustrated in Figure 3.6b. After an odd number of half-revs, ve, not shown in Figure 3.5 and Figure 3.6 but shown in Figure 3.4, switches directions. Also, the geocentric velocity at the beginning of the half-rev return is 68 opposite of that at the end, or v1+ = -v2-. A turn angle of 2 is required to achieve v2+, thus re-initiating the generic return. Due to symmetry, it is clear that 1 = 2 for this example. Figure 3.7 combines the two separate diagrams of Figure 3.6 into one by fixing the direction of ve and aligning the respective top and bottom half-rev `X's. This is equivalent to rotating the diagram in Figure 3.6b by 180 about an axis perpendicular to the plane of the paper (also the Earth's orbit), then overlapping the two diagrams. The profile view is included for clarity. Figure 3.7: Velocity diagram of a 1 half-rev return with ve fixed 3.5.2 Turn Angle Optimization Thus far, this section has demonstrated how to re-initiate a generic return utilizing gravity-assisted Earth flybys that may include full or half-rev return orbits. As mentioned previously, a free parameter is associated with each full-rev return. 69 Additional parameters that are constrained but not fixed are the number of flybys before re-initiation. For example, if a spacecraft is departing Earth on a 2 full-year return orbit and needs to wait three years before re-initiating, should the spacecraft use a gravity-assisted flyby at each of the following Earth encounters, or wait until the third encounter? This is addressed by solving the following optimization problem: Minimize the maximum required turning angle necessary to re-initiate a given generic return orbit using any sequence of 2 full or 1 half-rev returns subject to the constraint that exactly hj half-years elapse before the re-initiation. Note that the goal is to minimize the maximum required turn angle not the sum of the required turn angles. This is chosen because the primary focus is to find ballistic cyclers. It is acknowledged, in some cases, that a powered cycler23,24 in the simplified or real model may benefit from minimizing the sum rather than the maximum of the required turning angles. However, the difference is immaterial if searching for ballistic cyclers in a simple model. If hj=0, the only option is to immediately re-initiate using one flyby as shown in Figure 3.4. If hj=1, the only option is to use a half-rev return intermediate flyby before re-initiating as shown in Figure 3.7. It is arbitrary whether the above or below plane maneuver is used. Note that the choice is only arbitrary in the simple model stated. If the v is out of plane or the Earth's orbit is non-circular, the flybys that initiate the above and below plane half-revs may differ significantly. If hj=2, the options are two 70 half-rev returns or one full-rev return. In the case of using one full-rev return, due to the geometry, the maximum turn angle is minimized if the two turn angles are equal. This serendipitously occurs only if the intermediate v terminates at the half-rev `X.' This is identical to the case of using two half-rev returns. Thus, only two flybys are required when hj=2. If hj=3, the options are to use three half-revs, or one half-rev and one full-rev return. In either case, it is required to have at least one flyby with a turn angle equal to that required when hj=1. It is decided to choose the option that requires fewer flybys in cases when additional flybys do not reduce the maximum required turning angle. Therefore, when hj=1, 2, or 3, two flybys are required (fj=2), one to navigate v from the initial location to the half-rev `X,' and one to navigate it to the final re-initiation location. A similar argument can be made for hj 4. This is summarized in Table 3.2. 71 Table 3.2: Optimal number of flybys necessary to re-initiate a generic return hj 0 fj 1 Velocity diagram Time between flybys (yr) - 1 2 t2 - t1 = 2 2 t2 - t1 = 1 3 2 same as hj =1 t2- t1= 3/2 t2 - t1 = 1 t3 - t 2 = 1 4 3 5 4 t2 - t1 = 1 t3- t2= 1/2 t4 - t 3 = 1 6 4 t2 - t1 = 1 t3 - t 2 = 1 t4 - t 3 = 1 7 4 same as hj =5 t2 - t 1 = 1 t3- t2= 3/2 t4 - t 3 = 1 t2 - t1 = 1 t3 - t 2 = 1 t4 - t 3 = 1 t5 - t 4 = 1 8 5 hj even hj odd ... hj/2 + 1 2{INT(hj/4 + 1)} ... 72 tk-tk-1 = 1 k = 2...(f /2) tk-tk-1 = 1 tk-tk-1 = (1/2) MOD(hj,4) k = f/2+1 k = f/2+2... f tk-tk-1 = 1 ... k = 2...f Once the number of required flybys is known, it is desirable to space them along the full-rev circle in such a manner that the maximum turn angle is minimized. Looking at Figure 3.4, define a spherical coordinate system using latitude and longitude for the sphere of radius v. The origin of the coordinate system is the center of the sphere. The associated z-axis is aligned with ve making the full-rev circle a line of constant latitude. The x-axis is defined such that v- has a positive x component and no y component. Figure 3.8 illustrates this coordinate system with an example that requires five flybys (fj=5) to re-initiate the generic return. This coordinate system is also used in Section 2.7 Figure 3.8: Turn angle optimization for a re-initiation that requires five flybys The smallest turn angle that ensures a full-rev free-return is the flyby that moves v1- in the x-z plane along the zero longitude line to the full-rev circle. This angle, MIN, is found from Eq. (3.2) and is illustrated in Figure 3.8a. 73 MIN = cos -1 ( cos FR cos GR + sin FR sin GR ) (3.2) Depending on the magnitude of this minimum turn angle, there are two cases to consider. If v1- is sufficiently far from the full-rev circle, as seen in Figure 3.8a, then MIN is the minimized maximum turn angle. However, if v1- is sufficiently close to the full-rev circle, as seen if Figure 3.8b, then a unique longitude exists, 0 < < /2, such that the turning angle, b, is equal for all five flybys. In this case, b is the minimized maximum turn angle. The latitudes of the full-rev circle and the generic return geocentric velocity vectors are given by Eqs. (3.3) and (3.4). Equation (3.3) can be derived from Eq. (2.17) when vF =vB and vB =ve, as is the case for a 2 full-rev return to the Earth. FR = - sin -1 v ( 2ve ) (3.3) (3.4) T GR = 2 - cos -1 v v e 1- ( v ve ) All of the turning angles generalized for any fj > 2 can be calculated using Eqs. (3.5) (3.7). a = cos -1 ( cos 2 FR cos a + sin 2 FR ) where a = ( f j - 2) (3.5) In order to solve for b, Eq. (3.6) must be solved iteratively for . A unique solution exists for 0 < < /2 if MIN < a. 74 cos 2 FR cos cos b + - cos FR cos cos SR + cos 2 FR sin sin b + + sin 2 FR - sin FR sin GR = 0 ( ) b = ( - 2 ) / f j - 2 ( ) ( ) (3.6) b = cos -1 ( cos GR cos FR cos + sin GR sin FR ) (3.7) If fj = 1, the turning angle for the only required flyby is c. c = - 2 GR (3.8) If fj = 2, the turning angle for both required flybys is found by solving Eq. (3.7) with = /2. Equations (3.2) and (3.5) (3.7) are derived using the known latitudes and longitudes of the vectors that define the turning angles. These coordinates are summarized in Table 3.3. Equation (3.6) is derived by setting the angle between v1and v2- equal to the angle between v2- and v3- from Figure 3.8b. This requires that v2- is orthogonal to v1- subtracted from v3-. Table 3.3: Coordinates for heliocentric flyby velocities Figure 3.8a latitude v1v2v3- GR FR FR longitude (west) 0 0 a Figure 3.8b latitude longitude (west) GR 0 FR b+ FR Due to symmetry, Eqs. (3.5)-(3.8) are still valid if hj is odd and a half-rev return is required. However, caution is advised when solving for velocities after an odd number of half-rev returns because the velocity diagram switches orientation as indicated in Figure 3.6. 75 Given the value of hj, any arbitrary generic return can now be re-initiated using fj optimized flybys. The calculation of the minimized maximum required turning angle, MINIMAX-j, can be summarized as follows: 1) Calculate c from Eq. (3.8) 2) Get fj from Table 3.2 3) IF fj =1 MINIMAX-j = c ELSE IF fj =2 MINIMAX-j = cos-1(sin GR sin FR ) ELSE IF fj >2 Calculate MIN, a using Eqs. (3.2) and (3.5) IF MIN a MINIMAX-j = MIN ELSE IF MIN < a calculate b using Eqs. (3.6) and (3.7) MINIMAX-j = b 3.6 CYCLERS WITH MULTIPLE IDENTICAL GENERIC RETURNS As long as the total period of the cycler is an integer multiple of the synodic period, it is possible for one cycle of a given cycler to consist of multiple generic returns. If they are identical, the magnitude of v remains unchanged at all of the Earth flybys. As seen in Table 3.1, the spacing of multiple generic returns is a free parameter. This section addresses the logic for choosing this spacing. Given a total of h half-years allotted for full and half-rev returns and a total of s identical generic returns, the 76 purpose of this section is to determine how many half-years, hj, should be grouped with each generic return, sj. Note that by definition, h j =1 s j =h Once a specific generic return is grouped with an optimal number of half-years, the procedure described in the previous section can be applied independently to solve for the best flyby configuration and associated MINIMAX-j for that grouping. The goal is to minimize the maximum of MINIMAX-j for j=1s. The resulting angle is MAX. The logic is briefly outlined below: 1) Calculate c , MINIMAX-j using hj=INT(h/s) 2) IF c MINIMAX-j THEN h1s-1= INT(h/s) hs = INT(h/s) +MOD(h/s) ELSE h1 = h h2s = 0 3) MAX = MINIMAX-1 This is derived based on the following observation. If hj 1, MINIMAX-j will stay the same or decrease if the value of hj is increased. However for hj=0, MINIMAX-j may increase or decrease if the value of hj is increased, depending on the geometry of the generic return. Consider the geometry in Figure 3.8a. Because c < MIN, it is clear that the maximum required turning angle is lowest if no intermediate free-returns are used. In Figure 3.8b however, it is clear that intermediate flybys help to reduce the maximum required turning angle, MINIMAX-j. 77 As an example, if h=10 and s=3, the two possible cases are {h1=3, h2=3, h3=4} or {h1=10, h2=0, h3=0}. In the first case, the geometry of the generic return is similar to that shown in Figure 3.8b. Thus, the ten half-years are equally distributed among the three generic returns in order to minimize the total max turning angle. Since h3 > h1, the max turning angle associated with the third generic return is guaranteed to be less than or equal to the max turning angle associated with the first generic return. Therefore, as the logic step 3) indicates, MAX is encountered after the first generic return. In the latter case, the geometry is similar to that of Figure 3.8a. Thus, the max turning angle is lowest if no intermediate flybys are used to re-initiate the generic return. Therefore, all of the available ten half-years are grouped with the first generic return in an attempt to lower its associated max required turning angle and leave the second and third generic returns with no intermediate flybys. Again, the total max turning angle, MAX, is encountered after the first generic return. Note that for h 1, MIN is the lower bound for MAX. 3.7 ALGORITHM OVERVIEW An overview of the algorithm is shown in Figure 3.9. It searches for all cycler solutions with periods from 1 to pMAX synodic periods. A feasible cycler of the form p.h.s.i is one whose time of flight for the generic return is sufficiently large such that NMAX from Lambert's problem is positive. If NMAX =0, the only solution to Lambert's problem is the Earth's orbit. Also, a negative TOF is clearly not a feasible candidate. Equation (3.1) indicates that TOF is indirectly related to h and s. Therefore, in order to 78 analyze the maximum number of feasible cyclers, it is desirable to choose hMAX and sMAX sufficiently large. The algorithm simply ignores infeasible combinations of p, h, and s. Figure 3.9: Algorithm summary 3.8 RESULTS Table 3.4 lists an example of resulting cyclers using the algorithm described in Figure 3.9. Each column represents important characteristics to consider when evaluating a particular cycler trajectory. The Aphelion Ratio, AR, is the ratio of the maximum ecliptic-plane cycler aphelion radius to 1.52 AU, the radius of Mars. For AR > 1, it is possible for the cycler to intercept Mars without a powered maneuver. The 79 Turn Ratio, TR, is the ratio of the maximum physically allowable turning angle to the maximum required turning angle, MAX. The maximum allowed turning angle is based on a 200 km altitude Earth flyby. For TR > 1, all required flybys are physically attainable without powered maneuvers. Cyclers described in this dissertation are only guaranteed one Mars encounter each period. Therefore the duration between successive Mars encounters is p synodic periods. One approach that increases this frequency is to initiate a new cycler each synodic period. The obvious downside is the cost of each additional cycler vehicle. Thus p vehicles are required to ensure a one-synodic period duration between successive Mars encounters. The magnitude of the planet-centered velocities are important because they represent a relative measure of the maneuver requirements to taxi to and from the planets and an existing cycler. The geocentric velocity is also a relative measure for the maneuver requirement to initiate the cycler from Earth. Of course, these are the hyperbolic velocities at the sphere-of-influence boundary. The actual maneuver requirements will vary due to the gravity of the bodies. These velocities are inversely related to the duration of the Earth-Mars transit. The inbound cyclers, although not reported here, have a short Mars-Earth transit instead of EarthMars. These are easily found by adjusting the initiating time of the cycler. An additional p vehicles are required to ensure a one-synodic period duration between successive Mars-Earth inbound trips. Due to symmetry, the energy properties for inbound and outbound cyclers are identical.7 The last column shows the number of 80 gravity-assisted Earth flybys and the geocentric turning angle associated with each one. Depending on the specific needs of a mission, it may be desirable to have a large or small number of flybys. The ARMIN and TRMIN values used to generate Table 3.4 are somewhat arbitrary. However, they are chosen such that the resulting cyclers are good candidates for optimization using a more realistic solar system. In the true model, the radius of Mars varies from 1.38-1.67 AU. The lower value corresponds to an AR of 0.91, thus ARMIN is chosen to be 0.9. The TRMIN value of 0.85 is chosen to include the Aldrin Cycler1,13,14 in the results. A total of forty-four cyclers are shown in Table 3.4, of which twenty-four are entirely ballistic. The turn angle optimization described in previous sections is evidenced by the number, order, and values of the turn angles shown in the last column. For example, the two generic returns for Cycler 4.9.2.-1 have corresponding hj's of nine and zero respectively. The first generic return is re-initiated using all of the available half-years because the second can be re-initiated using no intermediate returns with only a 24 turn angle. The best configuration for the first re-initiation is to use the MIN turning angle of 83 to traverse down to the full-rev circle, followed by four equally spaced maneuvers with turning angles of 44.8 , and finally an 83 maneuver back to the re-initiation location. The velocity diagram for Cycler 4.9.2.-1 is similar to that shown in Figure 3.8a. Several of the cyclers presented in Table 3.4 are worth mentioning. The bold indicates the cyclers with the most promising characteristics. Cycler 2.5.1.+0 is an 81 example of a solution with a relatively short Earth-Mars transit time, perhaps desirable for a human-crewed mission. The benefits of the 94-day trip are offset by the relatively high terminal speeds. Cycler 3.1.1.+3, Cycler 3.1.2.+1, and Cycler 3.7.1.+1 have low terminal speeds and require two, three, and four flybys respectively. Cycler 4.0.3.+1 has a transit time of 160 days and relatively good terminal speeds using generic returns only. Cycler 4.3.1.-5 has remarkably low energy requirements at Earth and Mars. The speeds are low because the generic return portion of this cycler is very near a Hohman transfer. At Earth, the cycler has a v of 3.10 km/s compared to the Hohman value of 2.84 km/s, while at Mars the cycler has a v of 2.53 km/s compared to the Hohman value of 2.57 km/s. The Aphelion Ratio is 0.992, thus the cycler doesn't quite reach Mars in the simplified model. Cycler 4.5.1.-4, Cycler 4.5.2.-2, and Cycler 4.11.1.-2 also have promising energy characteristics. 82 Table 3.4: Two, three, and four-synodic period ballistic or near-ballistic cyclers ARMIN=0.9, and TRMIN=0.85 Cycler p.h.s.i 1. 0. 1.-1c 2. 1. 1.+2d 2. 3. 1.+1e 2. 5. 1.+0 3. 1. 1.+3 3. 1. 1.+2 3. 1. 2.+1 3. 1. 3.+0 3. 3. 1.+2 3. 5. 1.+2 3. 5. 1.+1 3. 5. 2.+0 3. 7. 1.+1 3. 9. 1.+0 4. 0. 3.+1 4. 1. 1.-6 4. 1. 1.-5 4. 1. 1.-4 4. 1. 2.-3 4. 1. 2.-2 4. 1. 4.-1 4. 3. 1.-5 4. 3. 1.-4 4. 5. 1.-4 4. 5. 1.-3 4. 5. 2.-2 4. 5. 3.-1 4. 6. 1.+4 4. 6. 3.+0 4. 7. 1.-3 4. 7. 1.-2 4. 8. 1.+3 4. 8. 1.+2 4. 9. 1.-3 4. 9. 1.-2 4. 9. 2.-1 4.10. 1.-3 4.10. 1.+2 4.11. 1.-2 4.12. 1.-2 4.12. 1.+1 4.13. 1.-1 4.14. 1.-1 4.14. 1.+0 a b c d e Aphelion Ratioa 1.47 0.95 1.08 1.44 1.07 1.43 1.07 1.43 1.19 0.94 1.43 1.43 1.07 1.43 1.07 0.94 1.15 1.44 0.94 1.43 1.43 0.99 1.26 1.07 1.44 1.07 1.43 0.91 1.43 1.20 1.77 0.96 1.31 0.94 1.44 1.44 0.92 1.03 1.07 0.97 1.16 1.44 1.12 1.49 Turn Ratiob 0.86 1.11 0.92 1.12 1.19 0.89 1.23 0.93 1.06 1.80 1.15 1.06 1.56 1.17 1.18 1.37 1.11 0.89 1.40 0.93 0.93 1.29 1.01 1.55 1.15 1.40 1.02 1.50 0.88 1.38 0.96 1.64 0.86 1.83 1.16 1.05 1.46 1.65 1.58 1.43 1.48 1.16 1.13 1.09 EarthMars (or aphelion) Earth v (km/s) Time (days) 146 6.5 207 4.1 143 94 174 115 181 123 141 231 115 121 175 116 160 256 173 137 250 132 129 268 154 196 137 191 130 154 105 163 120 164 76 256 137 132 263 131 195 268 93 137 199 66 5.4 7.8 3.6 5.4 3.4 5.1 4.3 2.7 5.4 5.2 3.6 5.4 4.3 2.7 4.1 5.5 2.6 5.2 5.1 3.1 4.7 3.6 5.5 3.4 5.1 6.8 6.4 4.3 6.6 7.7 12.5 2.7 5.5 5.2 10.2 8.9 3.6 11.6 10.8 5.5 14.7 14.1 Mars v (km/s) 9.7 2.0 5.3 9.9 4.6 9.2 4.6 9.1 6.8 1.5 9.2 9.2 4.6 9.2 4.9 1.6 6.1 9.3 1.5 9.2 9.2 2.5 7.6 4.7 9.3 4.6 9.2 2.1 9.5 6.8 11.4 3.1 10.7 1.6 9.3 9.2 3.6 5.0 4.7 4.8 8.2 9.3 9.4 12.7 Required Geocentric Turning Angle at each Flyby (deg) 84 92 93 54 93 95 93 95 94 70 73 83 71 72 85 92 94 95 92 95 95 93 94 71 73 81 87 46 60 72 74 37 37 69 72 83 30 31 70 25 27 72 22 25 92 93 54 93 95 93 95 94 70 73 83 71 45 85 92 94 95 92 95 95 93 94 71 73 81 87 46 60 72 74 37 37 45 45 45 30 31 45 25 27 30 22 25 71 73 81 87 46 60 72 74 37 37 45 45 45 30 31 45 25 27 30 22 25 71 73 81 87 46 60 72 74 37 37 45 45 45 30 31 45 25 27 30 22 25 37 37 45 45 45 30 31 45 25 27 30 22 25 69 72 83 30 31 70 25 27 30 22 25 25 27 30 22 25 72 22 25 24 84 84 24 16 16 24 24 12 12 12 70 73 83 71 45 85 70 73 83 71 45 45 72 24 24 16 16 54 54 Cycler reaches Mars if Aphelion Ratio 1 All flybys have minimum altitudes above 200 km if Turn Ratio 1 Aldrin cycler1,13,14 "Case 2" cycler described by Byrnes et al16 "Case 3" cycler described by Byrnes et al16 83 Details about the flyby maneuvers for a few of the discussed solutions are provided in Table 3.5 - Table 3.8. They have sufficient data to simulate one complete cycle plus the first leg of the second cycle for each described cycler. The ecliptic is the x-y axis plane and the initial position of the Earth is always on the x-axis. Table 3.5: Cycler 2.5.1.+0 Location time (days) vx (km/s) vy (km/s) vz (km/s) Earth Mars 0 6.50 4.35 0 a a a Table 3.6: Cycler 3.1.2.+1 Location time (days) vx (km/s) vy (km/s) vz (km/s) Earth Mars Earth Earth Earth Mars 0 0.71 3.32 0 a a a Earth 652 -5.19 Earth Earth Earth Mars 1018 1.40 1200 1565 1659 0 0 0 94 0 0 0 AU 181 0 0 0 AU 1083 1265 2348 2529 0 0 0 -1.40 -5.29 6.12 3.20 -0.98 4.55 -0.09 -1.48 -1.28 -3.59 -3.27 3.39 3.39 0.62 0 -1.41 -6.12 4.55 3.20 T rmars at t0 = [1.41 0.57 0] T rmars at t0 = [1.15 0.99 0] a initial v with respect to Earth a initial v with respect to Earth Table 3.7: Cycler 4.3.1.-5 Location time (days) vx (km/s) vy (km/s) vz (km/s) T Table 3.8: Cycler 4.5.2.-2 Location time (days) Earth Mars Earth 0 a Earth Marsb Earth Earth Marsb 0 -1.24 2.84 0 a a Earth 1474 -3.29 -0.75 -2.91 AU Earth Earth Earth Mars 1657 2022 3131 3322 3.29 -1.80 1.29 0.75 -4.04 0.62 -2.91 -0.50 0 0 0 0 268 0 0 0 2583 0.18 3131 2.42 3399 0 0 0 191 1109 0 0 0 3.38 -2.86 -0.50 vx (km/s) -0.71 vy (km/s) 3.34 vz (km/s) 0a a a -3.24 -2.16 3.09 AU 3.09 T rmars at t0 = [1.03 1.12 0] rmars at t0 = [ 0.93 1.20 0] a b a initial v with respect to Earth initial v with respect to Earth 0.008 AU from Mars (cycler aphelion) 84 a) Cycler 4.3.1.-5, sun-fixed b) Cycler 2.5.1.+0, sun-fixed c) Cycler 4.3.1.-5, Earth/Mars-fixed (TRP frame) d) Cycler 2.5.1.+0, Earth/Mars-fixed (TRP frame) e) Cycler 4.3.1.-5, near Earth view of c) f) Cycler 2.5.1.+0, near Earth view of d) Figure 3.10: Top-down views of Cycler 4.3.1.-5 and Cycler 2.5.1.+0 85 Figure 3.10 shows the top-down view of one cycle of the cyclers presented in Table 3.7 and Table 3.5. The dots and numbers sequentially label the planet encounters. A star next to a number indicates a flyby is required. Two stars indicate a flyby that reinitiates the next cycle. Parts c and d illustrate the same trajectories shown in parts a and b, but are plotted in a translating, rotating, and pulsating reference frame42 that fixes both Earth and Mars for all times. In this frame, the Earth is fixed at the origin, and Mars is fixed at unity on the x axis for all time. The trajectory plots the position of the spacecraft relative to Earth and Mars. At any specified time, one unit in the pulsating frame is equivalent to the instantaneous line of sight distance between Earth and Mars. In such a frame, a true cycler orbit is exactly periodic. This frame provides a good measure of periodicity when it is difficult or impossible to find a true cycler, as is the case for the realistic solar system. The dynamics of such frames are discussed further in Appendix A. Parts e and f are zoomed-in on the Earth several orders of magnitude from parts c and d respectively. Note that all half and full-rev returns are out of the ecliptic plane for both cyclers in Figure 3.10. The near-Hohman qualities of the Earth- Mars leg are responsible for the near-symmetry seen in part c. Additional insight available from the pulsating frame is the visualization of near-encounters with Earth or Mars. As seen in part c, during the generic return, the cycler comes very close to a second Mars encounter, followed by a short transit to Earth. Note that distances on the graph may be deceiving because the units pulsate by 86 definition in this frame. The near-encounter does, however, come within 0.06 AU from Mars. When expanding the results to include true ephemerides, this additional nearencounter will be constrained as a true-encounter, thus significantly reducing the number of required cycler vehicles. As mentioned above, 2p vehicles are typically required to sustain a short Earth-Mars transit and a short Mars-Earth transit each synodic period. However, if a cycler contains inbound and outbound transits during each cycle, then only p vehicles are required. This tremendous advantage combined with its Hohman-like energy characteristics make Cycler 4.3.1.-5 an excellent candidate to study in a more realistic solar system. Figure 3.11: Three Dimensional view of Cycler 3.1.2.+1 Figure 3.11 illustrates a 3D view of one cycle of Cycler 3.1.2.+1. The first leg leaves Earth en-route to Mars via a generic return orbit. After 1+ revolutions, an out-ofplane half-rev return connects two Earth flybys, then the second identical generic return 87 completes the last leg. The third flyby is required to start the process again. detailed energy and time characteristics of this cycler are shown in Table 3.6. The Figure 3.12: Number of cyclers found vs. period Figure 3.12 summarizes the number of cyclers found for values of p from 1 to 6. If Table 3.4 were expanded to include five- and six-synodic period cyclers, an additional 157 entries would be added. These are documented in Table 3.9 - Table 3.11. In total, 2502 cyclers are investigated, 116 of which are totally ballistic. The eighty-four near-ballistic cyclers are documented based on the arbitrary values of ARMIN = 0.9 and TRMIN=0.85. Cyclers that are integer multiples of previously recorded cyclers are not included in these numbers. For example, Cycler 4.6.2.+1 is not included in Table 3.4 or Figure 3.12 because it is simply two cycles of Cycler 2.3.1.+1. 88 Some of the cyclers found with this method have been previously documented. Cycler 1.0.1.-1, with a Turn Ratio of 0.86, is the Aldrin cycler.1,13,14 Cycler 2.1.1.+2 and Cycler 2.3.1.+1 are identical to the "Case 2" and "Case 3" cyclers described by Byrnes et al.16 The cyclers found by McConaghy et al17 are identical to those discussed in this paper with h=0 and s=1. Four of the six-synodic period cyclers are listed in Table 3.10. 89 Table 3.9: Five-synodic period ballistic (or near) cyclers ARMIN=0.9, and TRMIN=0.85 Cycler p.h.s.i 5. 1. 1.-7 5. 1. 2.-3 5. 1. 5.-1 5. 2. 1.+7 5. 2. 2.+2 5. 2. 5.+0 5. 3. 1.-7 5. 3. 1.-6 5. 3. 3.-2 5. 4. 1.+6 5. 4. 1.+5 5. 4. 3.+1 5. 5. 1.-6 5. 5. 1.-5 5. 5. 1.-4 5. 5. 2.-3 5. 5. 2.-2 5. 5. 4.-1 5. 6. 1.+5 5. 6. 1.+4 5. 6. 1.+3 5. 6. 2.+2 5. 6. 2.+1 5. 6. 4.+0 5. 7. 1.-5 5. 7. 1.-4 5. 7. 1.-3 5. 8. 1.+4 5. 8. 1.+3 5. 8. 1.+2 5. 9. 1.-4 5. 9. 1.-3 5. 9. 1.-2 5. 9. 2.-2 5. 9. 2.-1 5. 9. 3.-1 5.10. 1.+3 5.10. 1.+2 5.10. 2.+1 5.10. 2.+0 5.10. 3.+0 5.11. 1.-3 5.11. 1.-2 5.11. 2.+1 5.12. 1.+2 5.12. 1.+1 5.13. 1.-3 5.13. 1.-2 5.13. 2.-1 5.14. 1.+2 5.14. 1.+1 5.14. 2.+0 5.15. 1.-2 5.15. 1.-1 5.16. 1.+1 5.16. 1.+0 5.17. 1.-1 5.18. 1.+0 Aphelion Ratio 1.04 1.20 1.44 0.90 1.20 1.43 0.92 1.10 1.07 0.94 1.12 1.07 0.96 1.18 1.48 0.94 1.45 1.44 0.98 1.20 1.49 0.94 1.44 1.43 1.02 1.30 1.71 1.03 1.31 1.72 1.10 1.48 2.15 1.08 2.10 1.44 1.11 1.49 1.07 2.08 1.43 1.24 1.83 1.14 1.24 1.82 0.97 1.49 1.45 0.97 1.48 1.43 1.11 2.16 1.10 2.12 1.50 1.46 Turn Ratio 0.97 1.00 0.92 1.07 0.94 0.91 1.17 0.90 1.19 1.45 1.06 1.44 1.95 1.44 1.08 1.79 1.19 1.10 1.74 1.23 0.90 1.36 0.87 1.16 1.78 1.27 0.93 1.91 1.30 0.92 2.09 1.38 0.93 1.57 0.90 1.16 1.94 1.25 1.71 0.87 1.24 1.76 1.09 1.00 1.81 1.07 2.69 1.39 1.20 3.06 1.41 1.22 2.13 0.93 2.37 0.91 1.37 1.45 EarthMars or aphelion (days) 229 168 133 182 128 118 270 205 195 189 122 170 279 186 154 262 142 134 198 107 82 219 104 116 245 169 142 154 94 73 204 154 130 198 117 137 123 82 160 81 112 177 138 101 101 70 280 153 141 196 82 105 202 130 126 64 152 84 v at Earth (km/s) 5.0 4.7 5.2 4.5 5.2 5.3 3.8 5.5 3.6 4.9 7.0 3.8 4.3 6.2 8.0 3.0 5.9 5.3 5.4 7.7 9.8 3.3 6.4 5.4 4.8 7.0 9.1 6.1 8.6 11.0 5.6 8.0 10.5 4.0 7.9 5.5 6.9 9.8 4.3 8.6 5.7 6.6 9.5 9.6 8.1 11.5 4.3 8.1 5.9 5.3 9.8 6.4 5.6 10.5 6.8 12.6 8.1 9.6 v at Mars Required Geocentric Turning Angle at each Flyby (deg) (km/s) 4.3 7.0 9.2 1.3 7.1 9.2 1.4 5.7 4.7 1.9 6.3 4.7 2.2 7.0 10.3 1.7 9.5 9.3 2.7 7.6 11.0 1.7 9.5 9.2 3.6 8.5 11.9 4.3 9.1 12.6 5.7 10.3 13.8 4.9 12.9 9.3 6.2 10.9 4.9 13.0 9.3 7.8 12.5 7.5 8.3 13.2 2.3 10.3 9.5 2.6 10.9 9.5 5.8 13.9 6.0 14.6 10.4 10.6 93 94 95 92 94 95 92 93 93 63 64 75 52 53 54 63 66 78 49 49 50 58 60 73 52 53 54 40 41 42 40 42 45 60 65 72 35 36 59 62 66 41 43 47 32 34 37 42 66 28 32 60 39 45 29 35 42 32 93 94 95 92 94 95 92 93 93 63 64 75 52 53 54 63 66 78 49 49 50 58 60 73 52 53 54 40 41 42 40 42 44 45 45 45 35 36 59 62 45 41 43 47 32 34 30 30 66 28 30 60 30 30 26 25 22 22 67 28 85 37 28 28 28 37 37 37 47 63 64 75 52 53 54 63 66 78 49 49 50 58 60 73 52 53 54 40 41 42 40 42 44 45 45 45 35 36 59 62 45 41 43 47 32 34 30 30 66 28 30 60 30 30 26 25 22 22 47 61 52 53 54 63 66 78 49 49 50 58 60 73 52 53 54 40 41 42 40 42 44 45 45 45 35 36 59 62 45 41 43 47 32 34 30 30 66 28 30 60 30 30 26 25 22 22 61 68 66 35 35 35 86 84 46 46 46 40 41 42 40 42 44 45 45 45 35 36 59 62 45 41 43 47 32 34 30 30 66 28 30 60 30 30 26 25 22 22 40 42 45 60 65 72 35 36 59 62 66 41 43 47 32 34 30 30 66 28 30 60 30 30 26 25 22 22 67 65 46 46 59 62 60 59 62 60 47 32 34 30 30 66 28 30 60 30 30 26 25 22 22 47 37 42 66 28 32 60 39 45 26 25 22 22 29 35 22 22 42 32 90 Table 3.10: Six-synodic period ballistic (or near) cyclers. Part I ARMIN=0.9, and TRMIN=0.85 Cycler p- h-s -i 6. 0. 1.+9a 6. 0. 1.+8b 6. 0. 1.+7c 6. 0. 1.+6d 6. 1. 2.-4 6. 1. 3.-3 6. 1. 4.-2 6. 1. 6.-1 6. 2. 1.+8 6. 2. 1.+7 6. 2. 1.+6 6. 2. 2.+3 6. 2. 2.+2 6. 2. 3.+2 6. 2. 3.+1 6. 2. 4.+1 6. 2. 6.+0 6. 3. 1.-9 6. 3. 4.+1 6. 4. 1.+7 6. 4. 1.+6 6. 4. 1.+5 6. 4. 1.+4 6. 5. 1.-8 6. 5. 1.-7 6. 5. 1.-6 6. 5. 5.-1 6. 6. 1.+6 6. 6. 1.+5 6. 6. 1.+4 6. 6. 1.+3 6. 6. 2.+2 6. 6. 2.+1 6. 6. 5.+0 6. 7. 1.-7 6. 7. 1.-6 6. 7. 2.+3 6. 7. 3.-2 6. 7. 5.+0 6. 8. 1.+6 6. 8. 1.+5 6. 8. 1.+4 6. 8. 1.+3 6. 8. 1.+2 6. 8. 3.+1 6. 9. 1.-6 6. 9. 1.-5 6. 9. 1.-4 6. 9. 2.-3 6. 9. 2.-2 6. 9. 4.-1 a b Aphelion Ratio 0.92 1.03 1.17 1.34 1.09 0.95 1.07 1.44 0.94 1.08 1.24 1.07 1.43 0.94 1.43 1.07 1.43 0.92 1.07 0.98 1.13 1.33 1.58 0.95 1.11 1.31 1.44 1.02 1.20 1.45 1.78 1.19 1.77 1.43 0.98 1.17 0.91 1.08 1.43 0.91 1.08 1.30 1.62 2.09 1.07 1.03 1.26 1.57 0.96 1.47 1.45 "Cycler 6S8" c Turn Ratio 1.40 1.22 1.07 0.93 0.91 1.30 1.16 0.90 1.26 1.07 0.91 1.19 0.89 1.39 0.92 1.23 0.93 0.89 1.04 1.59 1.33 1.11 0.93 1.62 1.16 0.88 1.15 1.82 1.48 1.21 0.99 1.38 0.96 1.03 1.49 1.05 0.98 1.40 0.99 2.39 1.89 1.50 1.18 0.94 1.47 1.97 1.35 0.98 1.61 0.94 1.21 EarthMars (or aphelion) v at Earth v at Mars (km/s) (km/s) Time (days) 213 3.0 1.2 179 4.0 3.9 133 111 203 264 197 135 220 158 123 174 115 235 119 181 123 279 156 227 142 113 96 283 213 180 137 189 128 104 89 141 99 122 289 199 176 199 107 211 158 116 95 81 179 248 186 161 274 150 139 d Required Geocentric Turning Angle at each Flyby (deg) 86 85 85 84 93 92 93 95 92 93 94 93 95 92 95 93 95 91 93 70 71 71 72 47 47 48 73 58 59 60 61 72 74 85 47 47 62 62 62 51 53 54 55 57 77 32 33 34 54 56 67 93 92 93 95 92 93 94 93 95 92 95 93 95 91 93 70 71 71 72 47 47 48 73 58 59 60 61 72 74 85 47 47 62 62 62 51 53 54 55 57 60 32 33 34 54 56 45 17 5.0 6.0 4.9 3.1 3.8 5.4 3.3 4.3 5.4 3.6 5.4 2.6 5.2 3.4 5.1 5.7 4.5 3.6 4.7 5.9 7.0 6.2 8.4 10.4 5.5 3.9 5.2 6.4 7.7 4.3 6.6 5.1 6.7 9.1 5.1 4.1 6.1 2.9 4.3 5.7 7.2 8.6 3.4 7.3 10.0 12.4 3.8 7.2 5.7 6.7 8.7 5.4 1.7 4.8 9.3 1.7 5.0 7.6 4.6 9.2 1.5 9.2 4.6 9.1 1.8 5.0 2.4 6.0 8.5 10.6 2.4 6.6 9.8 9.3 3.5 7.1 9.6 11.7 6.8 11.3 9.2 3.1 7.8 1.5 5.0 9.4 1.0 5.0 8.3 10.8 13.1 4.6 4.5 9.1 12.4 2.0 10.0 9.4 100 74 57 39 74 57 39 57 39 39 39 47 46 32 31 24 16 93 70 71 71 72 47 47 48 73 58 59 60 61 72 74 85 47 47 62 62 62 51 53 54 55 57 60 32 33 34 54 56 45 47 47 48 73 58 59 60 61 72 74 85 47 47 91 62 62 51 53 54 55 57 60 32 33 34 54 56 45 91 73 77 51 53 54 55 57 77 32 33 34 67 69 45 32 32 33 34 67 69 67 67 69 56 56 56 32 73 77 77 77 46 45 19 19 19 19 46 46 46 46 32 31 24 16 93 24 16 93 16 16 "Cycler 6S9" "Cycler 6S7" "Cycler 6S6" described described by McConaghy et al 91 Table 3.11: Six-synodic period ballistic (or near) cyclers. Part II ARMIN=0.9, and TRMIN=0.85 Cycler p- h-s -i 6.10. 1.+5 6.10. 1.+4 6.10. 1.+3 6.10. 1.+2 6.10. 2.+2 6.10. 2.+1 6.10. 4.+0 6.11. 1.-5 6.11. 1.-4 6.11. 2.+2 6.12. 1.+4 6.12. 1.+3 6.12. 1.+2 6.13. 1.+5 6.13. 1.-5 6.13. 1.-4 6.13. 1.-3 6.13. 2.-2 6.13. 3.-1 6.14. 1.+3 6.14. 1.+2 6.14. 1.+1 6.14. 2.+1 6.14. 2.+0 6.14. 3.+0 6.15. 1.+4 6.15. 1.-4 6.15. 1.-3 6.15. 1.-2 6.15. 2.+1 6.16. 1.+2 6.16. 1.+1 6.17. 1.+3 6.17. 1.-3 6.17. 1.-2 6.17. 2.-1 6.18. 1.+2 6.18. 1.+1 6.18. 2.+0 6.19. 1.+2 6.19. 1.-2 6.19. 2.+0 6.20. 1.-4 6.20. 1.+1 6.20. 1.+0 6.21. 1.+1 6.21. 1.-1 6.22. 1.+0 Aphelion Ratio 0.94 1.15 1.44 1.89 0.94 1.43 1.43 1.09 1.38 0.98 1.00 1.26 1.67 1.08 0.90 1.18 1.58 1.10 1.46 1.08 1.44 2.09 1.07 2.08 1.43 1.11 0.95 1.32 1.94 1.11 1.20 1.78 1.14 1.04 1.59 1.48 0.94 1.44 1.43 1.19 1.20 1.47 0.93 1.07 2.08 1.29 1.63 1.43 Turn Ratio 2.31 1.76 1.34 1.02 1.84 1.17 1.06 1.75 1.17 1.61 2.14 1.57 1.15 0.94 3.44 1.92 1.21 1.65 1.06 1.93 1.34 0.94 1.59 0.85 1.10 0.90 3.00 1.57 0.94 1.27 1.67 1.08 1.06 2.91 1.24 1.30 2.32 1.34 1.17 0.97 2.09 1.18 1.09 1.94 0.94 1.01 1.19 1.35 EarthMars (or aphelion) v at Earth v at Mars (km/s) (km/s) Time (days) 219 3.3 1.7 137 4.9 6.4 104 86 231 115 121 219 173 191 232 120 92 79 276 198 160 202 143 158 104 81 175 91 119 74 285 179 147 117 128 89 69 241 160 149 219 104 116 64 195 79 183 160 81 57 158 105 6.4 8.1 2.7 5.4 5.2 8.2 11.0 6.1 3.7 5.5 7.4 16.7 5.4 9.2 12.4 5.0 6.1 4.3 6.4 8.6 3.6 7.4 5.2 17.2 6.3 10.5 14.3 7.7 5.2 7.7 17.9 7.5 12.5 7.3 3.3 6.4 5.4 18.8 9.4 10.6 12.8 4.3 8.6 20.3 12.7 6.4 9.6 12.2 1.5 9.2 9.2 6.2 10.7 2.9 2.7 7.8 11.1 9.7 1.5 7.9 12.5 5.4 9.6 4.9 9.6 13.1 4.6 12.7 9.2 10.4 2.6 10.0 14.8 6.4 7.1 11.7 11.3 4.8 12.6 10.1 1.7 9.5 9.2 12.5 8.2 11.1 5.0 4.9 13.0 14.4 12.9 9.5 Required Geocentric Turning Angle at each Flyby (deg) 50 52 54 56 69 72 83 33 33 48 51 53 55 22 25 26 27 55 60 52 54 57 70 75 79 22 25 26 28 39 53 56 17 21 26 48 50 54 72 17 23 35 29 52 57 14 27 54 45 45 45 45 45 45 45 33 33 48 36 36 36 22 25 26 27 55 30 30 30 30 30 30 30 22 25 26 28 39 26 25 17 21 22 45 22 22 22 17 22 35 20 20 20 14 18 18 45 45 45 45 45 45 45 33 33 48 36 36 36 22 25 26 27 55 30 30 30 30 30 30 30 22 25 26 28 39 26 25 17 21 22 45 22 22 22 17 22 35 20 20 20 14 18 18 45 45 45 45 45 45 45 33 33 48 36 36 36 22 25 26 27 55 30 30 30 30 30 30 30 22 25 26 28 39 26 25 17 21 22 45 22 22 22 17 22 35 20 20 20 14 18 18 45 45 45 45 45 45 45 33 33 48 36 36 36 22 25 26 27 55 30 30 30 30 30 30 30 22 25 26 28 39 26 25 17 21 22 45 22 22 22 17 22 35 20 20 20 14 18 18 50 52 54 56 69 72 83 33 33 48 36 36 36 22 25 26 27 55 30 30 30 30 30 30 30 22 25 26 28 48 26 25 17 21 22 48 22 22 22 17 22 35 20 20 20 14 18 18 48 51 53 55 22 25 26 27 55 30 30 30 30 30 30 30 22 25 26 28 48 26 25 17 21 22 50 22 22 22 17 22 35 20 20 20 14 18 18 22 25 26 27 55 60 52 54 57 70 75 79 22 25 26 28 48 26 25 17 21 22 50 22 22 22 17 22 35 20 20 20 14 18 18 48 53 56 17 21 22 50 22 22 22 17 22 35 20 20 20 14 18 18 17 21 26 50 50 54 72 17 23 35 20 20 20 14 18 18 35 29 52 57 14 18 18 14 27 54 35 46 50 47 45 31 31 72 72 48 47 46 24 24 24 92 3.9 CHAPTER CONCLUSIONS A systematic method for constructing ballistic and near-ballistic cyclers between Earth and Mars in a simple solar system is presented. The method combines the advantages of previous works into one more generalized approach, utilizing both the geometry associated with free-return velocity diagrams and the many solutions that arise from the multiple revolution Lambert's Problem. The method only requires It iterations on sub-problems and does not require an initial trajectory estimate. marches through the entire defined solution space of all feasible combinations of 2 full, 1 half, and identical generic return orbits that combine to form a cycler. Free parameters associated with each unique combination of free-returns are then chosen to minimize the maximum required turning angle, thus maximizing the probability that the cycler is ballistic. The resulting ballistic and near-ballistic cyclers exhibit a wide range of energy and time characteristics, making them good candidates for a variety of potential applications. The method is general in the sense that it encompasses many previously known Earth-Mars idealized cyclers. Additionally, twenty-four new ballistic cyclers are found with periods of four synodic periods or less. remarkably low terminal speeds. Several of the new cyclers exhibit In particular, the Earth-Mars transfer on Cycler 4.3.1.-5 is very near a Hohman. In addition, this near-ballistic four-synodic period cycler only requires four vehicles to maintain a one-synodic period frequency for both inbound and outbound transfers. 93 The method is limited by imposed constraints in the problem formulation, thus several extensions will be addressed in the next chapter. These include generalizing the definitions of half and full-rev returns to include n transfers where n is any integer, and considering cyclers with non-identical generic returns. 94 4 Idealized Free-Return Cyclers Composed of n Transfers and One or More Identical or Different Generic Returns 4.1 CHAPTER SUMMARY In this chapter a new technique is developed to globally identify and optimize all useful idealized Earth-Mars cyclers constructed with Earth-Earth free-returns patched by gravity-assisted flybys. Previous techniques that use solutions of the multiplerevolution Lambert problem are expanded. Using combinatorics, the entire solution space is investigated for combinations of generic, half-rev, and full-rev free-return transfers for cyclers with periods up to three synodic periods. All three types of freereturns are generalized to include the possibility for multiple revolutions. Multiple options for the short Earth-Mars and Mars-Earth legs that characterize the cyclers are included. The generalized technique requires the solutions to hundreds of thousands of minimax optimization problems, and results in 142 previously undocumented ballistic cyclers. In addition, the global search verifies and encompasses results from several earlier studies that looked at a variety of subsets of the current solution space. The final list is exhaustive for the defined class of cyclers, and is an excellent resource to seek cyclers in a more realistic solar system. 95 4.2 INTRODUCTION Recently, several studies have investigated Earth-Mars cyclers using Earth free- return trajectories patched together with gravity-assisted Earth flybys.13,15,16,17,29,30 The free-returns can be divided into three mutually exclusive categories: 1) Half-rev (oddn) transfers, 2) Full-rev (even-n) transfers, and 3) generic transfers. Reference 17 presents all cyclers consisting of a single generic return while Ref. 15 investigates several cyclers using two identical or non-identical generic returns patched by an Earth flyby. Reference 16 shows solutions for two-synodic period cyclers with a single generic return patched with 1 half-year and 2 full-year transfers. Reference 29, or Chapter 3 of this text, presents optimized solutions for all cyclers that use one or more identical generic returns patched by any combination of 1 half-year or 2 full-year transfers, including those discussed in Refs. 16 and 17. Reference 30 presents several improved versions of previously documented cyclers when the definitions for half- and full-rev returns are generalized to include multiple revolutions. These solutions are also tabulated in Section 4.3 of the current chapter. Clearly, there has been much recent success in finding Earth-Mars cyclers using patched free-return trajectories. However, the tremendous number of potential combinations of half-rev, full-rev, and generic returns has encouraged investigators to consider limited subsets of cyclers of this class. The current chapter seeks to generalize this search to include all subsets including several previously investigated and uninvestigated cycler classes. 96 Periodic constraints associated with cyclers restrict true solutions to exist only in simplified solar system models. Therefore, the references mentioned above primarily focus on techniques to find solutions in a circular-coplanar model. Additionally, Refs. 3, 4, 8, 15, and 16 find corresponding ballistic solutions in an ephemeris model. Refs. 13, 15, and 18 include ephemeris model solutions that use impulsive or low-thrust maneuvers respectively when necessary. The current chapter is limited to the circularcoplanar model, and the majority of the cyclers presented require no fueled maneuvers. With the exception of the VISIT9,10,11,12 cyclers, and two cyclers presented in Ref. 30, previously documented cyclers use generic return transfers for the Earth-Mars transit legs. The current study includes both generic returns and full-rev returns as transit legs. This increases the solution space and gives additional options for important cycler properties such as Earth-Mars transfer times and hyperbolic energy at Mars. While half-rev returns are useful in this study as loitering orbits, they are not considered as potential transit legs because most half-rev returns occur out of the ecliptic plane, and thus, have no opportunity to encounter Mars. The first section discusses the primary motivation for this chapter and presents a sampling of expected results. The second section defines an idealized Earth-Mars freereturn cycler and is followed by a section that outlines the problem statement. The next section gives background information on several combinatorics problems that will be useful in finding all feasible combinations of free-returns that can be patched together to form cyclers. The following section gives a detailed overview of the solution method 97 including algorithms, examples, and post-processing techniques. The Results section catalogues 174 purely ballistic cyclers and twenty-nine very-near-ballistic cyclers with forty-nine different values of v at the Earth. The method finds all useful one-, two-, and three-synodic period cyclers. The resulting patched trajectories are documented using a new nomenclature for free-return cyclers introduced in Ref. 25. Previously documented cyclers are noted and promising new cyclers are discussed. 4.3 MOTIVATION AND A SAMPLING OF EXPECTED RESULTS The primary motivation for this chapter is to extend the results of Chapter 3 with regard to the following two topics: 1) Remove the limitation that multiple generic returns must be identical. 2) Include the possibility for the half- and full-rev returns to make multiple revolutions around the sun. The first topic is deemed valuable based on the success in Ref. 15, where several promising two synodic period cyclers are discovered composed only of non-identical generic returns. It is reasonable to assume then, that other cyclers composed of nonidentical generic returns exist perhaps including n returns and/or with a period of three-synodic periods. The second topic is an application of the generalized free-return solutions given in Chapter 2. Applications for free-returns include any flyby missions that require consecutive encounters with the same body. In particular, cycler missions are excellent candidates because free-returns orbits are often used to satisfy the periodic time constraints.8,17,19 The half and full-rev free-returns in Chapter 3 are limited to and 2 98 transfers with half- and full-year times of flight respectively. The following analysis demonstrates how some of the resulting cyclers from Chapter 3 can be improved when including this more generalized definition for the n transfers. The remainder of the Chapter will develop from scratch a new technique to systematically search the entire newly defined solution space in a global search for idealized free-return Earth-Mars cyclers. A majority of the solutions from Chapter 3 require multiple consecutive 1 and 2 transfers that each must be patched with an Earth flyby. In many cases, several of the flybys become unnecessary if the n transfers are considered. The first four entries in Table 4.1 illustrate a few favorable examples. Take Cycler 4.11.1.-2 as an example. The second number, 11, indicates that the cycler must loiter at the Earth for eleven halfyears before re-initiating the next cycle. In the method outlined in Chapter 3, the optimized solution is to perform four one-year transfers with three half-year transfers in the middle, all patched by Earth flybys. Alternatively, using the generalized definition for a half-rev transfer outlined in Chapter 2, the seven free-return trajectories can be replaced with one 5.5-year half-rev return. Although the required turning angle increases to 107o, it still requires an Earth flyby altitude above 200 km. Because the half-rev loitering orbit is not used for the Earth-Mars transfer, the energy characteristics of the cycler remain the same. On the contrary, the last two entries in Table 4.1 use three-year full-rev transfers in the ecliptic plane for the Earth-Mars transfer legs. Thus, the v's at Mars are different, and significantly better in these cases, from the similar 99 cyclers reported in Chapter 3. Cycler 4.13.1.-1 has a 40% decrease in v at Mars while increasing the transit time by only 3 days. Cycler 4.11.1.-4 did not come close to Mars using the methods described in Chapter 3, thus it was not reported. However, using the general n transfers, a very attractive solution was found with a short transit time and low terminal speeds. Table 4.1 Examples of improved solutions to cyclers documented in Chapter 3. Cycler a previous/ new previous new previous new previous new previous new previous new previous new v Earth (km/s) 7.8 7.8 5.4 5.4 5.4 5.4 3.6 3.6 5.5 5.5 3.8 b v Mars (km/s) 9.9 9.9 9.2 9.2 9.2 9.2 4.7 4.7 9.3 5.6 4.8 Earth-Mars Transit time (days) 94 94 115 115 116 116 195 195 137 140 168 Sequence of n half-year transfers patched by flybys 2-1-2 5 2-1-2 5 2-2-1-2-2 9 2-2-1-1-1-2-2 11 2 -2-2-1-2-2-2 4-1-2-6 2-6-2-1 Turning Angle for 200 km alt. flyby (deg) 60 60 84 84 84 84 111 111 84 84 108 Max Turning Angle Req. (deg) 54 61 73 69 72 82 70 107 72 79 87 Cycler 2.5.1.+0 Cycler 3.5.1+1 a a Cycler 3.9.1.+0 a Cycler 4.11.1.-2 Cycler 4.13.1.-1 Cycler 4.11.1.-4 a a b Solutions documented in Chapter 3 , Solution found using methods outlined in Chapter 3 does not reach Mars. A full understanding of Table 4.1 requires a thorough reading of the preceding chapters of this text. However, the purpose of this section is not to detail the origin of the solutions, but rather to illustrate an example application of the mission planning tool developed in Chapter 2 as seen in Figure 2.22. The rest of this chapter will present a new technique that uses the solutions presented in Chapter 2 to the search the entire meaningful solution space of idealized free-return Earth-Mars cyclers. 100 4.4 IDEALIZED FREE-RETURN CYCLER DEFINITION Similar to the previous chapter, an idealized free-return Earth-Mars cycler is defined to be a cycler in an ideal solar system that is composed of a sequence of EarthEarth free-return trajectories patched together with Earth gravity-assisted flybys. The circular-coplanar solar system is described in Section 3.3. The following chapter will seek analogous "real-world" solutions based on the initial guesses from the simple model solutions. 4.5 PROBLEM SETUP The purpose of this chapter is to globally identify all useful ballistic idealized free-return Earth-Mars cyclers, where a cycler is characterized as ballistic if all of its flybys require altitudes greater than 200 km. The goal is to identify all possible combinations of free-return trajectories that can be patched together to form cyclers. This chapter is based heavily on the results from Ref. 30, or Chapter 2 of this text, where the three types of free-returns are discussed in detail. The primary result of Chapter 2 is a reliable and robust method for systematically obtaining all half-rev, fullrev, and generic returns that have a given value of v at Earth. Figure 4.1 is an example velocity diagram that visually displays all the freereturn solutions for a circular Earth and v=0.1838 AU/TU. It is identical to Figure 2.22 with the exception that each full-rev circle and set of half-rev points can only 101 represent one solution with a particular time of flight. Of course in the simple solar system model, if a spacecraft is on a one year full-rev return and does not perform a maneuver upon returning to Earth, then of course, it will return again after two years making it a two year solution also. However, for applications of this chapter, this case can be ignored because it is identical to the case of back-to-back one year solutions, which will be included in the march through the solution space. The same is true for half-rev points. Again, for purposes of this chapter, only one time of flight is associated with each set of half-rev points. Figure 4.1: Example v diagram with free-return solutions for |v|=0.1838 AU/TU 102 For cyclers composed only of Earth-Earth free-return trajectories, each leg begins and ends at the Earth. As Chpater 3 discusses, in order to patch all of the legs with flybys, the value of v at Earth must match for all incoming and outgoing legs. Due to symmetry, when the Earth is in a circular orbit, v at the end of a leg is identical to v at the beginning for any free-return. Figure 4.1 shows the outbound v solutions that lead to all three types of free-returns for a given v. Table 4.2 summarizes how to obtain the inbound x, y, and z components of v based on the outbound x, y, and z components of v for a particular leg. The main point is that all inbound and outbound magnitudes of v at Earth are identical. Thus, all inbound and outbound v vectors before and after each flyby for the entire cycler lie on the same v sphere. Table 4.2: Inbound v for free-returns a Free-return type Half-revolution Full-revolution Generic a ^ ^ v in x -v out x v out x -v out x v in y -v out y v out y v out y v in z v out z v out z v out z x in direction of rE , z in direction of v E Given a diagram similar to Figure 4.1 and the information in Table 4.2, all of the required turning angles on any v sphere can be easily computed for any arbitrary sequence of consecutive free-returns, and the v values automatically match. Note that for a particular full-rev circle, one degree of freedom is required, and for a given set of half-rev points, the above plane `o' or the or below plane `x' must be chosen to specify v out. A unique diagram similar to Figure 4.1 can be generated for any value of v, and 103 clearly the locations of the full-rev circles, sets of half-rev points, and generic dots will change for each different v. The times of flight associated with each generic dot will also vary. The problem addressed by this chapter is to find the appropriate values of v that have particular sequences of generic, half-rev, and full-rev returns that meet the criteria given in Table 4.3. Table 4.3: Criteria for a sequence of free-returns to be a ballistic cycler 1) 2) 3) Criteria all times of flight add up to an integer multiple of the synodic period all flybys patching the free-returns require altitudes greater than 200 km at least one of the legs crosses the orbit of Mars . This is a challenging problem because of the size of the solution space. For the example v of 5.5 km/s shown in Figure 4.1, there are 40 generic return dots, 13 full-rev circles, and 5 sets of half-rev points. If the search were restricted to cyclers containing only these solutions with sequences of g-g-f-h (generic-generic-full-half), then there are (40)(40)(13)(10)=208,000 possible unique combinations to consider. Of course this is a miniscule subset of all the combinations. If we considered cyclers with sequences of gg-g-f-f-h, then there are 83,200,000 unique combinations to consider, and this, again, is only a small subset. The first criterion from Table 4.3 is extremely prohibitive, meaning there may not be any combinations for a given v that add up to exactly an integer multiple of the synodic period. However, within a small tolerance, for instance 0.01 years, there may be several valid combinations. Therefore, the vast majority of the combinations can 104 immediately be eliminated by examining this timing constraint and its relationship to generic and n returns. A cycler is constrained to have a repeat time that is an integer multiple of the synodic period. If a cycler is to consist of generic returns and any combination of n transfers, then the total time of flight, TOF, in years for all generic returns is given by Eq. (4.1) where p is the integer number of synodic periods, is the synodic period, and y is the total number of half-years allotted for half- or full-rev transfers. TOF= p-y/2 (4.1) All values of y that yield a time of flight greater than 0.876 years for p=13 are included to compose an exhaustive list of times of flight allotted for generic returns. These values are shown below in Table 4.4. From Chapter 2, all generic returns with times of flight less than 0.876 years are simply the Earth's orbit. Limiting the search to p=13, or cyclers with repeat times of one, two, or three synodic periods, is very reasonable because only one Mars-Earth or Earth-Mars transit is guaranteed during each cycler period. Thus a three-synodic period cycler typically requires six vehicles to maintain an Earth-Mars transit and a Mars-Earth transit every synodic period. Thus, it is desirable for p to be small. 105 Table 4.4: Total TOF allotted for generic returns, = 2.135384 yra TOF b (allotted for generic returns) 0.9062 1.1354 1.2708 1.4062 1.6354 1.7708 1.9062 2.1354 2.2708 2.4062 2.7708 a p 3 1 2 3 1 2 3 1 2 3 2 y 11 2 6 10 1 5 9 0 4 8 3 TOF b (allotted for generic returns) 2.9062 3.2708 3.4062 3.7708 3.9062 4.2708 4.4062 4.9062 5.4062 5.9062 6.4062 p 3 2 3 2 3 2 3 3 3 3 3 y 7 2 6 1 5 0 4 3 2 1 0 based on Earth and Mars mean semi-major axes, b obtained from Eq. (4.1). From Table 4.4, for up to three-synodic period cyclers, there are 22 time of flight values that can be allotted for any number of generic returns. For each value of v, it is necessary to check all combinations of the generic return times of flight from a diagram similar to Figure 4.1 to the values given in Table 4.4. If a particular combination of k generic returns has a combined TOF within a desired tolerance to the ith TOF listed in Table 4.4, then all permutations of the k generic returns and n returns totaling yi half-years need further evaluation. returns are discarded. The idea is to find all v values that lead to ballistic cyclers. When considering all the possible combinations of the generic return solutions and all permutations of the remaining good combinations along with the n returns totaling y half-years over a range of v values, even at large intervals, there are seemingly an infinite number of sequences that need evaluation based on steps 2 and 3 from the criteria in Table 4.3. 106 All other combinations of the generic Also, for each sequence, the degree of freedom associated with each full-rev circle must be selected such that the maximum turning angle is minimized. This large problem can be reduced to a manageable size by efficiently finding all meaningful sequences. The details of this process are outlined in a later section titled "Solution Method." The next section, however, first gives some pertinent background on the field of combinatorics and explains how it will be useful in solving this complex problem. 4.6 COMBINATORICS Combinatorics is a branch of mathematics that studies the enumeration, combination, and permutation of sets of elements.43,44,45,46,47 Combinatorial optimization is the science of selecting particular sets of discrete variables in a manner that optimizes some performance index. In this well-established technical field, typical applications include airline scheduling, communication networks, management science, and any other business, science, or engineering application dealing with very large discrete optimization problems. The nature of most problems precludes the possibility of simply enumerating (generating) all possible choices and globally selecting the best one. Branch and bound methods, cutting plane techniques, and other classic approaches can be used to find locally optimal solutions. Similar to optimization of continuous variables, most techniques cannot guarantee a globally optimal solution. 107 The current problem of selecting particular sequences of direct returns is clearly a discrete optimization problem. However, if a few practical limitations are imposed, then the problem can be globally solved with reasonable computational power by generating and evaluating all meaningful combinations. Thus, the combinatorics problem becomes one of enumeration rather than a complicated local optimization scheme. The following subsections cover four common topics in combinatorial analysis that are encountered in solving the current cycler problem. An example is presented for each topic. Generating algorithms, for reference, are given in Appendix B. 4.6.1 General Counting Given a vector x with n positive integers, it is desired to generate a matrix X(n,m), where each column contains a unique ordered string of n integers. The ith entry can take the value 1x(i). There are x(1) choices for the first digit, x(2) choices for the second, etc. Thus, the total number of strings or columns of X is m=x(1)x(2)...x(n1)x(n). An example solution is shown below. GIVEN: RESULT: x = [ 2 1 4] T 1 1 1 1 2 2 2 2 X = 1 1 1 1 1 1 1 1 1 2 3 4 1 2 3 4 An odometer is a common application of a general counting mechanism with a slightly modified definition. In the case of an odometer, each entry ranges from 09 instead of 1x(i). 108 In the context of the cycler application, a general counting routine is necessary because, as seen in Figure 2.22, there may be more than one full-rev circle for a given time of flight. Although, not present in Figure 2.22, there also may be multiple sets of half-rev points for a given time of flight, and of course, each set has two solutions. This is explained further in a later section. 4.6.2 n-Tuples An "n-tuple", is a grouping of n elements without regard to order. For example, {1,2,4} and {1,4,2} represent the same 3-tuple. Given two integers, m and n, it is desirable to generate a matrix X(n,q), where each column is an n-tuple, with entries that range from 1m. The total number of n-tuples43 is q= (m+n-1)!/n!/(m-1)! The resulting X containing all the 3-tuples (n=3) of a set with 4 elements (m=4) is shown below. Each n-tuple is arbitrarily written in ascending order. A generating algorithm is given in Appendix B. 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 4 X = 1 1 1 1 2 2 2 3 3 4 2 2 2 3 3 4 3 3 4 4 1 2 3 4 2 3 4 3 4 4 2 3 4 3 4 4 3 4 4 4 With regard to the current application, an n-tuples routine is needed to find groupings of the generic return times of flight that add up to the total times of flight given in Table 4.4. 109 4.6.3 Beaded Necklace Problem A connected necklace is composed of n beads, and each bead is one of q colors. A vector x is given with q positive integers. The ith entry of x represents the number of beads in the necklace associated with the ith color. Thus, the necklace has x(1) beads of the 1st color, x(2) beads of 2nd color, ..., and x(q) beads of qth color. The problem is to find all the unique color sequences for the necklace. Namely, generate a matrix X(n,m), where each column contains a unique necklace. Note, the beads are situated around a circle, thus {1,2,3,4} is identical to {2,3,4,1}. Also, a necklace viewed from the top is the same as if viewed from the bottom, thus {1,2,3,4} is identical to {4,3,2,1}. If each bead is a different color, for n>2, there are (n-1)!/2 unique necklaces.43 The example below shows all unique necklaces containing 4 red beads, 2 blue beads, and 1 yellow bead. A generating algorithm is given in Appendix B. GIVEN: x = [ 4 2 1] , n = 7, q = 3 T RESULT: r r r r r r r r r r r r r r r r r r r r r y r r y b b X = r r y r b b r r r b b r r r r b r y b y b b b y r b r y b b b y b b y b The current problem of finding combinations of free-returns to form cyclers is a perfect application for the beaded necklace problem. Suppose a sequence of times of flight is identified to have a total period that is an integer multiple of the synodic period, for instance: TOF={1.465, 1.942, 1, 1, 0.5, 0.5} years. These times of flight add up to 110 three synodic periods, making any sequence of them eligible to form a cycler. It is necessary, then, to find all cyclical permutations of these free-returns in order to find the best order in terms of the maximum required turning angle. This is exactly analogous to the beaded necklace problem, with x = [1 1 2 2], n=6, and q=4. 4.6.4 Partitions of an integer A partition of n is a grouping of positive integers without regard to order that has a sum of n. The partition function, P(n), gives the number of partitions for a particular n. P(n) can be found using a number of methods, including the use of power series, recursive functions, and infinite series generating functions.43,44,45,46 The enumerating generating function for partitions is p ( x) = x kn =(1 + x + x 2 + )(1 + x 2 + x 4 + )(1 + x 3 + x 6 + ) n=0 k =1 (4.2) This can be used as a brute force method to calculate P(n), because p ( x ) = P ( n) x n n=0 Thus, the coefficient of xn in the expansion of the right side of Eq. (4.2) is the number of partitions of n. Of course, only a limited number of product terms from the far right side of Eq. (4.2) are necessary to find the coefficient of xn for a given n. This series product can also be used to explicitly generate each partition.43 Rewriting the enumerating generating function as p ( x) = (1 + x1 + x1+1 + x1+1+1 + )(1 + x 2 + x 2 + 2 + x 2 + 2 + 2 + )(1 + x 3 + x3+ 3 + x 3+ 3+ 3 + ) 111 Expanding each term, maintaining this form of the exponents, and grouping but not adding common powers of x gives p ( x) = {1} + { x1} + { x1+1 + x 2 } + { x1+1+1 + x1+ 2 + x 3 } + { x1+1+1+1 + x1+1+ 2 + x 2 + 2 + x1+ 3 + x 4 } + Each term represents one partition. Thus, the two partitions of two are 1-1 and 2; the three partitions of three are 1-1-1, 1-2, and 3; the five partitions of four are 1-1-11, 1-1-2, 2-2, 1-3, and 4, and so on. Therefore, Eq. (4.2) can be used to enumerate individual partitions and to find P(n). Other simpler methods of partition enumeration yield easily programmable algorithms.45,47 In regards to the current application, suppose a cycler performs one or more generic returns and has n half-years before needing to initiate the next cycle. Generating all of the partitions of n gives every possible combination of half- and fullrev returns that have a combined time of flight of n half-years. For example, if n=3, from above we see that the only three combinations of half-years that need consideration are 1-1-1, 1-2, and 3. For the scope of this study, n=11, is the largest integer that needs partitioning. The first eleven function evaluations for P(n) are {1, 2 , 3, 5, 7, 11, 15, 22, 30, 42, 56}, and the actual partitions for n=111 are given in Appendix B in Table B1. 4.7 SOLUTION METHOD In this section, the solution method is presented by examining in detail an example branch of the solution space tree, from the original selection of v at Earth to the final minimax optimization problem. As mentioned previously, the purpose of this 112 study is to find all feasible ballistic Earth-Mars free-return cyclers by marching through and evaluating all possible permutations of generic, half-, and full-rev returns. In total, the method requires the evaluation of billions of branches, with almost two million requiring the solution to a minimax problem. The main algorithm and all necessary post-processing steps are outlined. 4.7.1 1. Main Algorithm and Explanation Select a value for v at Earth. The algorithm examines all values of v at Earth considered desirable for a Earth-Mars cyclers. In this study, all v values from 2.510 km/s, at intervals of 0.0025 km/s are considered. The lower bound was chosen because a Hohman transfer to Mars requires 2.9 km/s, and the upper bound was chosen arbitrarily to be the largest desirable excess velocity at Earth. For illustration purposes, an example value of 5.5 km/s is chosen to be consistent with Figure 4.1 2. From Chapter 2, calculate and store the locations and pertinent information for all full-rev circles and sets of half-rev points for the current v. From Table 4.4, eleven half-years is the upper bound for times of flight allotted for n returns when considering cyclers of three synodic periods or less. Therefore, it is only necessary to find the n return solutions for times of flight up to 11 half-years. For the arbitrary v value of 5.5 km/s, these are illustrated in Figure 4.1. 113 3. Create a vector with 11 entries, representing 1-11 half-years. The ith entry of h is equal to the number of Earth to Earth solutions with a TOF of i/2 years for the current value of v. Call this vector h, where h represents half-year. Each set of half-rev points has an above-plane and below-plane solution. Thus, h(1), h(3), h(5), ..., h(11) are even-valued. For the current example, h=[2 1 0 2 2 2 2 2 2 4 2], meaning there are two 0.5-year solutions, one 1-year solution, zero 1.5-year solutions, two 2-year solutions, etc. As mentioned previously, in Figure 4.1, unlike Figure 2.22, any particular full-rev circle or set of half-rev points has only one associated time of flight. 4. From Chapter 2, calculate and store the locations and all pertinent information for all generic return solutions for the current v. In order to improve the accuracy of the solutions, the current study takes the linearly interpolated solutions obtained from the solution method in Chapter 2, and uses them as initial guesses in a one-dimensional solver that fixes v and iterates on TOF and in a Keplerian integration until the trajectory re-encounters Earth to a desired tolerance. For v =5.5 km/s, from Figure 4.1, there are forty solutions, each with a different time of flight. Table 4.5. The solutions are numbered and listed in 114 Table 4.5: Generic return solutions for |v|=5.5 km/s (0.1838 AU/TU) # 1 2 3 4 5 6 7 8 9 10 TOF (yr) 1.348 1.467 1.920 1.938 2.049 2.074 2.387 2.505 2.780 2.837 # 11 12 13 14 15 16 17 18 19 20 TOF (yr) 3.146 3.209 3.402 3.520 3.706 3.782 3.922 3.941 4.052 4.075 # 21 22 23 24 25 26 27 28 29 30 TOF (yr) 4.199 4.279 4.411 4.529 4.657 4.745 4.836 4.881 5.112 5.160 # 31 32 33 34 35 36 37 38 39 40 TOF (yr) 5.236 5.326 5.417 5.534 5.622 5.718 5.783 5.842 5.922 5.943 5. Use the "n-tuple" routine to find all n-tuples of the generic return solution set, for n=14. Limiting the search to only include cyclers with four or less generic returns is a practical limitation that dramatically decreases the runtime of the algorithm without sacrificing much in terms of results. A one-, two-, and three-synodic period cycler has a period of approximately 2.14, 4.27, and 6.41 years respectively. From Chapter 2, the shortest time of flight for a generic return that is different from the Earth's orbit is 0.876 years. Thus, the maximum number of generic solutions during a one-, two-, and three-synodic period cycler is two, four, and seven respectively. Thus, for a three-synodic period cycler, it is possible, although unlikely, to have up to seven generic returns. However, in this example, from Table 4.4, the shortest time of flight solution is 1.358 years. Thus, any combination of more than four of the solutions has a total time of flight greater than three synodic periods. This limitation is also deemed reasonable in retrospect, because the results will show no ballistic cyclers 115 consisting of four generic returns although all combinations were thoroughly investigated. Continuing with the example, there are (40+n-1)!/n!/(40-1)! n-tuples. This amounts to 40, 820, 11480, and 123410 combinations when n is 1, 2, 3, and 4 respectively, totaling 135750 combinations to consider. 6. Check the sum time of flight for each of the n-tuples against each time of flight given in Table 4.4. If there is a match to within a given tolerance, record the n-tuples and the matching information from Table 4.4. The t tolerance should be sufficiently large such that when v is incremented, no solutions are missed, but sufficiently small such that every n-tuple is not considered a good solution. Tolerance values of 0.01-0.04 yr are found to be reasonable. For the current example, of the 135750 n-tuples, there are 79 with t less than 0.04 yr. Table 4.6 lists a selected few. The value is the synodic period necessary for the particular combination of free-returns to be a cycler, and is obtained from = -t/p. Table 4.6: Good n-tuples n-tuple ... {3} {1,3} {1,6} {4,4} {1,1,12} {2,2,2,2} Total TOF a (yr) ... 1.9200 3.2678 3.4216 3.8758 5.9045 5.8671 Target TOF b (yr) ... 1.9062 3.2708 3.4062 3.9062 5.9062 5.9062 t (yr) ... 0.0139 -0.0030 0.0154 -0.0304 -0.0017 -0.0391 pb ... 3 2 3 3 3 3 yb ... 9 2 6 5 1 1 c ... 2.131 2.137 2.130 2.146 2.136 2.148 ... a ... b ... c ... ... ... ... from Table 4.5. from Table 4.4. the true Earth-Mars synodic period is = 2.135 yr. 116 7. All good n-tuples need evaluation. For the current example, of the 79 good combinations, the bold n-tuple from Table 4.6 is considered further. This row indicates that the first and sixth generic return solutions from Table 4.5 combine with six half-years allotted for n returns to a total time of flight near three synodic periods. The next task is to use the partition routine to find all integer partitions of six in order to find all combinations of n returns that have a total time of flight of six half-years. Table B1 in Appendix B lists all the partitions of six. In order for a specific partition of six to be a valid sequence of n returns for this problem, clearly, a solution must exist for each of the n returns in the sequence for the current value of v. Recalling step 3 above, the h vector for v=5.5 km/s is [2 1 0 2 2 2 2 2 2 4 2]. Thus, the partition {3,3} of six is not valid because h(3)=0, meaning there is no solution for a 3/2 year n return when v=5.5 km/s. Based on the current h vector and the y value from the current good n-tuple being evaluated, generate a list of all valid partitions of y. For the current example, this list is {6}, {5, 1}, {4, 2}, {4, 1, 1}, {2, 2, 2}, {2, 2, 1, 1}, {2, 1, 1, 1, 1}, and {1, 1, 1, 1, 1, 1}. 8. All valid partitions need evaluation. For the current example, of the eight valid partitions, {2, 2, 1, 1} is considered further. The same string represented in years rather than half-years is {1, 1, 0.5, 0.5}. The current n-tuple of generic returns is {1, 6}. Represented in times of flight in years, this is {1.348, 2.074}. Next, combine these to create one string representing a unique combination of 117 times of flight for free-returns that can be patched together to form a cycler. This combined string is {1.348, 2.074, 1, 1, 0.5, 0.5}. Use the beaded necklace routine to find all unique cycler permutations of the string. This is analogous to finding all the unique necklaces using one red bead, one blue bead, 2 yellow beads, and 2 green beads. Thus, x = [1 1 2 2] , n = 6, q = 4 T 1.35 1.35 1.35 1.35 1.35 1.35 1.35 1.35 1.35 1.35 1.35 1.35 1.35 1.35 1.35 1.35 2.07 2.07 2.07 0.5 2.07 2.07 0.5 2.07 0.5 0.5 0 1 1 1 0.5 1 1 1 1 0.5 2.07 1 0.5 2.07 0.5 2.07 0.5 2.07 2.07 0.5 1 2.07 0.5 X= 0.5 1 1 0.5 1 1 0.5 0.5 2.07 1 0.5 2.07 2.07 0.5 2.07 1 0.5 1 1 1 0.5 0.5 0.5 1 1 1 0.5 1 1 1 0.5 0.5 1 1 1 1 1 1 0.5 0.5 0.5 0.5 1 1 0.5 0.5 0.5 0.5 9. The 16 columns of X represent all unique cyclic permutations of the numbers {1.348, 2.074, 1, 1, 0.5, 0.5}. Each of the 16 columns needs evaluation. One effective method of reducing the size of the solution space is to ignore any solutions with back-to-back 0.5-year solutions. Nothing is lost because, as Chapter 3 explains, these cases are not feasible in the true solar system. Furthermore, back-to-back above or below plane 0.5-year solutions are simply a restricted case of a 1-year full-rev solution which will eventually be evaluated anyway. Returning to the current example, the bold column is considered further. Now using the h vector, create a new integer vector, xns, with entries representing the number of solutions for the ith TOF from the current cyclic permutation. The subscript, ns, represents number of solutions. Each generic return TOF will have a 1 associated with it, and the number of 118 solutions associated with the half- and full-rev TOFs are found in h. For the current example, this vector is , xns=[1 1 1 2 1 2]T, representing the number of solutions associated with each member of the cyclic permutation {1.348, 0.5, 1, 2.074, 1, 0.5}. Give xns as input to the general counting routine. For the current example the only sequences resulting from xns are the four columns of X. 1 1 1 X= 1 1 1 1 1 1 1 2 2 1 1 1 1 1 1 1 1 1 2 1 2 However, for a general case, there are x(1)x(2)...x(n) sequences to consider, which could range anywhere from one to several hundred. 10. Again, each of the columns of X, or each ordered combination that results from the general counting routine needs evaluation. For the sake of this example, the bold column is considered further. The active cyclic permutation is still {1.348, 0.5, 1, 2.074, 1, 0.5}. The sequence {1, 2, 1, 1, 1, 1} indicates to use the first solution for each of the legs except on the second leg use the second solution. Remember, all generic returns will have only one solution. All fullrev returns may have several associated full-rev circles. If there is more than one, the number in the general counting sequence indicates which circle to use, in ascending order of latitude. All half-rev returns may have multiple sets of half-rev points. If there is more than one set, the number in the general counting sequence indicates which set to use, again, in ascending order of latitude. An 119 above or below plane solution is indicated by an odd or even number respectively. Since the second leg in the current example is a half-rev return, the 2 in the general counting sequence indicates the below plane solution of the first set of half-rev points. Note, if there were two sets of half-rev points for the associated time of flight, a 3 or 4 would indicate the above or below plane solution respectively on the second set. Returning to the v sphere, we now have a specific ordered sequence of free-returns. In order to calculate turning angles required to patch the freereturns, the only remaining unknowns are the degrees of freedom associated with each full-rev return. The location on each full-rev circle needs specification. The following minimax optimization problem is posed: Choose the longitude value for each full-rev circle, if any, such that the maximum turning angle is minimized for the entire cyclic sequence of free-returns. Because each of the unknowns is an angle ranging from 0o360o, it is straight-forward to perform a global grid search at an appropriately small interval (15o works well) to select the best combination of unknowns to use as an initial guess for any non-linear minimax solver. For n degrees of freedom at an interval of d degrees, the global grid search requires floor(360/d)n maximum turning angle function evaluations. For the current example, there are two fullrev returns, thus two degrees of freedom are sought, requiring floor(360/15)2 or 576 function evaluations. The grid evaluates {0o, 0o}, {0o, 15o}, {0o, 30o}, ... , 120 {115o, 0o}, ... , {345o, 315o}, {345o, 330o}, {345o, 345o}. Identify the best combination. 11. For the current example, the bold pair of unknowns, {115o, 0o}, requires the smallest maximum turning angle out of all 576 evaluated. Use this set of unknowns as an initial guess in any non-linear minimax solver. By performing an initial global grid search, the local solution found by the solver is likely the global solution as well. For this example, the longitudes that minimize the maximum turning angle are {109.4o, 1.566o}. There are six turning angles required to patch the six legs into a cycler. The turning angles in degrees are {88.7o, 88.7o, 19.3o, 87.2o, 71.7o, 88.0o}. The ith turning angle in the sequence patches the ith leg with the (i-1)st leg, except the first turning angle patches the first and last legs. The incoming v for each of the legs is found using the outgoing v and Table 4.2. Figure 4.2 illustrates the incoming and outgoing v for each of the legs. The gravity assisted flybys rotate the incoming v on ith leg to become the outgoing v on (i+1)st leg. The minimax problem, in this example, chooses the v location of the third and fifth legs on the one-year fullrev circle such that the maximum of the required six turning angles is minimized. As is common with most minimax problems, a range of solutions may exist that have the same minimized performance index. However, the nonlinear minimax solver used for this study, VG12 from the Harwell Subroutine Library48, returns discrete minimized solutions. Thus, while the solution shown 121 in Figure 4.2 is indeed a local minimum, there may be other solutions in the same neighborhood with the same minimized maximum turning angle. Figure 4.2: Velocity diagram for the discussed example cycler 12. If the maximum turning angle is less than the maximum allowed turning angle and at least one of the legs (in the ecliptic) reaches the orbital radius of Mars, then record all of the information along with the t value from Table 4.6. For the current example, the fourth leg, or the 2.07 yr generic return, has an aphelion of 2.19 AU, well beyond the radius of Mars, and the maximum turning 122 angle is 88.7o, while the turning angle at Earth for v of 5.5 km/s and a 200 km altitude is 83.7o. Thus, the current cycler is not quite ballistic because it requires Earth flybys with altitudes below 200 km. Remember, from Table 4.6, the t value of 0.0154 yr indicates the current cycler exists only if the synodic period between Earth and Mars is 2.130 yr, which is slightly different from the more accurate value of 2.13538546 yr. 4.7.2 Post-processing After the algorithm has run for the entire range of v, a long list is compiled of the ballistic cyclers, each with their associated v, t, and other pertinent information. Table 4.7 illustrates an example list. When v is incremented by a small amount, the same cyclers will reappear with slightly different values for t. The similar cyclers are then placed into common bins as illustrated in Table 4.8. There are many ways to rearrange the original list into families of cyclers. The idea is to group cyclers that have identical sequences of free-returns, with the only difference being the slight difference in the generic return times of flight. The characterizing values for each family is an integer array of an ordered set of numbers that uniquely identify each family. The array should include the order and times of flight in half-years for each of the n returns, the order and integer valued qualities of each generic return (such as number of revolutions and if it is a fast or slow solution), which leg is used for the Earth-Mars transfer, and lastly the active column of the general 123 counting routine matrix indicating which of the solutions to use if there are multiple solutions for any of the n returns. Table 4.7: Original list of ballistic cyclers v (km/s) ... 5.32 5.32 5.32 ... 5.33 5.33 5.33 ... 5.34 5.34 5.34 ... 5.35 5.35 5.35 ... Table 4.8: Rearranged list of ballistic cyclers v (km/s) ... 5.32 5.33 5.34 5.35 5.32 5.33 5.34 5.35 5.32 5.33 5.34 5.35 ... t (yr) ... 0.020 -0.008 0.003 ... 0.022 -0.007 0.001 ... 0.024 -0.006 -0.001 ... 0.026 -0.005 -0.003 ... Cycler Family ... A B C ... A B C ... A B C ... A B C ... Characteristic Information ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... t (yr) ... 0.020 0.022 0.024 0.026 -0.008 -0.007 -0.006 -0.005 0.003 0.001 -0.001 -0.003 ... Cycler Family ... A A A A B B B B C C C C ... Characteristic Information ... ... ... ... ... ... ... ... ... ... ... ... ... ... Each of the families is then examined for a switching sign of t. If a switch occurs, the two boundary cyclers and all of the accompanying information are interpolated to a zero t. This final interpolated cycler has a period that is exactly an integer multiple of the Earth-Mars synodic period (to the accuracy of linear interpolation). From Table 4.8, cycler family C experiences a switching sign for t over the range shown from v of 5.32 to 5.35 km/s. Thus, cycler family C has a ballistic cycler for the true Earth-Mars synodic period with v = 5.335 km/s. 4.7.3 Using full-revolution returns for the Earth-Mars transfers Earth-Mars transfers must occur on legs in the ecliptic plane. Therefore, all generic returns and all n returns with longitudes of 0o or 180o are eligible to be a 124 transit leg if the aphelion is greater than the radius of Mars. The algorithm presented, however, only identifies cyclers using generic return transit legs. Half-rev returns exist in the ecliptic plane only for a handful of distinct values of v, and it is unlikely that those values also correspond exactly with generic returns that add up to an integer multiple of the synodic period. However, all full-rev returns can exist in the ecliptic plane if the longitude degree of freedom is specified as 0o or 180o. In order to find solutions with full-rev return transit legs, the existing algorithm needs a simple modification. For each full-rev return in a final sequence, check to see if the aphelion reaches Mars when constrained to be in the ecliptic. If true, constrain the degree of freedom associated with that full-rev return to be 0o and 180o respectively, then choose the remaining degrees of freedom to minimize the maximum turning angle of the sequence. Thus, the minimax problem is additionally solved twice, each time with (n-1) degrees of freedom, where n is the number of full-rev returns in the sequence. 4.8 RESULTS The described algorithm was modified to find all cyclers using generic and fullrev transit legs simultaneously. Of the hundreds of thousands of minimax problems posed and solved, Table 4.9 through Table 4.12 present all of the ballistic solutions. Again, this means that the patched trajectories encounter Mars and all required turning angles are less than that required for a 200 km altitude flyby. The solutions are valid 125 for a circular-coplanar Earth-Mars solar system with a synodic period of the true EarthMars system. Each solution is characterized by v at Earth and Mars, Earth-Mars transit times (outgoing), Mars-Earth transit times (incoming), the aphelion of the transit leg, the turn ratio or ratio of the maximum allowed to the maximum required turning angle, and a sequence of descriptor strings for each leg of the cycler. All legs reported are EarthEarth free-returns. The transit times and Mars v are calculated using the designated transit leg, as indicated by an uppercase descriptor letter. Note, for these cyclers only one Mars encounter is guaranteed during each cycler period. Therefore, a given cycler of this class must be designated as an inbound or outbound cycler. The transit aphelion is important because this must be greater than the radius of Mars in order for an encounter to be possible. A turn ratio, TR, greater than 1 indicates that all required flybys have altitudes greater than 200 km. The descriptor strings for each Earth-Earth leg originate in Ref. 25, where a standardized nomenclature for cyclers is presented. The purpose of the naming system is to provide an efficient means of describing these complicated trajectories uniquely. Given the definitions outlined in Ref. 25, and a descriptor string for each leg of an arbitrary cycler, all of its characteristics can be calculated and the entire trajectory can be systematically reproduced. The details are left out of the present discussion; however, a few items warrant mentioning. There are three types of descriptor strings, one for each of the three types 126 of free-returns. Each string type begins with a letter indicating either half-rev, full-rev, or generic return. An uppercase letter indicates that the transit times and Mars v were calculated using this leg. In all three cases, the first number in the parenthesis is the time of flight in years for that Earth-Earth leg. The number following the colon in the full-rev strings represent the number of revolutions by the spacecraft, thus the full-rev return is an M:N resonant orbit. All angled arguments are given in degrees. If more than one of the strings in a cycler description has a plus or minus value as one of its arguments, then all permutations of the plus and minus are valid. mentioned in Ref. 25 are left as normal text to ensure readability. For short hand purposes, when discussing a cycler from Table 4.9 through Table 4.13, the form vE xx...x will be used where x is the first letter of each leg descriptor. Additional information can be added if necessary to distinguish between any two cyclers. Table 4.9: Ballistic two-synodic period cyclers vE (km/s) 4.99 a 8.05 8.16 b 9.35 a a All subscripts tin aphel. tout vM TR (km/s) (days) (days) (AU) 5.10 150 150 1.64 2.65 10.02 93 93 2.19 1.45 10.06 92 92 2.20 1.08 10.52 85 85 2.21 1.70 b Leg 1 g(1.4612,526.02,Ll) g(1.4951,538.24,Ll) G(1.7708,277.48,U) G(1.7238,260.58,U) Leg 2 Leg 3 Leg 4 G(2.8096,651.46,U) G(1.7757,279.24,U) f(1:1,74.468,-180.000) f(1:1,74.244,143.198) h(0.5,0,U,+-15.756) f(1:1,74.244,-36.802) g(2.5469,916.9,L) documented in Ref. 15. documented in Chapter 3. 127 Table 4.10: Ballistic three-synodic period cyclers with generic transit legs. Part I. tout tin aphel. vM vE (km/s) (km/s) (days) (days) (AU) 3.34 3.34 3.36 3.36 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.42 3.42 3.42 3.42 3.42 3.42 3.42 3.42 3.42 3.42 3.42 3.42 3.42 3.42 3.45 3.64 3.64 3.77 3.77 a a a TR 1.24 1.16 1.11 1.26 1.42 1.42 1.22 1.22 1.22 1.22 1.22 1.22 1.22 1.22 1.22 1.22 1.22 1.22 1.41 1.27 1.22 1.22 1.22 1.22 1.22 1.22 1.22 1.22 1.22 1.22 1.22 1.22 1.22 1.62 1.18 1.76 1.63 1.63 1.16 1.16 1.16 1.16 1.16 1.16 1.16 1.16 1.16 1.16 1.16 1.16 1.16 1.81 1.16 1.45 1.16 1.03 1.03 1.16 1.09 1.09 1.17 Leg 1 g(2.4096,867.45,Ls) g(2.4096,867.45,Ls) G(2.9808,713.1,U) g(1.4433,519.59,Ll) g(1.4438,519.75,Ll) g(1.4438,519.75,Ll) g(1.4438,519.75,Ll) g(1.4438,519.75,Ll) g(1.4438,519.75,Ll) g(1.4438,519.75,Ll) g(1.4438,519.75,Ll) g(1.4438,519.75,Ll) g(1.4438,519.75,Ll) g(1.4438,519.75,Ll) g(1.4438,519.75,Ll) g(1.4438,519.75,Ll) g(1.4438,519.75,Ll) g(1.4438,519.75,Ll) g(1.37,493.2,Ls) g(1.37,493.2,Ls) g(1.37,493.2,Ls) g(1.37,493.2,Ls) g(1.37,493.2,Ls) g(1.37,493.2,Ls) g(1.37,493.2,Ls) g(1.37,493.2,Ls) g(1.37,493.2,Ls) g(1.37,493.2,Ls) g(1.37,493.2,Ls) g(1.37,493.2,Ls) g(1.37,493.2,Ls) g(1.37,493.2,Ls) G(2.9531,703.11,U) g(2.4845,894.42,Ll) g(2.4845,894.42,Ll) G(2.9062,686.21,U) G(2.9062,686.21,U) G(2.9062,686.21,U) G(2.9062,686.21,U) G(2.9062,686.21,U) G(2.9062,686.21,U) G(2.9062,686.21,U) G(2.9062,686.21,U) G(2.9062,686.21,U) G(2.9062,686.21,U) G(2.9062,686.21,U) G(2.9062,686.21,U) G(2.9062,686.21,U) G(2.9062,686.21,U) G(2.9062,686.21,U) G(5.9062,1406.22,U) G(2.9043,685.56,U) g(1.3659,491.72,Ls) g(1.4519,522.68,Ll) g(1.3622,490.38,Ls) G(4.9062,1046.22,U) g(1.4626,526.55,Ll) G(1.9531,343.11,U) G(1.9531,343.11,U) G(1.9531,343.11,U) G(1.9522,342.8,U) Leg 2 h(0.5,0,Ls,+-6.428) G(2.9966,718.76,U) g(3.4253,1233.11,Ls) h(0.5,0,Ll,+-6.477) f(1:1,83.444,0.000) G(2.9624,706.46,U) G(2.9624,706.46,U) G(2.9624,706.46,U) G(2.9624,706.46,U) G(2.9624,706.46,U) h(0.5,0,Ll,-6.556) h(0.5,0,Ll,-6.556) h(0.5,0,Ll,6.556) h(0.5,0,Ll,6.556) h(0.5,0,Ll,-6.556) h(0.5,0,Ll,-6.556) h(0.5,0,Ll,6.556) h(0.5,0,Ll,6.556) f(1:1,83.421,179.999) G(3.0362,733.02,L) G(3.0362,733.02,L) G(3.0362,733.02,L) G(3.0362,733.02,L) G(3.0362,733.02,L) h(0.5,0,Ls,-6.579) h(0.5,0,Ls,-6.579) h(0.5,0,Ls,6.579) h(0.5,0,Ls,6.579) h(0.5,0,Ls,-6.579) h(0.5,0,Ls,-6.579) h(0.5,0,Ls,6.579) h(0.5,0,Ls,6.579) g(2.9531,703.11,U) G(2.9217,691.79,U) h(0.5,0,Ll,+-7.005) h(3.5,3,U,+-2.022) f(1:1,82.747,156.413) f(1:1,82.747,-156.413) h(0.5,0,U,-7.253) h(0.5,0,U,7.253) h(0.5,0,U,-7.253) h(0.5,0,U,7.253) h(0.5,0,U,-7.253) h(0.5,0,U,-7.253) h(0.5,0,U,-7.253) h(0.5,0,U,-7.253) h(0.5,0,U,7.253) h(0.5,0,U,7.253) h(0.5,0,U,7.253) h(0.5,0,U,7.253) h(0.5,0,U,+-7.278) g(3.5018,1260.65,L) g(1.4481,521.32,Ll) G(4.9543,1063.54,U) G(5.044,1095.84,L) h(0.5,0,U,+-8.731) G(1.9718,349.83,U) g(1.9531,343.11,U) g(1.9531,343.11,U) g(1.9531,343.11,U) g(1.9522,342.8,U) Leg 3 G(2.9966,718.76,U) f(1:1,83.572,1.223) g(1.4433,519.59,Ll) G(2.9624,706.46,U) f(1:1,83.444,179.992) h(0.5,0,U,-6.556) h(0.5,0,U,-6.556) h(0.5,0,U,6.556) h(0.5,0,U,6.556) G(2.9624,706.46,U) G(2.9624,706.46,U) G(2.9624,706.46,U) G(2.9624,706.46,U) G(2.9624,706.46,U) G(2.9624,706.46,U) G(2.9624,706.46,U) G(2.9624,706.46,U) G(3.0362,733.02,L) f(1:1,83.421,3.535) h(0.5,0,L,-6.579) h(0.5,0,L,-6.579) h(0.5,0,L,6.579) h(0.5,0,L,6.579) G(3.0362,733.02,L) G(3.0362,733.02,L) G(3.0362,733.02,L) G(3.0362,733.02,L) G(3.0362,733.02,L) G(3.0362,733.02,L) G(3.0362,733.02,L) G(3.0362,733.02,L) h(0.5,0,U,+-6.631) f(1:1,82.995,-180.000) G(2.9217,691.79,U) h(0.5,0,U,-7.253) h(0.5,0,U,7.253) f(3:2,86.818,-175.066) f(3:2,86.818,0.461) f(1:1,82.747,-174.308) f(1:1,82.747,0.456) f(1:1,82.747,2.010) f(1:1,82.747,0.004) f(1:1,82.747,-174.404) f(1:1,82.747,-173.959) f(1:1,82.747,3.410) f(1:1,82.747,4.371) f(1:1,82.747,1.674) f(1:1,82.747,176.621) Leg 4 h(0.5,0,U,+-6.428) Leg 5 Leg 6 4.53 4.53 4.52 4.56 4.53 4.53 4.53 4.53 4.53 4.53 4.53 4.53 4.53 4.53 4.53 4.53 4.53 4.53 4.59 4.59 4.59 4.59 4.59 4.59 4.59 4.59 4.59 4.59 4.59 4.59 4.59 4.59 4.54 4.59 4.59 4.63 4.63 4.63 4.63 4.63 4.63 4.63 4.63 4.63 4.63 4.63 4.63 4.63 4.63 4.63 4.67 4.63 4.80 6.71 6.74 6.82 9.14 9.15 9.15 9.15 9.16 188 188 185 191 182 182 182 182 182 182 182 182 182 182 182 182 182 182 193 193 193 193 193 193 193 193 193 193 193 193 193 193 180 175 175 171 171 171 171 171 171 171 171 171 171 171 171 171 171 171 171 171 199 147 161 138 123 120 120 120 120 188 188 185 191 182 182 182 182 182 182 182 182 182 182 182 182 182 182 193 193 193 193 193 193 193 193 193 193 193 193 193 193 180 175 175 171 171 171 171 171 171 171 171 171 171 171 171 171 171 171 171 171 199 147 161 138 123 120 120 120 120 1.62 1.62 1.62 1.62 1.62 1.62 1.62 1.62 1.62 1.62 1.62 1.62 1.62 1.62 1.62 1.62 1.62 1.62 1.62 1.62 1.62 1.62 1.62 1.62 1.62 1.62 1.62 1.62 1.62 1.62 1.62 1.62 1.62 1.62 1.62 1.62 1.62 1.62 1.62 1.62 1.62 1.62 1.62 1.62 1.62 1.62 1.62 1.62 1.62 1.62 1.63 1.62 1.63 1.81 1.82 1.82 2.17 2.17 2.17 2.17 2.17 G(3.0195,727.04,L) f(1:1,83.444,180.000) f(1:1,83.444,103.758) f(1:1,83.444,-1.701) f(1:1,83.444,-173.985) f(1:1,83.444,1.379) f(1:1,83.444,2.175) h(0.5,0,U,-6.556) h(0.5,0,U,6.556) h(0.5,0,U,-6.556) h(0.5,0,U,6.556) f(1:1,83.444,98.343) f(1:1,83.444,-175.616) f(1:1,83.444,90.849) f(1:1,83.444,-175.770) f(1:1,83.421,-0.003) f(1:1,83.421,1.682) f(1:1,83.421,176.774) f(1:1,83.421,2.456) f(1:1,83.421,4.542) f(1:1,83.421,1.657) h(0.5,0,L,-6.579) h(0.5,0,L,6.579) h(0.5,0,L,-6.579) h(0.5,0,L,6.579) f(1:1,83.421,4.744) f(1:1,83.421,-87.155) f(1:1,83.421,4.125) f(1:1,83.421,-88.190) h(0.5,0,L,-6.579) h(0.5,0,L,6.579) h(0.5,0,L,-6.579) h(0.5,0,L,6.579) f(1:1,83.421,0.808) f(1:1,83.421,4.560) f(1:1,83.421,-88.182) f(1:1,83.421,2.896) h(0.5,0,L,-6.579) h(0.5,0,L,6.579) h(0.5,0,L,-6.579) h(0.5,0,L,6.579) h(0.5,0,U,-6.556) h(0.5,0,U,6.556) h(0.5,0,U,-6.556) h(0.5,0,U,6.556) f(1:1,83.444,-177.279) f(1:1,83.444,88.162) f(1:1,83.444,-174.467) f(1:1,83.444,96.297) h(0.5,0,U,-6.556) h(0.5,0,U,6.556) h(0.5,0,U,-6.556) h(0.5,0,U,6.556) h(0.5,0,U,+-7.005) f(1:1,82.747,-90.000) f(1:1,82.747,23.587) f(1:1,82.747,-23.587) f(1:1,82.747,0.704) 3.77 3.77 3.77 3.77 3.77 3.77 3.77 3.77 3.77 3.77 3.77 3.77 3.77 a a a a a a a a a a a f(1:1,82.747,100.227) f(1:1,82.747,3.956) h(0.5,0,U,-7.253) h(0.5,0,U,-7.253) h(0.5,0,U,7.253) h(0.5,0,U,7.253) h(0.5,0,U,-7.253) h(0.5,0,U,-7.253) h(0.5,0,U,7.253) h(0.5,0,U,7.253) f(1:1,82.747,11.426) f(1:1,82.747,2.974) f(1:1,82.747,176.668) f(1:1,82.747,176.800) f(1:1,82.747,1.042) f(1:1,82.747,1.221) f(1:1,82.747,1.674) f(1:1,82.747,2.030) f(1:1,82.747,3.117) f(1:1,82.747,2.992) h(0.5,0,U,-7.253) h(0.5,0,U,7.253) h(0.5,0,U,-7.253) h(0.5,0,U,7.253) h(0.5,0,U,-7.253) h(0.5,0,U,7.253) h(0.5,0,U,-7.253) h(0.5,0,U,7.253) 3.78 3.78 3.80 4.15 4.15 4.53 5.13 5.22 5.22 h(0.5,0,Ll,+-7.322) G(3.0922,753.18,L) f(1:1,81.269,2.243) g(1.9718,349.83,U) h(2.5,2,U,+-2.359) f(1:1,79.947,170.217) f(1:1,79.947,-170.217) g(2.5017,900.61,L) h(0.5,0,U,-10.053) h(0.5,0,U,10.053) f(1:1,79.947,-9.783) f(1:1,79.947,9.783) f(1:1,80.129,179.998) a a 5.22 5.22 a documented in Chapter 3 . 128 Table 4.11: Ballistic three-synodic period cyclers with generic transit legs. Part II. tout tin aphel. TR vM vE (km/s) (km/s) (days) (days) (AU) 5.25 5.25 5.30 5.30 5.30 5.30 5.30 5.30 5.30 5.31 5.32 5.33 5.33 5.33 5.33 5.33 5.33 5.33 5.66 5.66 5.66 5.66 5.66 5.66 a a Leg 1 G(1.9491,341.67,U) G(1.9491,341.67,U) g(1.4646,527.25,Ll) g(1.4646,527.25,Ll) g(1.4646,527.25,Ll) g(1.4646,527.25,Ll) g(1.4646,527.25,Ll) g(1.4646,527.25,Ll) g(1.4646,527.25,Ll) g(1.4647,527.3,Ll) g(1.3497,485.89,Ls) g(1.3496,485.84,Ls) g(1.3496,485.84,Ls) g(1.3496,485.84,Ls) g(1.3496,485.84,Ls) g(1.3496,485.84,Ls) g(1.3496,485.84,Ls) g(1.3496,485.84,Ls) G(1.9062,326.21,U) G(1.9062,326.21,U) G(1.9062,326.21,U) G(1.9062,326.21,U) G(1.9062,326.21,U) G(1.9062,326.21,U) G(1.9062,326.21,U) G(1.9062,326.21,U) G(1.9062,326.21,U) G(1.9062,326.21,U) G(1.9062,326.21,U) G(1.9062,326.21,U) G(1.9062,326.21,U) G(1.9062,326.21,U) G(1.9062,326.21,U) G(1.9062,326.21,U) G(1.9062,326.21,U) G(1.9062,326.21,U) G(1.9062,326.21,U) G(1.9062,326.21,U) G(1.9062,326.21,U) G(1.9062,326.21,U) G(1.9062,326.21,U) G(1.9062,326.21,U) G(3.9062,686.21,U) G(3.9062,686.21,U) G(3.9062,686.21,U) G(1.8987,323.54,U) G(1.8987,323.54,U) G(1.8987,323.54,U) g(2.5077,902.78,L) G(1.8803,316.9,U) G(1.8699,313.18,U) g(3.1633,1498.78,L) g(1.4788,532.39,Ll) g(2.7886,1363.89,U) G(2.6667,600.01,U) G(2.8508,306.27,U) G(2.6347,588.48,U) g(1.8367,1021.21,U) g(2.1398,1130.33,L) G(1.7031,253.11,U) G(1.7025,252.9,U) Leg 2 f(1:1,79.896,135.679) f(1:1,79.896,127.982) G(1.9416,338.97,U) f(2:1,87.296,-4.285) G(1.9416,338.97,U) f(1:1,79.79,-0.014) G(1.9416,338.97,U) G(1.9416,338.97,U) G(1.9416,338.97,U) G(3.9414,698.92,U) G(4.0565,740.32,L) f(1:1,79.727,180.000) f(2:1,87.1,-180.000) G(2.0566,380.38,L) G(2.0566,380.38,L) G(2.0566,380.38,L) G(2.0566,380.38,L) G(2.0566,380.38,L) f(2:1,85.562,138.594) f(1:1,79.1,180.000) f(1:1,79.1,180.000) f(1:1,79.1,-179.999) f(1:1,79.1,180.000) f(1:1,79.1,-180.000) f(1:1,79.1,179.998) f(2:1,85.562,90.000) f(2:1,85.562,-90.000) f(2:1,85.562,90.000) f(2:1,85.562,-90.000) f(3:2,81.663,77.842) f(3:2,81.663,-149.095) f(1:1,79.1,158.489) f(1:1,79.1,158.489) f(1:1,79.1,158.489) f(1:1,79.1,-158.489) f(1:1,79.1,-158.489) f(1:1,79.1,-158.489) f(1:1,79.1,160.993) f(1:1,79.1,-158.808) f(1:1,79.1,137.943) f(1:1,79.1,155.720) h(4.5,4,U,+-9.414) h(2.5,2,U,+-4.858) f(1:1,79.059,158.513) f(1:1,79.059,-158.513) g(2.5074,902.67,L) g(2.5074,902.67,L) f(1:1,78.919,180.000) G(3.8984,683.44,U) g(3.5259,1269.32,L) g(4.5362,1633.03,L) G(3.2429,807.44,L) G(4.9273,693.83,U) G(3.6176,942.33,U) g(3.7395,1706.21,U) g(3.5554,1279.94,L) g(2.7715,1357.74,U) G(3.5695,925.01,U) f(1:1,71.455,180.000) h(1.5,1,U,+-9.517) g(4.7037,2053.31,Ls) Leg 3 g(1.9571,1064.54,U) g(1.9571,1064.54,U) f(3:2,82.487,118.851) G(1.9416,338.97,U) f(1:1,79.79,180.000) G(1.9416,338.97,U) f(1:1,79.79,166.207) f(2:1,87.296,108.211) f(1:1,79.79,163.866) f(1:1,79.764,180.000) f(1:1,79.751,1.228) G(2.0566,380.38,L) G(2.0566,380.38,L) f(3:2,82.41,2.487) f(1:1,79.727,1.173) f(1:1,79.727,2.542) f(2:1,87.1,1.573) f(1:1,79.727,0.488) h(2.5,2,U,+-4.768) f(1:1,79.1,113.787) f(1:1,79.1,144.338) h(2.5,2,U,-4.768) h(2.5,2,U,4.768) f(1:1,79.1,114.484) f(1:1,79.1,-150.526) h(0.5,0,U,-10.900) h(0.5,0,U,10.900) h(0.5,0,U,-10.900) h(0.5,0,U,10.900) h(0.5,0,U,-10.900) h(0.5,0,U,10.900) h(0.5,0,U,-10.900) h(0.5,0,U,-10.900) h(0.5,0,U,-10.900) h(0.5,0,U,10.900) h(0.5,0,U,10.900) h(0.5,0,U,10.900) h(0.5,0,U,-10.900) h(0.5,0,U,10.900) h(3.5,3,U,-8.316) h(3.5,3,U,8.316) Leg 4 h(0.5,0,U,-10.104) h(0.5,0,U,10.104) f(1:1,79.79,-180.000) f(1:1,79.79,114.028) f(1:1,79.79,179.973) f(2:1,87.296,166.206) f(1:1,79.79,108.210) f(2:3,86.601,-180.000) Leg 5 f(1:1,79.896,-52.281) f(1:1,79.896,11.936) Leg 6 9.16 9.16 9.17 9.17 9.17 9.17 9.17 9.17 9.17 9.20 9.24 9.26 9.26 9.26 9.26 9.26 9.26 9.26 9.26 9.26 9.26 9.26 9.26 9.26 9.26 9.26 9.26 9.26 9.26 9.26 9.26 9.26 9.26 9.26 9.26 9.26 9.26 9.26 9.26 9.26 9.26 9.26 9.30 9.30 9.30 9.28 9.28 9.28 9.33 9.35 9.40 6.15 11.35 3.78 6.57 13.07 7.04 5.01 11.15 10.75 10.76 120 120 118 118 118 118 118 118 118 118 136 135 135 135 135 135 135 135 113 113 113 113 113 113 113 113 113 113 113 113 113 113 113 113 113 113 113 113 113 113 113 113 112 112 112 112 112 112 111 109 107 203 100 166 115 81 108 131 156 83 82 120 120 118 118 118 118 118 118 118 118 136 135 135 135 135 135 135 135 113 113 113 113 113 113 113 113 113 113 113 113 113 113 113 113 113 113 113 113 113 113 113 113 112 112 112 112 112 112 111 109 107 203 100 166 115 81 108 131 156 83 82 b 2.17 2.17 2.17 2.17 2.17 2.17 2.17 2.17 2.17 2.17 2.18 2.19 2.19 2.19 2.19 2.19 2.19 2.19 2.17 2.17 2.17 2.17 2.17 2.17 2.17 2.17 2.17 2.17 2.17 2.17 2.17 2.17 2.17 2.17 2.17 2.17 2.17 2.17 2.17 2.17 2.17 2.17 2.18 2.18 2.18 2.17 2.17 2.17 2.18 2.17 2.17 1.71 2.69 1.53 1.70 3.17 1.72 1.57 2.34 2.22 2.22 1.02 1.02 1.27 1.13 1.13 1.13 1.12 1.12 1.12 1.13 1.00 1.12 1.01 1.00 1.00 1.00 1.00 1.00 1.26 1.24 1.24 1.24 1.24 1.24 1.24 1.22 1.22 1.22 1.22 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.16 1.16 1.10 1.10 1.03 1.25 1.20 1.20 1.34 1.26 1.26 1.36 1.30 1.39 1.10 1.10 1.48 1.45 1.01 1.27 1.13 1.34 1.16 1.10 f(1:1,79.79,104.278) f(1:1,79.79,105.098) f(1:1,79.727,0.000) f(1:1,79.727,2.763) f(2:1,87.1,4.041) f(2:3,86.381,4.726) f(1:1,79.727,2.781) f(1:1,79.727,3.757) h(2.5,2,U,-4.768) h(2.5,2,U,4.768) f(1:1,79.1,0.002) f(1:1,79.1,0.001) h(0.5,0,U,-10.900) h(0.5,0,U,10.900) f(2:1,85.562,-90.000) f(2:1,85.562,90.000) f(1:1,79.1,-78.303) f(1:1,79.1,32.035) f(1:1,79.1,-21.511) f(1:1,79.1,21.511) f(1:1,79.1,-158.489) f(2:3,84.614,-55.762) f(1:1,79.1,-145.301) f(1:1,79.1,49.280) f(1:1,79.1,23.051) f(2:3,84.614,64.883) f(2:1,85.562,-61.041) f(2:1,85.562,61.041) f(1:1,79.727,-0.456) f(1:1,79.727,2.288) f(1:1,79.1,-63.066) f(1:1,79.1,27.556) f(1:1,79.1,0.000) f(1:1,79.1,-0.003) 5.66 5.66 5.66 5.66 5.66 f(1:1,79.1,-13.387) f(1:1,79.1,0.743) 5.66 5.66 5.66 5.66 5.66 a a f(2:1,85.562,-166.157) f(1:1,79.1,-16.932) f(1:1,79.1,-89.902) f(1:1,79.1,11.318) f(2:1,85.562,12.898) f(1:1,79.1,20.291) f(1:1,79.1,-32.081) f(1:1,79.1,32.081) f(1:1,79.1,-21.511) f(1:1,79.1,10.189) 5.66 5.66 5.66 5.66 5.66 5.66 5.66 5.66 5.68 5.68 b b a a h(0.5,0,U,-10.941) h(0.5,0,U,10.941) f(2:1,85.196,0.005) f(1:1,78.919,-0.888) g(2.5074,902.67,L) f(1:1,78.419,-0.011) f(1:1,79.059,-21.487) f(1:1,79.059,21.487) f(1:1,78.919,-0.002) f(1:1,78.919,0.000) 5.68 5.75 5.75 5.75 5.78 6.01 6.17 6.21 6.59 7.56 7.95 8.72 8.82 8.99 9.60 9.92 9.94 a f(1:1,72.976,0.238) f(1:1,72.638,144.363) G(2.2664,455.89,L) g(1.7031,253.11,U) f(1:1,71.455,0.000) h(1.5,1,U,+-9.517) documented in Chapter 3 documented in Section 4.3. 129 As discussed in the Solution Method section, the number of potential free-return combinations grows rapidly as the total period of the cycler increases. Thus, as expected, there are zero, a few, and hundreds of ballistic one-, two-, and three-synodic period cyclers found respectively. In theory, the process can be repeated for total periods of four or more synodic periods. However, the number of permutations is computationally prohibitive, and furthermore, the solutions are not as attractive as the lower period cyclers because the frequency of a guaranteed Mars encounter is too low. Table 4.9 lists the ballistic two-synodic period cyclers. Of the five, two were documented previously in Ref. 15 and two were documented in Chapter 3. Note the numbers may differ slightly because the synodic period in most previous studies was approximated to be 15/7 years.3,13,17,16,19,20 The new cycler, 8.05gGf, is composed of two generic returns and one full-rev return. The relatively high v at Earth and Mars is compensated by the quick transfer time of 93 days. The first entry, 4.99gG, is the notable `S1L1' cycler with favorable energy characteristics discovered first by McConaghy et al. in Ref. 15. Table 4.10 and Table 4.11 list all of the ballistic three-synodic period cyclers with generic transit legs. In total there are 141 listed. The plus or minus arguments in the descriptor strings are only applied when nothing else in any of the strings changes except that value. The plus minus notation can not be used in many cases, and thus, there are many cyclers that appear to be duplicates because they have identical v, time, and turn angle characteristics. However, upon close inspection, a small difference is 130 seen, such as the last argument in one of the full-rev strings. These subtle differences may be important when looking for analogous solutions in a more realistic solar system. The majority of the cyclers listed in Table 4.10 and Table 4.11 are previously undocumented. Twenty-four of the cyclers were found in Chapter 3 and four were found in Section 4.3. Several of the most promising solutions are bolded. With regard to eventually finding solutions in the real solar system, cyclers with fewer legs are generally more favorable because there are fewer constraints in the problem formulation. Thus, all of the bold cyclers have two or three legs only. The first six bold cyclers from Table 4.10 have remarkable low values of v at both Earth and Mars. Cyclers 3.64gGg, 3.77Gh, and 3.78Gg, in particular, have favorable turning angle characteristics. Cycler 3.77Gh is an example that uses a half-rev return that is not a simple half-year transfer. The first argument of the second string is a 3.5 indicating it is a 7 half-year transfer. Other bold promising cyclers that use non-trivial half-rev returns are 5.22Ggh and 5.66Gfh. The bold cycler 3.77Ghf is an example that has good energy characteristics and uses a full-rev return that is not simply a 1:1 resonant orbit. Cycler 5.30gGf uses a 3:2 resonant orbit and has short transit times with relatively good turning angles. Cycler 5.66Gfh uses a 2:1 resonant orbit, and also has short transit times and even better turning angles. Cycler 9.94Gg has extremely short transit times of 82 days and is just two legs. Of course, in general, the transfer times are inversely related to the v values. 131 Table 4.12: Ballistic three-synodic period cyclers with full-rev transit legs. vE (km/s) 3.66 3.66 3.66 3.77 3.77 5.30 5.30 5.30 5.30 5.33 5.33 5.33 5.33 5.66 5.66 5.66 5.66 5.66 5.75 tin in/out tout vM (km/s) (days) (days) 4.66 173 199 out 4.66 173 199 out 4.66 199 173 in 4.71 4.71 5.44 9.21 9.21 9.21 5.45 9.22 9.22 9.22 9.32 9.32 9.32 9.32 9.32 9.36 170 200 207 118 136 136 143 136 118 118 141 141 113 141 113 111 200 170 143 136 118 118 207 118 136 136 113 113 141 113 141 142 out in in out in in out in out out in in out in out out aphel. TR (AU) 1.63 1.63 1.63 1.63 1.63 1.66 2.18 2.18 2.18 1.66 2.18 2.18 2.18 2.19 2.19 2.19 2.19 2.19 2.19 Leg 1 Leg 2 Leg 3 F(3:2,87.27,0.000) F(3:2,87.27,0.000) F(3:2,87.27,180.000) F(3:2,86.818,0.000) F(3:2,86.818,180.000) F(3:2,82.487,180.000) g(1.9416,338.97,U) f(1:1,79.79,180.000) F(2:1,87.296,180.000) F(3:2,82.41,0.000) g(2.0566,380.38,L) f(1:1,79.727,3.079) F(2:1,87.1,0.000) h(2.5,2,U,+-4.768) h(0.5,0,U,-10.900) h(0.5,0,U,10.900) h(0.5,0,U,10.900) h(0.5,0,U,-10.900) F(2:1,85.196,0.000) f(1:1,79.79,-180.000) F(2:1,87.296,180.000) f(1:1,79.79,-180.000) f(1:1,79.727,2.509) F(2:1,87.1,0.000) f(1:1,79.727,4.139) f(1:1,79.1,-159.239) f(1:1,79.1,20.761) f(1:1,79.1,159.239) f(1:1,79.1,-20.761) F(2:1,85.562,180.000) F(2:1,85.562,0.000) F(2:1,85.562,180.000) F(2:1,85.562,0.000) Leg 4 Leg 5 1.19 g(2.4062,866.21,Ls) f(1:1,82.955,87.388) 1.18 g(2.4062,866.21,Ls) h(0.5,0,Ls,+-7.045) 1.18 g(2.4062,866.21,Ls) h(0.5,0,Ls,+-7.045) 1.16 g(2.9062,686.21,U) 1.16 g(2.9062,686.21,U) 1.27 1.13 1.12 1.12 1.00 1.01 1.00 1.00 1.24 1.19 1.19 1.19 1.19 g(1.4646,527.25,Ll) g(1.4646,527.25,Ll) g(1.4646,527.25,Ll) g(1.4646,527.25,Ll) g(1.3496,485.84,Ls) g(1.3496,485.84,Ls) g(1.3496,485.84,Ls) g(1.3496,485.84,Ls) g(1.9062,326.21,U) g(1.9062,326.21,U) g(1.9062,326.21,U) g(1.9062,326.21,U) g(1.9062,326.21,U) h(0.5,0,U,+-7.253) h(0.5,0,U,+-7.253) g(1.9416,338.97,U) F(2:1,87.296,0.000) g(1.9416,338.97,U) g(1.9416,338.97,U) g(2.0566,380.38,L) F(2:1,87.1,180.000) g(2.0566,380.38,L) g(2.0566,380.38,L) F(2:1,85.562,180.000) f(1:1,79.1,159.239) f(1:1,79.1,-159.239) f(1:1,79.1,-159.239) f(1:1,79.1,154.905) g(2.5074,902.67,L) h(0.5,0,Ls,+-7.045) h(0.5,0,Ls,+-7.045) 1.34 g(1.8987,323.54,U) Table 4.12 lists the twenty-eight ballistic cyclers found that have full-rev transit legs. Only three-synodic period cyclers were found. The full-rev transit leg is indicated by the uppercase "F" at the beginning of one of the descriptor strings. Unlike cyclers with generic transit legs, those listed in Table 4.12 have differing inbound and outbound transfer times. The symmetry associated with generic returns (also known as symmetric returns7) causes both transfer times on a generic transit to be equal. In the case of a fullrev return, Earth arrival and departure locations on the orbit are identical, thus the symmetry is lost, and the inbound and outbound times are no longer the same. Therefore, the inbound/outbound designation is not arbitrary for cyclers with full-rev transit legs. There is a clear advantage to designate them as one or the other depending on which transfer time is the shortest. An additional column is added in Table 4.12 to make this preferred designation. 132 Again, the cyclers in Table 4.12 with the more favorable characteristics are bolded. Cyclers 3.66gfF and 3.77ghF both have very low v at Earth and Mars with reasonable transit times and turning angles. Cycler 5.30ggF also has good energy characteristics, turning angles, and an excellent inbound transit time. Note the 64 day difference in the outbound and inbound times. Cycler 5.75ggF has a short transfer time and relatively good turning angles. Table 4.13 results from reducing the minimum turn ratio from 1.00, the requirement for a ballistic cycler, to 0.95. It finds several cyclers with a variety of characteristics. Cycler 6.44Gg has a very long transit time resulting in a low v at Mars. Table 4.14 gives a summary of the numbers and types of cyclers documented. If the turn ratio minimum is reduced further, the famous one-synodic period Aldrin cycler13 is found with vE = 6.40 km/s, vM = 9.68 km/s, tin = tout = 145 days, TR=0.91, aphelion=2.22 AU, and the only descriptor string is G(2.1354, 7.1338,L). It is noteworthy that the Aldrin cycler is more favorable with the realistic synodic period than the assumed 15/7 years from Chapter 3. Using the true synodic period the turn ratio for the Aldrin cycler is 0.91. However, if the approximated synodic period is used, the turn ratio is 0.86. 133 Table 4.13: Near-ballistic three-synodic period cyclers tin aphel. tout vE vM (km/s) (km/s) (days) (days) (AU) 5.10 5.10 5.10 5.10 5.30 5.30 5.30 5.30 5.30 5.30 5.30 5.30 5.30 5.30 5.33 5.33 5.33 5.33 5.33 5.33 5.33 5.33 5.33 6.44 7.51 7.51 9.14 9.14 9.14 9.16 9.17 9.17 9.17 9.17 9.17 9.17 9.17 9.17 9.17 9.17 9.18 9.26 9.26 9.26 9.26 9.26 9.26 9.26 9.26 3.74 12.70 10.00 124 124 124 129 118 118 118 118 118 118 118 118 118 118 118 135 135 135 135 135 135 135 135 262 90 153 124 124 124 129 118 118 118 118 118 118 118 118 118 118 118 135 135 135 135 135 135 135 135 262 90 96 2.17 2.17 2.17 2.18 2.17 2.17 2.17 2.17 2.17 2.17 2.17 2.17 2.17 2.17 2.17 2.19 2.19 2.19 2.19 2.19 2.19 2.19 2.19 1.54 3.16 2.23 TR 0.98 0.98 0.98 0.99 0.95 0.95 0.95 0.95 0.95 0.95 0.95 0.95 0.95 0.95 0.95 0.97 0.97 0.97 0.97 0.97 0.97 0.96 0.96 0.95 Leg 1 Leg 2 Leg 3 g(1.4623,526.43,Ll) f(1:1,80.186,-175.289) f(1:1,80.186,96.849) g(2.3908,860.68,U) h(2.5,2,U,-2.942) h(2.5,2,U,2.942) f(1:1,79.79,106.484) f(1:1,79.79,109.822) f(1:1,79.79,-174.412) f(1:1,79.79,179.385) f(1:1,79.79,-71.397) f(1:1,79.79,-70.024) f(1:1,79.79,3.423) f(1:1,79.79,2.554) g(2.5028,901,L) f(1:1,79.727,1.353) f(1:1,79.727,-69.877) f(1:1,79.727,-173.216) f(1:1,79.727,-177.039) f(1:1,79.727,116.598) f(1:1,79.727,116.040) f(1:1,79.727,-72.833) f(1:1,79.727,0.000) Leg 4 Leg 5 Leg 6 g(1.4623,526.43,Ll) h(0.5,0,Ll,+-9.814) g(1.4623,526.43,Ll) g(1.4623,526.43,Ll) G(2.0154,365.54,L) g(1.4646,527.25,Ll) g(1.4646,527.25,Ll) g(1.4646,527.25,Ll) g(1.4646,527.25,Ll) g(1.4646,527.25,Ll) g(1.4646,527.25,Ll) g(1.4646,527.25,Ll) g(1.4646,527.25,Ll) g(1.4646,527.25,Ll) g(1.4646,527.25,Ll) g(1.4648,527.34,Ll) g(1.3496,485.84,Ls) g(1.3496,485.84,Ls) g(1.3496,485.84,Ls) g(1.3496,485.84,Ls) g(1.3496,485.84,Ls) g(1.3496,485.84,Ls) g(1.3496,485.84,Ls) g(1.3496,485.84,Ls) h(0.5,0,Ll,-9.814) h(0.5,0,Ll,9.814) f(1:1,80.175,0.006) G(1.9416,338.97,U) G(1.9416,338.97,U) G(1.9416,338.97,U) G(1.9416,338.97,U) G(1.9416,338.97,U) G(1.9416,338.97,U) h(0.5,0,Ll,-10.210) h(0.5,0,Ll,-10.210) h(0.5,0,Ll,10.210) h(0.5,0,Ll,10.210) G(1.9385,337.87,U) G(2.0566,380.38,L) G(2.0566,380.38,L) h(0.5,0,Ls,-10.273) h(0.5,0,Ls,-10.273) h(0.5,0,Ls,10.273) h(0.5,0,Ls,10.273) G(2.0566,380.38,L) G(2.0566,380.38,L) G(1.9815,353.35,U) f(1:1,80.186,146.62) g(1.4623,526.43,Ll) g(1.4623,526.43,Ll) f(1:1,80.175,-180.0) h(0.5,0,U,10.210) h(0.5,0,U,-10.210) h(0.5,0,U,-10.210) h(0.5,0,U,-10.210) h(0.5,0,U,10.210) h(0.5,0,U,10.210) G(1.9416,338.97,U) G(1.9416,338.97,U) G(1.9416,338.97,U) G(1.9416,338.97,U) h(0.5,0,L,+-10.258) h(0.5,0,L,-10.273) h(0.5,0,L,10.273) G(2.0566,380.38,L) G(2.0566,380.38,L) G(2.0566,380.38,L) G(2.0566,380.38,L) h(0.5,0,L,10.273) h(0.5,0,L,-10.273) G(1.9815,353.35,U) G(1.9815,353.35,U) f(1:1,79.79,0.000) f(1:1,79.79,-175.353) f(1:1,79.79,4.570) f(1:1,79.79,0.000) f(1:1,79.79,109.484) f(1:1,79.79,-176.752) f(1:1,79.79,106.448) f(1:1,79.79,-177.304) f(1:1,79.727,-173.55) f(1:1,79.727,3.464) f(1:1,79.727,1.372) f(1:1,79.727,-61.225) f(1:1,79.727,2.096) f(1:1,79.727,-62.018) f(1:1,79.727,0.000) f(1:1,79.727,0.000) h(0.5,0,U,-10.210) h(0.5,0,U,10.210) h(0.5,0,U,-10.210) h(0.5,0,U,10.210) h(0.5,0,U,-10.210) h(0.5,0,U,10.210) h(0.5,0,U,-10.210) h(0.5,0,U,10.210) h(0.5,0,L,10.273) h(0.5,0,L,-10.273) h(0.5,0,L,-10.273) h(0.5,0,L,10.273) h(0.5,0,L,-10.273) h(0.5,0,L,10.273) h(0.5,0,L,10.273) h(0.5,0,L,-10.273) g(2.087,1111.33,L) G(4.3191,1194.88,L) g(1.4891,536.07,Ll) G(2.9171,330.15,U) f(2:1,80.151,129.048) 0.99 g(1.4891,536.07,Ll) g(2.9171,330.15,U) F(2:1,80.151,180.000) 0.99 Table 4.14: Summary of results Total cycler period (synodic periods) 1 2 3 1 2 3 1 2 3 3 Type of transit leg generic generic generic full-rev full-rev full-rev generic generic generic full-rev Ballistic/ Number of cyclers found Number of cyclers found near ballistic (previously documented) (previously undocumented) ballistic 0 0 ballistic 4 1 ballistic 28 113 ballistic 0 0 ballistic 0 0 ballistic 0 28 near 1 0 near 0 0 near 0 27 near 0 1 134 Clearly, this type of study is only possible with the assistance of the modern computer. Using the algorithm and post-processing techniques described, a FORTRAN code running on a 1.7 GHz processor required approximately 20 CPU hours to find and interpolate all the solutions. The method described is only one of many techniques that could be devised to solve this complicated problem, and no claims are made about the efficiency of this particular method. However, any other solution method searching for all free-return ballistic Earth-Mars cyclers with periods up to three synodic periods should lead to the same list of resulting cyclers. 4.9 CHAPTER CONCLUSIONS In total, all values of v at Earth are examined from 2.510 km/s at intervals of 0.0025 km/s. At each value, every feasible permutation of all half-rev, full-rev, and generic returns is investigated. Of the trillions of combinations examined, nearly two million minimax problems are posed and solved. There are thirty-three previously documented and 170 previously undocumented promising cyclers found, including forty-nine different values of v at Earth. The solution characteristics span the full spectrum of possibilities. Although the majority of the presented cyclers are new, the results of this study encompass the results of several earlier studies that looked at a variety of subsets of the current solution space. The global generalized search for idealized Earth-Mars cyclers results in a database of all useful free-return cyclers with periods of three or less synodic periods. 135 The exhaustive list represents a significant milestone in the on-going search for ballistic cyclers using a true ephemeris model. The mean eccentricities of Earth and Mars (0.0167 and 0.0934 respectively) in the true solar system are non-trivial. These, along with a mean inclination difference of 1.851o, represent a significant departure from the assumptions of a circular-coplanar model. Other studies have shown that, while analogous solutions may exist in the more accurate model, the characteristics of the final optimized trajectories are quite different. Therefore, each of the solutions presented in this chapter, regardless of the performance characteristics in the simple model, can be reasonably considered as candidates for initial guesses in the search for cyclers in a more realistic model. Although, a variety of solution methods are possible, previous work and current efforts have indicated that an iterative continuation process that slowly "walks" a solution from the simple model to the more accurate model gives promising results. This will be addressed in the next chapter. 136 5 Finding Cyclers In an Accurate Solar System 5.1 CHAPTER SUMMARY A planetary cycler trajectory is a periodic orbit that shuttles a spaceship indefinitely between two or more planets. Recent interest in developing a cycler-type architecture as a viable alternative to the traditional approach to a human-crewed Mars mission has led to the discovery of many previously unknown Earth-Mars ballistic cycler orbits. However, due to the complexity of the problem, most of the solutions are found in an idealized circular-coplanar model. Of course, this begs the question of their existence in a more accurate model. This problem has been successfully addressed for individual cyclers by a number of researchers; however, the current study seeks to find accurate ephemeris solutions for an entire class of idealized cyclers. In this chapter, accurate ephemeris cyclers are sought for each of the promising parent cyclers documented in Chapter 4. An efficient constrained optimization problem is defined to accomodate long-duration, ballistic, zero-sphere-of-influence, patched conic trajectories. A continuation algorithm is then developed to transition circular-coplanar solutions to accurate ephemeris solutions. Finally, the algorithm is applied to 203 promising circular-coplanar parent cyclers for twenty-one launch windows each, equaling 4263 cases. In total, there are nine parent cyclers (one of them is the wellknown Aldrin cycler) that have at least one launch date with a total maneuver requirement of less than 1 m/s over seven full cycles. Additionally, thirty-nine and 137 seventy-four parent cyclers have at least one launch date with a maneuver requirement of less than 10 and 300 m/s respectively. Several of the most promising cyclers have consistently low requirements for all launch windows considered. In general, the study demonstrates the broad feasibility for accurate ephemeris cyclers. 5.2 INTRODUCTION Because the geometry of the true solar system is not exactly repeatable, true periodic cycler trajectories do not exist. However, the relative geometry of the inner solar system does repeat to a reasonable degree. Thus, quasi-periodic orbits over a finite number of cycles are sought when searching for cycler orbits in an accurate solar system. Although similar in nature, the general problem of finding realistic cyclers is significantly more complicated than the problem of finding cyclers in a circularcoplanar model. The repeat time for the relative geometry of Earth and Mars in the circular-coplanar case is one synodic period, while the true model only has an approximate repeat time of seven, or more accurately, fifteen synodic periods. The solution space for a seven- or fifteen-synodic period cycler trajectory is extraordinarily large versus the more reasonable one- two- or three-synodic period time frames associated with the circular-coplanar model. A potential fifteen-synodic period trajectory may contain on the order of fifty legs and as many flybys and powered maneuvers. Chapter 4 demonstrates the enormity of the solution space for a simple three-synodic period trajectory using an enumerative approach. 138 Clearly, the same method is not feasible when considering a fifteen-synodic period (or greater) cycler. The most obvious approach is to use solutions from the simple model as a starting point in the search for realistic cyclers. Previous efforts to identify realistic cyclers have focused on the circularcoplanar model, very accurate high fidelity models, and several models in between. References 3, 4, and 8 use an idealized resonance between Earth, Mars, and Venus, including both the circular-coplanar model and a model accounting for the mean eccentricities and inclinations of the planets. In the case of the Earth and Mars, Ref. 3 documents several ballistic cyclers in each model with general repeat times of four synodic periods or greater. Reference 9 extends the VISIT cycler concept to include a twenty year propagation using an accurate ephemeris. Reference 13 demonstrates the real world feasibility of the Aldrin cycler, also known as an escalator orbit, by minimizing required maneuvers in an accurate multi-conic optimizer. Reference 14 gives a low-thrust version of the same cycler optimized in an accurate ephemeris model. Reference 16 shows a seven cycle propagation of a near-ballistic two-synodic period cycler. Reference 15 gives hundreds of launch opportunities and associated characteristics for the S1L1 cycler using a true ephemeris. The ballistic S1L1 solutions with favorable characteristics were obtained using a continuation method that slowly increases the complexity of the solar system model until finally a true ephemeris is used. Reference 14 presents a true ephemeris version of a powered Earth-Mars cycler using low-thrust and a direct optimization technique. Although still in an idealized 139 model, Ref. 49 demonstrates the preliminary design of solar sail cyclers using optimal control theory. References 22, 24, and 50 summarize much of the previous work and propose several metrics associated with Earth-Mars cycler and hybrid cycler concepts when evaluating the overall design of a powered cycler architecture. The advantages and disadvantages of ballistic, low-thrust, and impulsive thrust cyclers and hybrid cyclers are discussed. In general, the basic structure of solutions associated with circular-coplanar models are reflected in the analogous solutions using an accurate ephemeris. The nonrepeating geometry of the true model, however, does have a significant, often unpredictable, effect on the important characteristics associated with cyclers, such as hyperbolic excess velocities at Earth and Mars, and the interplanetary transit times. As demonstrated in Ref. 15, a given parent cycler can produce an unlimited number of accurate ephemeris cyclers, each with varying characteristics, depending on the desired launch date and propagation time. The purpose of this chapter is to develop a robust method that is capable of finding true-ephemeris counterparts to the circular-coplanar parent cyclers presented in the previous chapters. The goal is of course to identify purely ballistic cyclers, meaning all manevuers are achievable with flybys only. However, because many of the presented circular-coplanar ballistic (or near-ballistic) solutions already require flybys that are near (or violate) the minimum altitude boundary of 200 km, it is not expected that each simple solution has ballistic counterparts in a true ephemeris model. Thus, it 140 is necessary to include the possibility for propulsive maneuvers. The solution method is selected based on efficiency and robustness because it must accommodate over 200 parent cyclers, each with a variety of launch dates. As a result, the selected method is direct and optimizes impulsive maneuvers only. Further optimization of any given cycler is considered beyond the scope of this study, although it is encouraged as future work. Examples include higher fidelity gravity models, alternative impulsive optimization, low-thrust capability, and indirect methods. In total, twenty-one different launch windows are considered for each of the 203 parent cyclers presented in Chapter 4, requiring 4263 cases. Total powered maneuver requirements are presented for each launch window for all one- and two-synodic period parent cyclers, and a select group of three-synodic period parent cyclers. Seventy-four parent cyclers are found to have at least one solution requiring total maneuvers less than 300 m/s over the seven cycle duration, and thirty-nine of those solutions require less than 10 m/s. Reproducible trajectories are presented for the minimum v launch date for each of the selected seventy-four parent cyclers. Because Chapter 4 includes most of the useful and previously published free-return class of Earth-Mars cyclers, this chapter is a broad and robust survey of the true ephemeris cyclers of the same class. The method presented proves to be a reliable preliminary design tool for accurate ephemeris cyclers. It demonstrates the existence of hundreds of previously unpublished ballistic or very near-ballistic accurate ephemeris cyclers. 141 5.3 PROBLEM DEFINITION AND ASSUMPTIONS Final trajectories are sought that use a realistic ephemeris file6, JPL's DE405, to obtain the positions and velocities of the planets. The mean obliquity5 of the ecliptic is used to rotate all vectors so they are referenced to the mean equinox and ecliptic plane of J2000. As long as the final trajectories use an ephemeris file, it is completely valid to use intermediate solutions that propagate the planets using Kepler's equation. Similar to the assumptions of the previous chapters, the motion of the cycler spacecraft is assumed to be governed by the sun only, thus Keplerian propagation is sufficient. Also, the trajectories are patched using an instantaneous and zero-sphere-of-influence patched conic flyby with a minimum altitude of 200 km. However, unlike the previous assumptions, flybys at Mars are included. In the circular-coplanar model, these were ignored because of the low mass of Mars, but more importantly, so that cyclers could be constructed using Earth-Earth free returns only. It is expected that this extra degree of freedom may alleviate some of the problems associated with the non-ideal orbits of Earth and Mars. The inertial positions of Earth and Mars approximately repeat every fifteen years, and several previous studies have used this as a baseline to determine the duration of a true cycler propagation.13,15,16,18 Earlier studies however, assumed (with more accuracy) that the basic repeat time of the true solar system is thirty-two years.3,4,8 A more detailed look at the relative geometries of the true orbits of Earth and Mars shows that there are several approximate repeat times worth considering. It is 142 desirable to find in the true solar system an X:Y resonance between Earth and Mars, meaning the Earth makes exactly X revolutions for every Y Mars revolutions, where X and Y should be as close to integers as possible. If a resonance existed where X and Y were exactly integers, then the relative geometry of Earth and Mars would be exactly repeatable (under the assumption that the sun is the only attracting body). Of course perfect Earth-Mars resonances in the true solar system do not exist. Table 5.1 lists several pertinent Earth-Mars resonances based on their mean periods.5 Table 5.1: Earth-Mars resonances accurate X:Y resonance (Earth revs: Mars revs) 15.000: 7.9755 17.000: 9.0389 30.000: 15.9510 32.000: 17.0144 34.000: 18.0778 45.000: 23.9265 47.000: 24.9899 60.000: 31.9020 64.000: 34.0288 77.000: 40.9409 94.000: 49.9798 approximate X:Y resonance (Earth revs: Mars revs) 15:8 17:9 30:16 32:17 34:18 45:24 47:25 60:32 64:34 77:41 94:50 |residual| (revs) 0.0245 0.0389 0.0490 0.0144 0.0778 0.0735 0.0101 0.0980 0.0288 0.0591 0.0202 Also, the total duration for the propagation must be an integer multiple of the repeat time of the parent cycler in the circular-coplanar model. Table 5.2 gives the total duration for a given number of cycles for cyclers with repeat times of one, two, three, and four synodic periods respectively. 143 Table 5.2: Integer multiples of circular-coplanar cycler repeat times Number of cycles 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1- synodic period (yr) 2.14 4.27 6.41 8.54 10.68 12.81 14.95 17.08 19.22 21.35 23.49 25.62 27.76 29.90 32.03 2- synodic period (yr) 4.27 8.54 12.81 17.08 21.35 25.62 29.90 34.17 38.44 42.71 46.98 51.25 55.52 59.79 64.06 3- synodic period (yr) 6.41 12.81 19.22 25.62 32.03 38.44 44.84 51.25 57.66 64.06 70.47 76.87 83.28 89.69 96.09 4- synodic period (yr) 8.54 17.08 25.62 34.17 42.71 51.25 59.79 68.33 76.87 85.42 93.96 102.50 111.04 119.58 128.12 Each entry of Table 5.2 is compared with each of the first numbers in the second column of Table 5.1. The bold entries are suggested as reasonable total durations. Certain bold entries are better than others as indicated by how close to integers their values are and the magnitude of the associated residuals from Table 5.1. For example, for three-synodic period cyclers, 32.03 is very near 32 and the 32:17 resonance has one of the smallest residuals listed; therefore, five cycles appears to be a prudent choice for the total duration of an accurate ephemeris version of a three-synodic period cycler. Although it is expected that the general structure and pattern of a thirty-two year propagation is repeatable, it is not guaranteed. Even if the total time is constrained to be exactly thirty-two years, the resonance is not perfect, and furthermore, no considerations have been made for feasibility of the flybys that would eventually patch multiple thirty-two year segments. Therefore, it is assumed that non-trivial powered 144 maneuvers may be necessary to ultimately patch the multiple cycle segments. Thus, when considering how to choose the total duration of a multiple cycle segment, it is desirable to include as many cycles as possible. The method developed in this chapter favors efficiency over performance in some instances because of the large number of cases that need evaluation. Therefore, all cyclers will be propagated for seven cycles, as it is the only row in Table 5.1 with all bold entries. Additionally, this choice is consistent with previous studies. 13,15,16,18 For further optimization of a particular cycler, it is recommended to use fifteen cycles (thirty-two years) for a one-synodic period cycler, eleven cycles (forty-seven years) for a two-synodic period cycler, either five or ten cycles (thirty-two or sixty-four years) for a three-synodic period cycler, and seven or eleven cycles (sixty or ninety-four years) for a four-synodic period cycler. As discussed in Chap. 4, only one short Earth-Mars or Mars-Earth transit is guaranteed per cycle. Thus, it is decided to search for accurate ephemeris solutions for cyclers with a repeat time of three synodic periods or less. This includes all circularcoplanar cyclers documented in this dissertation except the four-, five-, and six-synodic period cyclers from Chapters 3 and 4. Only outbound cyclers, or cyclers with short Earth to Mars legs, will be considered. Minor modifications are necessary to evaluate inbound cyclers, and it is reasonable to expect similar results. In summary, the main objective is to develop an efficient and robust method to optimize seven continuous cycles of a given circular-coplanar parent cycler using 145 patched conic two-body motion between true ephemeris planet locations. In general, it is assumed that the basic structure of the seven-cycle sequence is repeatable indefinitely. 5.4 SOLUTION METHOD There are many different approaches to consider when searching for true ephemeris solutions. This section will discuss the advantages and disadvantages of several options and explain in detail the selected method. 5.4.1 Continuation Method The basic idea of a continuation51 method, also referred to as imbedding or a homotopy method, is to solve a sequence of several sub-problems where the fidelity of the model in question is increased for each sub-problem until the desired level of complexity is achieved. The solution to each sub-problem becomes the initial guess for each successive sub-problem. A continuation method is ideal for the Earth-Mars cycler problem because the solution to the first sub-problem (circular-coplanar case) is already available. However, one of the shortcomings of a continuation method is that solutions to each sub-problem must exist in order to continue along the path. The solution to the last sub-problem is the only one of interest, yet it may not be possible to get there even if a solution exists because the problem may become infeasible at one or more of intermediate steps. 146 In most cases, the gap between the circular-coplanar model and the accurate ephemeris model prohibits an immediate jump to the latter. Therefore, the continuation method is deemed most appropriate, and other considerations are made to address its apparent shortcomings. 5.4.2 Non-Analytic Solutions Several attempts were made to use the analytic solutions for the interplanetary transfers presented in Chap. 2. These include the full-rev return solutions, half-rev return solutions, and the generic (not necessarily return) transfers. These solutions are extremely fast, accurate, and relatively straight-forward to implement. However, each case requires at least two integer valued inputs. Each circular-coplanar cycler has a unique set of integer identifiers as specified in the formal cycler nomenclature.25 These integer values fix the structure of each leg of the trajectory. However, as the complexity of the model moves toward the accurate case, it is feasible and very probable that the trajectory as a whole could benefit from minor changes in this structure. For instance, a short period one revolution transfer from Mars to Earth that exists in the circular-coplanar case may cease to exist when the launch and arrival dates become free parameters and move away from their original values. Of course, there is always a transfer that connects two points, but changing the value of the integer inputs (such as number of revolutions and indicating if it is a short period or long period solution) is a difficult if not insurmountable obstacle when dealing with a gradientbased optimizer. 147 Solving the transfers analytically with a defined fixed structure works very well for cyclers with favorable turning angle requirements, such as the S1L1 cycler.15 However, because the method developed for this study must be capable of optimizing any circular-coplanar cycler regardless of its characteristics, the non-analytic approach is a better choice. In this method, a specific transfer is found by numerically searching for the spacecraft velocity vector that leads to an intercept of the desired planet. This approach eliminates the integer programming problem, and the solution structure is free to morph as necessary. A major benefit to this approach is the relative ease of gradient calculations. Although tedious, partial derivatives of the final states with respect to the unknown velocities and times can be calculated analytically. The same calculations become much more difficult when Lambert's equation is part of the solution process. The analytic derivatives are discussed in a later section. 5.4.3 Gradient Method For the problem at hand, a gradient method is favored over a monte-carlo or genetic algorithm 52 approach based on convergence properties and computational requirements. There are two major advantages of a genetic algorithm: a GA searches a global solution space and requires no gradient information. For individual cyclers, some success is achieved using a genetic algorithm53 and the analytic transfer solutions. This is certainly a valid approach to trajectory optimization 54,55 , however, it is not considered practical for this study because it is not efficient when considering all the cases and launch dates that need consideration. 148 Although gradient methods give rise to local solutions, the fact that the solutions in the circular-coplanar case are ballistic gives confidence that the analogous solutions in the accurate model will be good local solutions if not global. Two standard sequential quadratic programming packages are used to optimize the free parameters. The first package is VF13 from the Harwell Subroutine Library.56 This code works well with cyclers with favorable turning characteristics, but often fails when the constraints become infeasible. The second package tried is the SNOPT (Sparse Nonlinear OPTimizer) driver from the Stanford Business Software Inc.57,58 SNOPT proves to be more robust for purposes of this study. It has an "non-linear elastic mode" that automatically minimizes a weighted sum of the absolute values of the constraint violations (in addition to an existing objective if any) when the constraints become infeasible. This feature provides robustness to the system because the algorithm continues to search for a "good solution" even if the optimality conditions are not satisfied or the constraints can not be met. The continuation method can therefore continue "walking" toward the accurate ephemeris model even if favorable solutions disappear along the path. Details regarding the SNOPT input and the constraint and free parameter choices are in later sections. 5.4.4 Multiple Shooting Method Reference 51 gives a detailed overview of direct multiple shooting methods in the context of other optimization techniques. This section discusses the benefits of 149 multiple shooting methods regarding accurate ephemeris cycler optimization and gives a detailed explanation of the specifically developed method. The long duration trajectories associated with accurate ephemeris cycler propagation lends to a multiple shooting approach. If each leg of a multiple-decade patched trajectory is numerically dependent on the final conditions of the previous leg, very small adjustments to the initial conditions of the first leg will lead to highly erratic and non-linear perturbations at the end of the long trajectory. To avoid the large sensitivities and cumulative numerical errors, a multiple shooting approach defines each leg of a patched trajectory to be completely independent from all other legs. Specific to the current problem, the seven parameters chosen to completely define each leg are the initial planet, final planet, initial time, final time, and the three components of the initial planet-referenced spacecraft velocity. Except for the specification of the initial and final planets, each of the stated parameters is unknown and free to be optimized. Thus, there are 5n unknowns, where n is the number of legs for a particular cycler propagation. A diagram is given in Figure 5.1. 150 Figure 5.1: Trajectory leg diagram The spacecraft initial position is fixed to the planet's initial position, and a planet's position and velocity is uniquely a function of time only via an ephemeris file. The ephemeris can be as accurate as possible, or it can be based on any set of fictional or mean orbital elements referenced to J2000, depending on the desired solar system model complexity. The sequence of planets remains fixed during the optimization procedure and is selected initially based on multiple cycles of the parent circularcoplanar cycler. Because each leg is independent, multiple shooting methods require boundary constraints to enforce continuity between the segments. Thus, there are n-1 scalar time constraints that directly result from using a multiple shooting approach. Although extra constraints require additional gradient calculations, an added benefit of a multiple shooting method is the favorable structure of the associated gradient matrix, also known 151 as the Jacobian. Again, because each leg is independent, any unknown variable affects only the constraints associated with one leg. Therefore the Jacobian, in general, is expected to be sparse with block matrices along the diagonal. This feature contributed to the selection of SNOPT57 as the primary optimizer because it was designed explicitly for large sparse problems with many active constraints. A second benefit of SNOPT is its favorable handling of linear constraints. The time constraint that enforces continuity is tf(i) = t0(i+1). This is a simple linear constraint that SNOPT easily solves during each major iteration of the optimization process. Therefore, disconnecting the times and enforcing a simple linear constraint gives the tremendous benefit of a sparse Jacobian, yet for all practical purposes, the constraint is non-existent because SNOPT solves it completely during every iteration. This is a significant benefit in terms of the computational cost and accuracy of determining the Jacobian, a topic covered in a later section. 5.4.5 Ballistic Flybys with Powered Maneuvers at the Sphere of Influence A ballistic gravity-assisted flyby is capable of rotating a spacecraft's planetcentered velocity as described in detail in Chap. 2. The magnitude, however, is constrained to be equal before and after the flyby. A second inequality constraint places an upper bound on the angle of rotation based on the flyby speed at the sphere of influence, the gravitational parameter of the flyby body, and the minimum acceptable flyby radius. These two constraints are given in Eqs. (5.1) and (5.2). They are enforced at each of the n-1 flybys for a trajectory with n legs. 152 v - (i -1) = v + (i ) (5.1) (5.2) req avail where, v T (i -1) v + (i ) - req = cos -1 v - (i -1) v+ (i ) (5.3) plan avail = 2 sin -1 2 plan + rp min v small (5.4) The problem as stated is a non-linear root finding problem with 5n unknowns, 3n + 2(n-1) equality constraints and (n-1) inequality constraints. It is not an explicit optimization problem because there is no defined objective function. However, as mentioned in the previous section, if all of the constraints cannot be met, SNOPT automatically enters an "elastic mode" that minimizes the constraint infeasibilities. In this case, a specified weight is irrelevant because there is no explicit performance index. One approach to avoid constraint infeasibilities is to introduce four additional unknowns per leg (an intermediate time and three components of a v), and attempt to minimize an explicit objective function defined to be the sum of the magnitudes of all the intermediate maneuvers. Although this method is valid and may yield more favorable trajectories, it is prohibitive in the current problem because it almost doubles the dimension of the parameter vector. A second more efficient approach is to take advantage of the "elastic mode" of SNOPT, and redefine the constraints, if possible, to directly measure the powered v necessary to make each flyby feasible. In this approach, the equality and inequality 153 constraints given in Eqs. (5.1) and (5.2) are replaced by one conditional equality constraint given by Eq. (5.5). v+ (i ) - v- (i -1) if req avail v0 (i ) = 2 2 v+ (i ) + v- (i -1) - 2v+ (i ) v- (i -1) cos(req - avail ) if req > avail (5.5) This approach does not increase the number of unknowns, eliminates the inequality constraints, and does not introduce an explicit performance index. A summary of the parameters and newly defined constraints are given below in Table 5.3. Table 5.3: Summary of parameters and constraints a Parameter Number v+(i) t0(i) tf(i) a Constraint rf(i) = rf plan (i) tf(i-1) = t0(i) v(i)=0 Number 3n n-1 n-1 5n-2 Type nonlinear equality linear equality nonlinear equality 3n n n 5n i indicates the ith leg , n = number of legs A diagram explaining the origin of Eq. (5.5) is given in Figure 5.2. Given the unknowns associated with any two successive legs, the trajectories are propagated according to Figure 5.1. At the end of the (i-1)st leg, v-(i-1) can be deduced, and v+(i) is simply three of the given unknowns. Their magnitudes can be calculated and are not necessarily equal. The required turning angle is found via the definition of a dot product and is given by Eq. (5.3). The maximum available turning angle is derived from basic properties of hyperbolic trajectories and is given in Eq. (5.4). It is desirable to use the smallest value of v because it is inversely related to the available turning 154 angle. If v-(i-1) < v+(i), then the powered maneuver should occur before the flyby, otherwise it should occur afterwards. All maneuvers take place during heliocentric flight just outside the planet's sphere of influence, which is currently modeled with a zero radius. A powered maneuver during the hyperbolic phase of a flyby is considered too risky and is beyond the scope of this study despite the acknowledged potential for fuel savings. Figure 5.2: Powered v required at the SOI of a flyby. 155 In Case A of Figure 5.2., the required turning angle can be achieved using a gravity-assisted flyby alone, while the powered maneuver is simply to account for the difference in magnitudes. In Case B, the flyby provides the maximum turning angle available, meaning the flyby has a periapse altitude of 200km. In order to optimize the effects of the flyby, it is performed in the plane that contains v-(i-1) and v+(i), assuming the angle between them is greater than 0 and less than . The remaining scalar v is calculated using the law of cosines. 5.4.6 Analytic Gradient Calculations The need to calculate partial derivatives is a common problem encountered in astrodynamics and trajectory optimization. More generally, almost all root-finding or optimization methods require some form of gradient information relating sensitivity of the constraints and the objective with respect to the free parameters. The partial first and second derivative matrices are referred to as the Jacobian and Hessian, respectively. A variety of root-finding and optimization techniques exist that require the full calculation of one, both, or neither or the two matrices at each major iteration. Methods also exist to provide estimated updates to both matrices so that full derivative calculations are not necessarily required at the every iteration. Some methods do not require gradient information, such as monte-carlo methods and genetic algorithms. However, for problems with well-behaved solutions spaces, gradient information is extremely helpful in driving the system towards a solution. 156 There are three primary methods of calculating partial derivatives. Numerical approximation is a brute force approach that determines sensitivity by perturbing a particular parameter and re-calculating a particular constraint. The selection of the perturbation size is a tuning parameter that requires significant attention in order to achieve the most accurate derivative possible. 59 In general, each forward or central differenced derivative requires one or two additional constraint function calls respectively. Although sufficient for most applications, this method is generally the least accurate and most expensive of the three methods. Despite the cost in accuracy and CPU time, the method is very easy to to implement across a variety of problems, and therefore is the most common of the three. The second approach is to calculate the partial derivatives analytically. This can be a very difficult and tedious task depending on the complexity of the constraints and their relationships to each parameter. The expressions must be checked and re-derived each time a constraint or parameter is introduced or redefined in a particular problem. However, once implemented, analytic gradients are the most accurate available, and in most cases, are significantly faster than approximating the derivatives numerically.60 In order to calculate the full Jacobian using central differences (without taking advantage of sparseness), 2k additional calls to the constraint function are required where k is the number of unknown parameters. However, the analytic derivatives can be calculated alongside the constraints in the original call to the function using no additional calls. 157 Unless the analytical derivatives are extraordinarily expensive to compute, they are almost always faster and more accurate. A third general method to calculate derivatives is automatic differentiation, a hybrid of the second approach. This method calculates analytic derivatives by decomposing each constraint into a sequence of the most elementary intrinsic mathematical functions, and then uses the basic chain rule from calculus to accumulate the derivative from the beginning to the end of the constraint function calculation. Once a system 61 is in place, this approach is much more flexible than directly calculating the analytic derivatives, and also benefits from the improved accuracy and efficiency over numerical derivatives. In general, however, explicit analytic derivatives are expected to be more accurate because there is less accumulation of round-off error. For best results, SNOPT requires the user to calculate the Jacobian at every major iteration, and the Hessian is intrinsically approximated. Although SNOPT has accompanying files to incorporate automatic differentiation, this feature is bypassed because the general structure of the constraints and unknowns of the current problem is fixed and it is possible to derive the analytic Jacobian. Because this study seeks to optimize thousands of different cases, the run-time benefit of using analytic derivatives is crucial. In this section, analytic expressions for the non-zero partial derivatives of the constraints with respect to the unknowns for the defined accurate ephemeris cycler problem are derived. This gradient matrix, or the Jacobian, consists of 5n-2 rows and 158 5n columns, because from Table 5.3, there are 5n unknowns and 5n-2 constraints. This totals 25n2-10n entries, thus for a propagation of an accurate ephemeris cycler with 35 legs there are 30,275 Jacobian entries. Fortunately a multiple shooting approach is used, and the vast majority of these entries are zero. In general, there are five scalar constraints associated with each leg as summarized in Table 5.3. The first leg is the only exception because there is no prior leg to patch. The only non-zero partial derivatives of Ci, the constraints associated with a particular leg, with respect to Ui, the unknowns associated with a particular leg, are given in Eq. (5.6). C1 r f (1) = U1 v + (1) r f (i ) v + (i ) Ci = U i 01x 3 i = 2 n v0 (i ) v + (i ) r f (1) t0 (1) r f (i ) t0 (i ) t0 (i ) t0 (i ) 0 r f (1) t f (1) r f (i ) t f (i ) 03 x 3 Ci = 0 , U i -1 01x 3 i = 2 n v0 (i ) 0 v + (i -1) 03 x1 0 v0 (i ) t0 (i -1) 03 x1 t0 (i ) t f (i -1) v0 (i ) t f (i -1) (5.6) Equation (5.6) is derived intuitively by a close evaluation of Figure 5.1 and the relationships between the constraints and parameters given in Table 5.3. The partial derivatives can be simplified as follows: 159 r f ( i ) v + ( i ) r f ( i ) t0 ( i ) r f ( i ) t f ( i ) t t0 ( i ) t f ( i -1) t0 ( i ) t0 ( i ) v0 ( i ) = = = = = (r f ( i ) - rPLAN f ( i ) ) v + ( i ) (r f ( i ) - rPLAN f ( i ) ) t0 ( i ) (r f ( i ) - rPLAN f ( i ) ) t f ( i ) (t0 ( i ) - t f ( i -1) ) t f ( i -1) (t0 ( i ) - t f ( i -1) ) t0 ( i ) = = (r f ( i ) ) v + ( i ) r f ( i ) t0 ( i ) = r f ( i ) v 0 ( i ) v 0 ( i ) v + ( i ) = r f ( i ) v 0 ( i ) I = r f ( i ) v 0 ( i ) = v f ( i ) - v PLAN f ( i ) = v - ( i ) = -1 =1 = v0 ( i ) v - ( i -1) v - ( i -1) v 0 ( i -1) = v0 ( i ) v f ( i -1) v - ( i -1) v 0 ( i -1) = v0 ( i ) v f ( i -1) v - ( i -1) t0 ( i -1) v + ( i -1) v0 ( i ) t0 ( i -1) v0 ( i ) t f ( i -1) = = = v0 ( i ) v - ( i -1) v - ( i -1) v + ( i -1) v0 ( i ) v - ( i -1) = = v0 ( i ) ( v f ( i -1) - v PLAN f ( i -1) ) v - ( i -1) v0 ( i ) v - ( i -1) t0 ( i -1) v - ( i -1) t0 ( i -1) v0 ( i ) v - ( i -1) v - ( i -1) t f ( i -1) (a f ( i -1) - a PLAN f ( i -1) ) Thus, Eq. (5.6) becomes C1 r f (1) r f (1) = v - (1) U1 v 0 (1) t0 (1) r f (i ) r f (i ) v - (i ) t0 (i ) Ci v 0 (i ) Ci = 03 x 3 = U i 01x 3 1 0 , U i -1 01x 3 i = 2 n v0 ( i ) i = 2n v f (i -1) 0 0 A v 0 ( i -1) v + (i ) v0 (i ) where A = v - (i -1) A 03 x1 0 v f (i -1) t0 (i -1) 03 x1 -1 A ( a f (i -1) - a PLAN f (i -1) ) (5.7) The final structure of the full Jacobian is given below. ( C1 U1 )3 x 5 ( C2 U1 )5 x 5 ... . ( C2 ( C3 U 2 )5 x 5 U 2 )5 x 5 ... ( C3 U 3 )5 x 5 ... 0 ... ... 0 ( Cn -1 U n - 2 )5 x 5 ( Cn -1 U n -1 )5 x 5 ( Cn U n -1 )5 x 5 ... ( Cn U n )5 x 5 (5n - 2) x (5n ) 160 The remaining few derivatives are separated into three general categories. The corresponding partials are derived in the following three sections. 5.4.6.1 Partials of the final state with respect to the initial state62 In general, the spacecraft state, x=[r v]T, is governed by a first order differential equation: dx(t ) = f ( x(t ), t ) dt Taking the partial derivative with respect to the state at an initial time gives dx(t ) f ( x(t ), t ) = x(t0 ) x(t0 ) dt d x(t ) f ( x(t ), t ) x(t ) = dt x(t0 ) x(t ) x(t0 ) and can be rewritten as d (t , t0 ) f ( x(t ), t ) (t , t0 ) = dt x(t ) x(t ) where (t , t0 ) = x(t0 ) (5.8) The partial derivative of the state at a final time with respect to the state at an initial time is therefore (t,t0) by definition. The matrix is often referred to as the state transition matrix, and can be obtained by numerically integrating Eq. (5.8) with the initial condition of (t0,t0) = I6x6. This is true for any force model that is a function of the state and time only. 161 For simple Keplerian motion, if the state is expressed as the classic orbital elements = [a e i M]T, instead of position and velocity, x, then the state transition matrix is analytic and has the same entries as the identity matrix except 6 ,1 (t , t0 ) = -3(t - t0 ) a 5 2 Thus, for Keplerian orbits, the partial of the final cartesian state with respect to the initial cartesian state can be calculated analytically without integrating the state transition matrix numerically. This expression is given in Eq. (5.9). x(t ) x(t ) (t ) (t0 ) = x(t0 ) (t ) (t0 ) x(t0 ) = (t0 ) x(t ) (t , t0 ) x(t0 ) (t ) (5.9) Ref. 62 gives detailed derivations for the partials of the cartesian coordinates with respect to the classic orbital elements and the inverses respectively. Both partial derivative matrices have entries with singularities when eccentricity or inclination is zero. This is a dilemma for the current application because every cycler has at least one Earth to Earth generic leg that takes place in the ecliptic, and most half-rev returns have small if not zero eccentricities. As a result, an unconventional set of orbital elements, , can be defined such that the partials are well behaved at low eccentricities and inclinations. References 63 and 64 define two such sets. The latter is chosen for the current application because it gives FORTRAN code for the necessary partials. This non-singular set is expressed as = a e sin ( + ) e cos ( + ) sin ( i / 2 ) sin ( ) sin ( i / 2 ) cos ( ) + + M T 162 Thus the final form of Eq. (5.9) is x(t ) x(t ) (t0 ) = x(t0 ) (t0 ) x(t0 ) (5.10) where the first term on the right side of the equation has the state transition matrix for the new orbital element set embedded and is found directly with the equations given in Ref. 64. The second term on the right is found by inverting the partial of x(t) with respect to (t0) when t = t0. Note that there are several typographical errors in the equations in Appendix 1 of Ref. 64; however, the FORTRAN code in Appendix 2 of Ref. 64 is correct. The equations for x/ and x/ found in Refs. 62 and 64, respectively, are well documented and require several pages of calculations. For brevity, they are not rewritten here. 5.4.6.2 Partials of the final state with respect to the initial time For the current problem as defined in Table 5.3, the state of a spacecraft on a given leg is a function only of the of the initial state and the time elapsed since t0. The initial position is a function of t0 only, and the initial velocity is a function of t0 and v+ only. x(t ) = f [r0 (t0 ), v 0 (t0 , v + ), t - t0 ] The partial of x with respect to t0 is x(t ) x(t ) r0 x(t ) v 0 x(t ) (t - t0 ) = + + t0 r0 t0 v 0 t0 (t - t0 ) t0 (5.11) 163 where r0 = v PLAN 0 , t0 v 0 (t - t0 ) = a PLAN 0 , and = -1 t0 t0 and x(t ) x(t ) M (t ) = (t - t0 ) M (t ) (t - t0 ) where M (t ) = M 0 + / a 3 (t - t0 ) (5.12) Equation (5.12) is true because the only orbital element affected by t-t0 is the mean anomaly, M. The term x(t)/M(t) is given in Section 7.1.2 of Ref. 62 and also is identical to x(t)/6(t0) from Eq. (5.10) that is obtained using the equations in Ref. 64. The final form of the partial derivative of the final state with respect to the initial time is given in Eq. (5.13). x(t f ) t0 = x(t f ) r0 v PLAN 0 + x(t f ) v 0 a PLAN 0 - x(t f ) M (t ) / a3 (5.13) 5.4.6.3 Partials of the powered v constraint The only two terms from Eq. (5.7) that remain unknown are v0 (i ) v - (i -1) and v0 (i ) v + (i ) The expression for v0(i) is given in Eq. (5.5). Dropping the i subscripts, these partial derivatives can be written as v+ - v- v v+ - v- v = v req avail v 1 v v v - v v cos( req - avail ) - v sin( req - avail ) v - v 0 if req avail if req > avail v0 v 164 where v = v T v v = v T / v v Differentiating Eqs. (5.3) and (5.4) and dropping the i subscripts gives - plan rp min avail T = 4 v small 2 2 v small ( plan + rp min v small ) avail =0 v big req v v = - v I 3 x 3 v v T - 3 v v vT v 1 - - + v- v+ 2 plan 1- 2 plan + rp min v small 2 where v if v - v + v small = - v + if v - > v + rp min Earth = 6578.0 km rp min Mars = 3598.5 km 5.4.7 Constraint infeasibilities and post-processing considerations As stated in Table 5.3, all legs of the patched trajectory have one associated vector constraint, and all legs except the first have two associated scalar constraints. If the current set of unknowns is in a region where all the constraints cannot be satisfied, SNOPT enters an "elastic mode" that seeks to reduce the constraint infeasibilities. The linear time continuity constraints are always feasible and easily satisfied, and no explicit objective is defined. Thus, the elastic mode is "equivalent to minimizing the sum of the 165 nonlinear constraint violations."57 "composite objective" becomes J= i =1 n v0 (1) = 0 Thus, when SNOPT enters this mode, the new ( v 0(i ) + rx f (i ) + ry f (i ) + rz f ( i ) ) (5.14) The final solution is therefore not guaranteed to intercept the desired planet. However, a final flyable trajectory must, at a minimum, have position continuity between each leg. One approach to reduce the residuals is to simply weight the position constraints (and associated derivatives) in the original formulation inside the optimization procedure. This is equivalent to adjusting the composite objective function to J= i =1 n v0 (1) = 0 v0 (i ) + W rx f (i ) + ry f (i ) + rz f ( i ) ( ) (5.15) For sufficiently large values of W, the solution will effectively ignore the v requirements, focusing only on the position violations, and vice-versa for sufficiently small values. Clearly, this parameter requires tuning to identify what value leads to the best solutions. A second method to achieve position continuity is to post-process the final trajectory by adding intermediate maneuvers to remove the residuals entirely. These can be independently optimized, but the final magnitudes must be added to the total cycler cost. After much trial and error, a hybrid of both approaches is selected. A variety of test trajectories are optimized using a range of weight factors, W. The final solutions, 166 each with various magnitudes for the position residuals, are post-processed for continuity, and the total cost is evaluated to choose the best weight factor. Canonical variables are used for calculations in this study such that the gravitational parameter of the sun is unity. Thus, the magnitudes of positions and velocities are similar. Each component of Eq. (5.15) is therefore expected to be of equal magnitude when W is on the order of one. After many trade studies, the best value for W appeared to be exactly 1, and this value was used for the thousands of subsequent runs. The serendipitous best value of 1 is indicative of the benefits of using canonical units for astrodynamic applications. In order to reduce the effect of the post-processed intermediate maneuvers on the overall fuel costs, the time of each maneuver is independently optimized over the course of the given leg. This one dimensional optimization problem is globally solved with a simple grid search over 100 equally spaced intervals, paying close attention to avoid n transfer angles. As a whole, trajectories are evaluated based on the total accumulated v following the post-processing. Although the intermediate maneuvers are not included in the original optimization problem, this hybrid approach gives good results and benefits tremendously from the increase in speed versus including the post-processed maneuvers in the original problem formulation. 167 The general solution method described in this section is used to optimize over 200 different parent cycler trajectories, each with twenty-one different launch windows. As a result, it is designed to be robust, efficient, and capable of accommodating all types of long-duration patched ballistic transfers. Presumably, any number of existing ballistic interplanetary trajectory optimization codes65,66,67 could be used to further optimize any of the solutions. In the initial stages of this study, COPERNICUS67 was used for trade studies to identify promising options for the constraint and parameter definitions. COPERNICUS is a high fidelity general trajectory optimization package currently under development at the University of Texas at Austin. 5.4.8 Algorithm Description As Section 5.4.1 explains, a continuation approach is selected to "walk" solutions from the circular-coplanar model to an accurate ephemeris model. There are many potential paths to consider when transitioning a solution from a simple model to a more accurate one. Similar to the method used in Ref. 15, the approach that generally gives the best results consists of three major steps. These are outlined in Figure 5.3. 168 0) Circular-coplanar model (mean values for Earth and Mars at J2000 for a, , , and ; but e=i=0) 1) Use continuation method to "walk" towards mean eccentricities for Earth and Mars with nstep equally spaced "steps" 2) Use continuation method to "walk" towards mean inclinations for Earth and Mars with nstep equally spaced "steps" 3) Make one final "step" from the now mean orbital element model directly to the accurate ephemeris Figure 5.3: Path from simple to real model It is very important to fix the initial values of a, , , and for Earth and Mars to their respective mean values at J2000, otherwise, the model changes too rapidly.68 Table 5.4 gives the mean values for the orbital elements of Mars and Earth at J2000.5 Table 5.4: Mean Elements at J20005 Classic Orbital Element a e i (deg) (deg) (deg) (deg) Earth 1.00000101812E+00 1.67086171540E-02 0.00000000000E+00 0.00000000000E+00 1.02937348083E+02 -2.47089957222E+00 Mars 1.52367934749E+00 9.34006199474E-02 1.84972647778E+00 4.95655237028E+01 2.86494710278E+02 1.93730406472E+01 While Figure 5.3 gives an overview of how to transition a single solution to a more accurate model, Figure 5.4 gives a broader algorithm that describes the overall process. Analogous ephemeris solutions are sought for each circular-coplanar parent cycler in twenty-one independent runs, each associated with a different launch window. The launch windows have units of synodic periods after J2000. Thus, a cycler with an 169 associated LaunchWindow value of i will have an original epoch time in between i and i+1 synodic periods after J2000. Twenty-one is chosen in order to observe three complete cycles of the approximate seven synodic period repeat time of the true inertial positions of Earth and Mars as discussed in Section 5.3. FOR ParentCycler =1 to 203 FOR LaunchWindow = 1 to 21 1) 2) 3) 4) 5) 6) 7) Set Earth and Mars orbital elements to the mean values at J2000 Set Earth and Mars inclination and eccentricity to zero Propagate simple model from J2000 forward LaunchWindow synodic periods. INPUT the beginning phase angle for current simple model ParentCycler Propagate simple model forward in time until desired phase angle achieved Record epoch time INPUT seven cycles of the simple model ParentCycler beginning at the epoch time. (steploop -1) FOR steploop = 1 to 6 1) 2) 3) 4) 5) 8) 9) nstep = 3 (gives nstep values of 1,3,9,27,81,243) "WALK" mean eccentricities with nstep steps (each step is 1 run of SNOPT) "WALK" mean inclinations with nstep steps (each step is 1 run of SNOPT) "STEP" to the true ephemeris model (this is 1 run of SNOPT) Store the total v, the full trajectory characteristics, and the current value of nstep END steploop loop Of all nstep runs, keep the best solution for current ParentCycler and LaunchWindow POST-PROCESS this best solution 1) 2) Optimize times during each leg for intermediate v to enforce position continuity Add each intermediate v to overall cycler v 10) Write to file the final true eph. sol. for current ParentCycler and LaunchWindow END LaunchWindow loop END ParentCycler loop Figure 5.4: General algorithm Trade studies are done using different values of nstep, or the number of continuation steps used to "walk" the solution to the accurate ephemeris. The results 170 are inconsistent, meaning the nstep value that leads to the best solution varies depending on the parent cyler and launch window. In order to account for this tuning necessity, each parent cycler and launch window is independently "walked" using several different number of steps. Several other loops can be added to consider other tuning variables such as input values to SNOPT or the v weighting scalar, W, as discussed in the prior section. However, these tuning loops significantly increase the run-time of the algorithm and only provide marginal solution improvement, depending on the specific tuning variable. Therefore, the default input tuning values for SNOPT are generally left unchanged, and the v weighting scalar of 1 is used for the final runs. If the algorithm as stated in Figure 5.4 is modified such that the nstep loop only runs once with a value of nstep=5, a value found to be an excellent compromise between efficiency and performance, then the total runtime on a 3.2 GHz processor is approximately 1 day (roughly 3 days on a 1 GHz processor). The code was written in FORTAN and compiled with a flag to optimize for speed. Roughly, a 50% increase in solution performance can be achieved if the full algorithm is run including the nstep loop. However, the increase in performance comes at substantial cost. The full runtime is approximately 4 days running continuously on 15 computers with an average processor speed of 2 GHz (roughly 120 days on a 1 GHz processor). The results of the full run are presented in the next section. 171 5.5 RESULTS A full run of the algorithm presented Figure 5.4 requires the evaluation of 203 parent cyclers, each with twenty-one different launch dates. Each parent cycler and launch window pair is then "walked" to a true ephemeris model 6 different times with nstep values of 1, 3, 9, 27, 81, and 243, respectively. The nstep values are chosen somewhat arbitrarily to be powers of three, where the primary goal is to efficiently sample a full range of values. In total, the SNOPT driver is called (203) x (21) x [2(1+3+9+27+81+243)+1] times equaling 3,107,727 calls. The values inside the bracket account for the number of steps taken for eccentricity, inclination, and finally the last step to the true ephemeris model. The lowest total v solution for each ParentCycler-LaunchWindow pair is written to a file. This includes (203) x (21) or 4,263 full reproducible trajectories varying from fifteen to forty-nine legs. Obviously, the full details of each trajectory can not feasibly be archived in this document. However, a summary of the total v required for the full seven cycle propagation is presented for a selected set of parent cyclers in Figure 5.5 and Figure 5.6. Total v values are presented for each launch window for seventy-seven of the 203 parent cyclers evaluated. All one- and two-synodic period parent cyclers, regardless of the solution characteristics, are included in the results. All three-synodic period parent cyclers with at least one launch date with a total v requirement of 300 m/s or less are also included. 172 The one- and two- synodic period parent cyclers are plotted together in Figure 5.5a. In total, 1 one-synodic period parent cycler and 5 two-synodic period parent cyclers are evaluated. The remaining seventy-one parent cyclers have a three-synodic period simple model repeat time. These are sorted according to their simple model Earth hyperbolic excess speed and specific sequence of free-returns. The shorthand names for each of the parent cyclers are extended to uniquely differentiate each one from those presented in Table 4.9 - Table 4.13. The first number is the simple model Earth hyperbolic excess speed, and the last number indicates the simple model repeat time in the synodic periods. The sequence of letters in between indicate the type and order of the free-returns that is an abbreviation of the formal cycler nomenclature developed in Ref. 25. The `+' or ` ' following each `h' indicates whether the half-rev return is above the or below the ecliptic plane. A capital letter indicates the Earth-Mars leg occurs on this Earth-Earth- free return. If the capital letter is an F, a zero or follows to indicate the longitude of the full revolution return, noting that it must be zero or to exist in the ecliptic plane. The last number prior to the `#' is the repeat time of the cycler in synodic periods. The unique number that follows the `#' is also included for ease of reference to a particular parent cycler. 173 Figure 5.5: Total v for optimized ephemeris cyclers. Part I 174 Figure 5.6: Total v for optimized ephemeris cyclers. Part II 175 In general the results are promising. There are nine, thirty-nine, and seventyfour parent cyclers that have at least one launch date that requires a total v of less than 1, 10, and 300 m/s respectively. Several of the most promising cyclers have consistently low v requirements for all launch windows considered. Most notable is cycler 4.991gG2(#83) as seen in Figure 5.5a. Also known as the "S1L1" cycler, this two-synodic period cycler is essentially ballistic for all launch dates. This is consistent with the findings in Ref. 15. Several three-synodic period cyclers, such as those with the prefix 5.658 as seen in Figure 5.6, also exhibit relatively flat lines that hover near a v of zero. The only one-synodic period cycler investigated is cycler 5.399G1(#1) or more commonly known in the literature as the "Aldrin cycler." The results of the true ephemeris versions of the Aldrin cycler are seen in Figure 5.5a. Remarkably, a seven cycle propagation with a launch date of August 6, 2003 (admittedly already passed) is found that is completely ballistic. The previously published true ephemeris (outbound) solution with impulsive maneuvers had a Nov. 1996 launch date with a total v of 1.73 km/s for seven cycles.13 Looking across all launch windows, this appears to be consistent with the magnitudes of v requirements seen in Figure 5.5a. The clear pattern of this extremely simple cycler is evidence that the true ephemeris geometry of the Earth and Mars does indeed repeat on the order of every fifteen years (or more accurately every thirty-two years) as discussed in Section 5.3. This previously unpublished pattern gives clear preference for the launch opportunities that re-occur 176 every fifteen years with v requirements around 0.5 km/s. This repeating pattern is more difficult to identify for the two- and three-synodic cyclers because their structure is much more complicated, and the durations are two or three times longer. A similar less predictable pattern would be expected if the Aldrin cycler were also propagated for fourteen or twenty-one synodic periods. The Aldrin cycler is very favorable in the sense that only one vehicle is required to maintain an outbound Earth-Mars opportunity every synodic period.13 However, it requires powered maneuvers in the circular-coplanar model because of high excess speeds at the planets. Therefore, the discovery of a seven cycle ballistic version of the Aldrin cycler is a significant find. Although the exact ballistic opportunity does not appear to repeat, there is a clear pattern of a preferred trajectory structure. Further investigation of this structure is an excellent candidate for future work. Table 5.5 gives detailed characteristics for the minimum v launch date solutions for each parent cycler documented in Figure 5.5 and Figure 5.6. In general, the average Earth-Mars transit times are inversely related to the excess speeds at the planets. Cycler #23 has the lowest excess speed of any solution, with an average v at Earth and Mars of 4.12 km/s and 4.79 km/s respectively, giving an average transit time of 187 days. One of the most promising high energy cyclers is the two-synodic period cycler 8.049gGf2 (#188). It requires a total v of 420 m/s and has an average transit time of 95 days. The corresponding launch date is July 2042 (allowing plenty of preparation time!). The near-ballistic three-synodic period cyclers are too numerous to 177 discuss individually. Table 5.5, Figure 5.5, and Figure 5.6 demonstrate that ballistic (or very near) three-synodic period cyclers are quite common in the accurate ephemeris model. A variety of launch dates and energy and time characteristics are available to any mission planner willing to accept that short Earth-Mars encounters only occur every three synodic periods. Table 5.5: Characteristics of smallest v solutions for selected parent cyclers Cycler 6.399G1 4.991gG2 8.049gGf2 8.165Gfh-f2 8.165Gfh+f2 9.353Gg2 3.406gGff3 3.406gfGf3 3.418gGff3 3.418gfGf3 3.639gGf3 3.768Gh-3 3.768Gh+3 3.768Gh-f3 3.768Gh-fff3 3.768Gfh-ff3 3.768Gfh+ff3 3.784Gg3 5.125gGgf3 5.301gGff3 5.301gGff3 5.301gGff3 5.301ggFf3 5.301gfGf3 5.301gFgf3 5.301gfGff3 5.219Ggh-3 5.219Ggh+3 5.219Ggfh-f3 5.225Ggg3 5.333gGf3 5.333ggF3 5.333gGff3 5.333gGff3 5.333ggFf3 5.333gfGf3 5.333gFgf3 # 1 83 188 192 193 195 19 20 33 34 49 54 55 56 62 64 65 74 89 94 96 97 98 99 100 102 111 112 113 117 122 123 126 127 128 129 130 total search v space (m/s) (sp) 0 0 420 1678 2612 253 144 5 99 7 20 2 0 0 15 38 7 7 74 210 247 4 185 272 150 206 87 0 0 4 6 0 145 268 201 89 2 2 272 1 11 19 20 19 20 5 5 8 1 20 14 11 8 17 20 17 17 1 11 21 6 9 16 1 9 9 16 16 19 16 19 13 16 10 13 12 15 21 avg. avg. avg. avg. avg. v v v v launch E-M date transit EE+ MM+ Cycler (days) (km/s) (km/s) (km/s) (km/s) Aug-03 Jun-25 Jul-42 Oct-43 Sep-41 Oct-43 Aug-12 Aug-11 Jan-19 Dec-03 Apr-43 Nov-30 Dec-24 Jul-18 May-37 Feb-44 Jul-37 Jul-37 Aug-03 Jul-25 Oct-46 Sep-14 Apr-21 May-36 Feb-02 May-21 Apr-20 Aug-35 Apr-35 Sep-41 Jun-35 Jan-42 Jun-29 Mar-34 Feb-23 Jul-29 Jul-25 Jan-32 Sep-46 143 6.02 165 5.37 95 8.65 109 7.78 107 8.02 101 9.12 191 4.92 175 4.84 187 4.11 222 5.39 227 6.11 209 5.15 177 4.70 183 4.53 200 4.56 173 4.37 195 4.58 195 4.59 176 4.79 118 6.33 114 6.48 120 5.81 113 7.24 130 5.62 122 6.05 114 7.00 114 7.22 119 5.51 117 6.64 128 5.53 124 5.57 117 5.50 153 7.74 156 6.72 151 6.70 154 7.77 116 6.42 140 5.96 149 6.01 6.63 5.37 8.65 7.87 8.40 9.20 4.91 4.84 4.12 5.39 6.11 5.15 5.49 4.99 5.00 5.27 4.70 4.74 5.29 6.33 6.48 5.81 7.24 5.60 6.05 7.00 7.23 6.20 6.79 5.95 5.58 6.36 7.72 6.70 6.72 7.76 6.42 5.96 6.02 9.33 5.48 9.96 9.33 5.48 9.96 5.333gGfff3 5.333gfGff3 5.658Gfh+f3 # 131 132 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 161 162 164 165 166 167 168 169 170 171 172 175 176 177 178 179 180 184 189 total search v space (m/s) (sp) 122 27 4 6 6 30 20 5 15 9 3 9 10 9 15 11 7 7 47 11 10 271 27 5 9 9 14 9 15 10 8 6 1 0 3 5 51 239 13 14 10 13 17 15 14 7 14 14 16 17 1 1 17 7 21 2 11 10 17 7 21 20 17 14 17 10 13 4 7 16 3 9 19 17 20 19 launch date Jul-29 Sep-30 Sep-22 Nov-28 Oct-37 Jun-33 Mar-31 May-16 Mar-31 Mar-31 May-35 Oct-37 Aug-03 Jul-03 Oct-37 Apr-16 Mar-46 Jul-05 Jul-24 Nov-22 Aug-37 Jan-16 Mar-46 Nov-43 Sep-37 Mar-31 Oct-37 Nov-22 Feb-29 Nov-09 Mar-16 Jun-35 Nov-07 Sep-20 Dec-41 Oct-37 Sep-42 Oct-41 avg. avg. avg. avg. avg. v v v v E-M transit EE+ MM+ (days) (km/s) (km/s) (km/s) (km/s) 154 148 111 117 111 143 114 140 113 113 120 112 118 118 112 109 119 113 146 118 110 148 113 119 115 113 112 118 119 110 111 113 113 108 113 109 220 131 7.78 6.68 6.91 6.27 6.50 6.23 6.08 5.96 6.09 6.09 6.42 6.49 6.75 6.74 6.50 6.51 6.25 6.25 6.74 6.84 6.52 6.80 6.05 6.10 6.47 6.07 6.52 6.83 6.44 6.68 6.52 6.58 7.03 7.39 6.82 6.93 6.50 8.26 7.77 6.65 6.72 6.29 7.27 6.23 6.37 6.02 6.42 6.42 6.36 7.43 7.16 7.05 6.96 6.65 6.71 6.25 6.76 7.48 6.52 6.68 6.38 6.17 6.39 6.64 7.31 7.50 7.15 6.63 6.40 6.44 7.33 7.39 7.33 7.35 6.51 7.77 10.20 9.77 9.57 9.52 9.46 9.39 9.58 9.42 9.58 9.57 9.43 9.41 9.44 9.44 9.43 9.57 9.55 9.74 10.14 9.63 9.85 9.82 9.45 9.57 9.42 9.47 9.42 9.62 9.65 9.79 9.61 9.43 9.78 9.96 9.90 9.56 6.20 6.71 10.20 9.77 9.57 9.52 9.46 9.39 9.58 9.42 9.58 9.57 9.43 9.41 9.44 9.44 9.43 9.57 9.55 9.74 10.14 9.63 9.85 9.82 9.45 9.57 9.42 9.47 9.42 9.62 9.66 9.79 9.61 9.43 9.78 9.96 9.90 9.56 6.20 6.71 10.34 10.34 5.658Gfh-f3 10.01 10.01 5.658Gfh-3 10.73 10.72 5.658gFh-3 5.12 5.19 4.81 5.56 5.16 5.16 5.25 4.86 5.03 5.19 4.85 4.84 4.69 9.64 9.59 9.46 9.48 9.44 9.28 9.53 9.46 9.02 9.59 9.54 9.15 9.57 6.24 9.61 9.68 9.40 9.49 5.12 5.19 4.77 5.56 5.16 5.16 5.25 4.86 5.03 5.19 4.85 4.84 4.69 9.64 9.59 9.46 9.48 9.44 9.28 9.53 9.46 9.02 9.59 9.54 9.15 9.57 6.23 9.61 9.68 9.40 9.49 5.658Gfh+3 5.658gFh+3 5.658Gffh-3 5.658Gffh+3 5.658Gfh-f3 5.658Gfh+f3 5.658Gfh-f3 5.658Gfh+f3 5.658Gfh-ff3 5.658Gfh+ff3 5.658Gfh-ff3 5.658gfh-fFpi3 5.658Gfh+ff3 5.658gfh+fFpi3 5.658Gfh-ff3 5.658Gfh-ff3 5.658Gfh+ff3 5.658Gfh+ff3 5.658Gfh-fff3 5.658Gfh+fff3 5.658Gffh-ff3 5.658Gffh+ff3 5.679Gfh-f3 5.679Gfh+f3 5.751ggF3 5.751Ggff3 6.205gG3 7.954Gg3 3.406gGh+fh-3 23 5.658gfh-fFzero3 160 5.658gfh+fFzero3 163 5.219Ggfh+f3 114 10.09 10.09 5.751Ggf3 10.20 10.20 5.751Gfgf3 178 If the average excess speeds before and after a flyby are not identical, either powered maneuvers are included or one of the transits is on the first or last leg. The corresponding v- or v+ does not exist and thus is not included in the average. This is seen more clearly in Appendix C, where the full information for each transit leg is included, rather than just the averages. Also in Appendix C, the full reproducible trajectories for each solution in Table 5.5 are documented. This consists of the information required to reproduce seventy-seven full cycler trajectories. Each one corresponds to the minimum v launch window for the selected parent cycler. Although not documented, the remaining 4186 full cycler solutions that result from applying the algorithm in Figure 5.4 are archived electronically. For an electronic version of Appendix C or the undocumented solutions, contact the author. Table 5.6 gives the average v over all the launch dates for each of the parent cyclers documented in Figure 5.5 and Figure 5.6. The entries are sorted in ascending order. This table gives a metric for the flexibility of each parent cycler with regard to launch opportunities. Impressively, there are twenty different parent cyclers that have an average cost over all twenty-one launch periods of less than 100 m/s per seven cycles. In some cases such as cycler 5.658Gfh+fff3(#170), there are as many as fortynine legs over the course of a forty-five year trajectory. It is quite remarkable that so many solutions exist that satisfy (or nearly satisfy) the constraints for each of the fortyeight flybys. 179 As expected, the general performance of an accurate ephemeris cycler is closely related to the turn ratio of the circular-coplanar parent cycler, where the turn ratio is the ratio of the maximum allowed turning angle to the maximum required turning angle. This is seen by comparing the turn ratios given in Table 4.9 - Table 4.13 with the average v rankings in Figure 5.6. Table 5.6: Solutions sorted by average v over all the launch dates Parent Cycler 5.658Gfh+ff3 5.658Gfh-f3 4.991gG2 5.658Gfh+ff3 5.658Gfh+f3 5.658Gfh-ff3 5.658Gfh-ff3 5.658Gfh+fff3 5.658Gfh-fff3 5.658Gffh+ff3 5.658Gffh-ff3 5.658Gfh+ff3 5.679Gfh+f3 5.333gfGf3 5.658Gfh-f3 5.658Gfh-ff3 5.751Ggf3 5.219Ggfh+f3 5.679Gfh-f3 5.658Gfh+f3 # 162 155 83 158 156 157 159 170 169 172 171 168 176 129 145 166 177 114 175 146 avg. tot. v (km/s) 24 28 29 30 37 37 37 41 41 61 65 69 74 75 82 82 88 91 95 100 Parent Cycler 5.658Gfh-3 5.658Gfh-ff3 5.219Ggfh-f3 5.219Ggh+3 3.768Gfh+ff3 5.658gfh-fFzero3 5.658Gfh+ff3 5.219Ggh-3 5.301gGff3 5.751Ggff3 3.768Gfh-ff3 5.225Ggg3 3.639gGf3 5.658Gffh-3 5.658Gfh+3 5.658Gffh+3 5.658Gfh-f3 5.658Gfh+f3 5.751Gfgf3 # 147 165 113 112 65 160 167 111 96 179 64 117 49 151 149 152 153 154 180 avg. tot. v (m/s) 106 106 108 122 126 129 131 140 141 141 148 153 158 161 166 174 180 207 214 238 Parent Cycler 5.751ggF3 6.205gG3 5.301gFgf3 5.125gGgf3 5.658gFh+3 5.301gGff3 5.333ggFf3 5.333gfGff3 5.658gFh-3 3.768Gh-fff3 5.333gGff3 5.301gfGf3 3.418gfGf3 3.418gGff3 7.954Gg3 3.768Gh-3 3.768Gh+3 5.333gGff3 3.768Gh-f3 5.301ggFf3 # 178 184 100 89 150 97 128 132 148 62 127 99 34 33 189 54 55 126 56 98 avg. tot. v (km/s) 273 315 346 361 382 393 394 403 411 428 455 469 507 526 536 544 551 595 597 687 Parent Cycler 5.301gGff3 9.353Gg2 5.333gFgf3 5.301gfGff3 8.049gGf2 5.333gGfff3 3.784Gg3 5.333gGf3 3.406gGff3 3.406gfGf3 5.333ggF3 5.658gfh-fFpi3 5.658gfh+fFpi3 3.406gGh+fh-3 8.165Gfh-f2 6.399G1 8.165Gfh+f2 # 94 195 130 102 188 131 74 122 19 20 123 161 164 23 192 1 193 avg. tot. v (m/s) 709 714 720 732 818 823 869 871 942 992 1176 1430 1752 2181 2471 3297 3612 5.658gfh+fFzero3 163 Figure 5.7 - Figure 5.10 give two examples of full trajectory plots representative of the solutions documented in Table 5.5. In general, the trajectories are difficult to visualize because of the typical complexity and duration. The example trajectories presented are two of the simplest. The first one is the seven cycle ballistic propagation of the Aldrin cycler, and the second is one of the promising high energy two-synodic period cyclers. Figure 5.8 and Figure 5.10 illustrate the two trajectories plotted in the 180 translating, rotating, and pulsating frame as described in Appendix A. Contrary to the exactly periodic circular-coplanar cyclers seen in Figure 3.10 parts c and d, these trajectories are indeed only quasi-periodic in the accurate ephemeris model. Figure 5.7: Ballistic Aldrin cycler 6.399G1(#1), launch: Aug 6, 2003 Figure 5.8: Aldrin cycler from Figure 5.7 plotted in TRP frame 181 Figure 5.9: High energy cycler 8.049gGf2(#188), launch: July 26, 2042, vtot=420 m/s, avg. transit 95 days Figure 5.10: High energy cycler from Figure 5.9 plotted in TRP frame 182 In Figure 5.7, note that the trajectory begins and ends almost on opposing sides of the sun. Initially, seven cycles of the idealized cycler have a total duration very near the fifteen-year approximate inertial repeat time of Earth and Mars. Because this total duration is not fixed in the optimization procedure, the solution is free to change as necessary. In most cases, the total duration remains in the neighborhood of the original, thus increasing the likelihood that the general pattern is repeatable. However, as seen in Figure 5.7, there are times when the optimizer arrives at a much lower cost by departing from the original total duration. Although these patterns are less likely to be repeatable, it decided for the broad purposes of this survey, to favor the lowest cost cyclers instead. 5.6 CHAPTER CONCLUSIONS A robust and efficient method is presented to optimize long duration ballistic trajectories with multiple flybys. Analytic gradients are derived, and post-processed intermediate maneuvers are optimized to ensure position continuity. The method is applied to a continuation algorithm developed to "walk" several cycles of existing circular-coplanar cycler solutions towards analogous accurate ephemeris solutions. The algorithm is then applied to each of the 203 circular-coplanar parent cyclers presented in Chapter 4 for a variety launch windows. Three main conclusions are drawn. First, efficiency is the main driver in the development of the constraint function and the parameter vector. The resulting system, complete with analytic gradients, is a stand-alone system that could prove useful in the preliminary design of any longduration, multiple-flyby, ballistic, patched conic trajectory. 183 Secondly, the continuation algorithm demonstrates the feasibility of transitioning simple model solutions to accurate ephemeris solutions. Again, the method is applied to several cycles of Earth-Mars cyclers, but could be applied to any similar class trajectory with a known idealized solution. Finally, and most importantly, this study demonstrates the existence of hundreds of so-called "real world" ballistic cyclers. It is a broad survey of ephemeris cyclers that originate from the general class of idealized cyclers that are composed of Earth to Earth free returns. Although several intermediate steps are necessary, such as finding solutions in an idealized model, the ultimate goal of cycler trajectory research, in general, is to obtain flyable cycler trajectories. While the solutions presented in this chapter are still subject to approximations, such as Keplerian motion, and the zerosphere-of-influence flybys, the next step of using the most accurate model possible is expected to be minor compared to the step demonstrated in this chapter. The solutions documented here conservatively represent at least of an order of magnitude increase in the number of known accurate ephemeris cycler trajectories. Many solutions have ballistic (or near-ballistic) launch opportunities every synodic period. In addition to removing the mentioned assumptions, extensions from this work include further customized optimization of individual solutions, such as the Aldrin cycler and the two-synodic period cyclers. On a larger scale, the basic problem could could be reformulated to define new classes of cyclers, such as those that incorporate Venus and or Mars flybys in the original idealized model. 184 6 Conclusions 6.1 DISSERTATION SUMMARY The Earth-Mars cycler problem is approached by first defining the class of cyclers that are of interest. In general, the goal is to find perpetually repeating trajectories that have short inbound and outbound transits between Earth and Mars that can be maintained with gravity-assisted flybys only. The problem is immediately simplified by excluding flybys at other planets such as Venus. Because of the nontrivial inclination and eccentricity of Mars, the relative geometry of Earth and Mars only approximately repeats every fifteen years. This solution space is deemed too large; thus, the problem is simplified further by assuming a circular-coplanar model for the orbits of Earth and Mars. In this model, the repeat time for the geometry is a much more manageable ~15/7 years. Because the mass ratio of Mars to Earth is roughly 1:9, the scope of the problem is narrowed again by assuming that flybys are only useful at Earth instead of Mars. Because this is clearly a preliminary design, the common assumption that trajectories are modeled as patched conics with zero-sphere-ofinfluence flybys is also made. These assumptions narrow the search considerably, but still leave room for an entire class of undiscovered cyclers. This class is defined to consist only of Earth to Earth free-return trajectories. Because of the zero-SOI assumption, the Earth to Mars legs or vice-versa can exist as simple intercepts in the middle of a Earth-Earth free return legs. 185 Following an introduction, the second chapter fully investigates the concept of every conceivable type of free-return trajectory. General definitions for full-rev returns, half-rev returns, and generic returns are proposed that include the possibility for multiple revolutions of the sun. A systematic method is developed to succinctly identify all three types of free-returns that exist with a common hyperbolic excess speed. The solutions are expressed geometrically, and the resulting velocity diagram is a mission-planning tool with potential applications beyond Earth-Mars cycler trajectories. The following two chapters present two different approaches to finding useful combinations of the three types of free-returns that can be patched together to form cyclers. In the first approach, the problem is simplified further to include only cyclers with 1 half-rev returns, 2 full-rev returns, and one or more identical generic returns. This simplification allows the full solution space for total periods of up to six synodic periods to be examined. It results in the confirmation of several previously published cyclers and the discovery of hundreds more. In total, twenty-four previously unpublished ballistic cyclers with repeat times of four synodic periods or less are documented in Chapter 3. Chapter 4 removes the assumptions on the free-returns such that cyclers can be composed of any generalized n return and up to four different or identical generic returns. Combinatorial analysis is used to generate literally trillions of cyclic permutations of free-return trajectories, and the feasible sequences are optimized. Only 186 cyclers with three- or less synodic period repeat times are considered. The method results in the documentation of 203 ballistic or very near-ballistic cyclers. Although most of the cyclers are new, the global search encompasses all of the previously published cycler solutions that fall within its defined class. The final main chapter deals with the obvious next step of finding cyclers in a more accurate solar system. An efficient and robust optimization method, complete with analytic gradients, is developed that could prove useful for trade studies or preliminary design for any long-duration, ballistic, multiple-flyby trajectory. A continuation method is then developed to "walk" the circular-coplanar solutions to accurate ephemeris solutions. At each intermediate step, the optimization method is applied, attempting to reduce any powered requirements to zero. The continuation method is applied to all of the parent cyclers documented in Chapter 4 for a variety of launch dates. This broad survey of accurate ephemeris cyclers results in thirty-nine parent cyclers with at least one launch date that has a total maneuver requirement of less than 10 m/s over seven full cycles. Twenty parent cyclers are found to have an average total requirement over all the launch dates of less than 100 m/s. The results include a ballistic launch opportunity for the Aldrin cycler, several ballistic or near-ballistic promising two-synodic period accurate ephemeris cyclers, and an abundance of ballistic three-synodic period accurate ephemeris cyclers. In general, a broad feasibility for ballistic or near-ballistic accurate ephemeris cyclers is established. 187 6.2 GLOBAL CLAIMS In the realm of optimization, it is somewhat bold to identify a solution as global. In particular it is important to clearly define the solution space if globality is indeed claimed. The word appears in the title of the dissertation primarily in reference to a global search over the entire solution space for a defined class of cyclers. In some instances, global also refers to solutions to specific optimization problems. Table 6.1 gives a summary of any global claims in the context of each main chapter of the dissertation. Table 6.1: Summary of global claims Chapter 2 Documents a method to globally find all free-return solutions to any celestial body in a closed Keplerian orbit Documents a method to globally find all free-return solutions with a particular outgoing v to any celestial body in a closed Keplerian orbit Chapter 3 Globally finds all families of idealized Earth-Mars freereturn cyclers in the defined solution space: -Including all half-year returns -Including all full-year 2 returns -Including all identical generic free-returns -Total repeat times of 6 or less synodic periods -Transits occur on generic returns only. Globally optimizes each family such that the maximum turning angle of the sequence of associated flybys is minimized Chapter 4 Globally finds all families of idealized EarthMars free-return cyclers in the defined solution space: -Having v at Earth less than 10 km/s -Including all possible odd-n freereturns (except no back-to-back 1/2 year solutions) where each set of half-rev points is associated with just one time of flight -Including all possible even-n freereturns where each full-rev circle is associated with just one time of flight -Including up to four identical or nonidentical generic returns -Total repeat times of 3 or less synodic periods -Transits occur on generic or full-rev legs. For each family, a global grid search is performed at small intervals for the entire solution space of the degrees of freedom associated with any full-rev return in the sequence. The best solution is input as a first guess into a local minimax optimizer. Chapter 5 A global search is performed in the sense that accurate ephemeris solutions are sought for all parent cyclers documented in Chap. 4. However, strictly local methods are used to optimize seven cycles of each parent cycler for multiple launch windows. The documented solutions are therefore not global. Further optimization is encouraged. 188 6.3 GENERAL CONCLUSIONS There are two main conclusions based on this work. First, the documented trajectories, both in the circular-coplanar model and in the more accurate ephemeris model, represent significant progress in Earth-Mars cycler research. A global search exhausts all possible solutions for an entire defined class of cyclers in the idealized model, and accurate ephemeris versions of each of the promising parent cyclers are optimized for a variety of launch windows. Hundreds of launch opportunities for ballistic ephemeris cycler solutions result. In general, the global nature of this study frees future researchers to look elsewhere and define new classes of cyclers, at least in terms of the idealized model. Secondly, the overall method, from defining the original problem, to finding idealized solutions, to finding analogous ephemeris solutions, demonstrates the feasibility of a generalized approach to designing cyclers. A similar approach could be used to find cyclers of different classes. In addition, several useful intermediate tools that have stand-alone capability are developed along the way. These include, but are not limited to, a detailed method to obtain all possible free-return orbits, several combinatorial algorithms used to enumerate large sets of numbers, and an efficient optimization method, complete with analytic gradients, for long-duration, multipleflyby, patched-conic, ballistic trajectories. These tools, although developed specifically to aid in the search for Earth-Mars cyclers, could be applied to any number of other astrodynamic or generally scientific applications. 189 6.4 MOST PROMISING CYCLERS Because only one interplanetary transit leg is guaranteed each cycle, the one- and two-synodic period cyclers are clearly favorable over the three- or more synodic period cyclers (that make up a majority of the solutions documented in this dissertation). Therefore, it is generally recommended that future work focuses on these solutions. In the circular-coplanar case, there are only five fully ballistic cyclers found with these repeat times, or six if the near-ballistic Aldrin cycler is included. In the more realistic model, the S1L1 cycler (4.991Gg2), 8.049gGf2, and the Aldrin cycler (6.399G1) debatably have the most promising characteristics. Further optimization of these and the other two-synodic period cyclers documented in Chapters 3 and 4 is suggested. 6.5 FINAL REMARKS This dissertation is not a comparative analysis of Earth-Mars transportation systems. The primary goal is not to sell the idea of cyclers versus any other existing system. Rather, the direction and purpose of this undertaking has been driven more by the idea of extracting the full potential from the cycler concept. That is, to fully investigate the free-return cycler trajectory problem, both from a fresh perspective and in the context of existing work. The ensuing endeavor proved to be an excellent dissertation topic. While there is no lack of supporting literature on Earth-Mars cyclers, the topic had certainly not 190 been exhausted, and still, much work remains. However, it is stated with confidence, that this dissertation documents all potentially useful cyclers in the limited class that is addressed. The general interpretation of cycler trajectories is yet to be determined. One thing is for certain: the cycler concept leads to a heavily constrained optimization problem and there are many clever approaches to solving it. This dissertation represents just a few. The practical side of cyclers is clearly a visionary topic, especially in the context of the current political landscape, the unending debate about the role of humans in space, and the extraordinary funding that is required to support it. However, granted that the timescale is quite debatable, extrapolating the progress of human technology and exploration, it seems appropriate to assume that humans will eventually colonize Mars and perhaps other regions of the solar system. Barring some global catastrophic event, when this vision materializes, accurate ephemeris cycler orbits, like those developed in this dissertation, will be a viable approach to obtain and sustain such a scenario. 191 Appendix A: Dynamics of a Translating, Rotating, and Pulsating (TRP) Reference Frame Figure A1: Translating, Rotating, and Pulsating Reference Frame Define a rotating pqw coordinate frame with the origin at r1 and the p direction unit vector always pointing towards r2. The unit vectors are defined as: ^ p p r2 - r1 = p r2 - r1 ^ q ^ ^ q p k = q p k ^ ^ ^ ^ ^ w p q An arbitrary vector, aijk (where the superscript indicates it is expressed in the ijk frame), can be expressed in the pqw frame using the transformation matrix, R. a pqw = R T aijk ^ ^ ^ where: R = [p | q | w ] 192 Now, define a tuv reference frame with pulsating unit vectors that are scaled by the non-constant distance p such that the location of r2 is always [1,0,0]T. An arbitrary vector, aijk can be expressed in the tuv frame as: atuv = 1 T ijk R a p Alternatively, noting that R is orthogonal and thus RT=R-1, an arbitrary vector, atuv, can be expressed in the ijk frame as: aijk = pRatuv Both the distance p and the transformation matrix R are functions of time. Differentiating twice with respect to the inertial ijk frame, an expression below is obtained. ( i a)ijk = pR ( t a)tuv + pRatuv + pRatuv + 2 pRatuv + 2 pR ( t a)tuv + 2 pR ( t a)tuv The pre-super-script indicates a derivative with respect to that frame. Both sides of the equation above are expressed in the ijk frame. In order to express them in the tuv frame, multiply both sides by RT/p. Below is the general expression for the inertial second derivative of a vector expressed in the tuv frame. The expressions for the first and second derivatives of the transformation matrix are given. 193 2p p 2p I 3 x 3 ( t a)tuv + R T R + R T R + I 3 x 3 atuv ( i a)tuv = ( t a)tuv + 2R T R + p p p where: 2 ^ p = p p-p p p 2 2 3 ^ p = p p - 2p p p + p (2 p - pp ) p 2 ^ q = q q-qq q 2 2 3 ^ q = q q - 2q q q + q (2 q - qq ) q (A1) ^ ^ ^ ^ ^ w = p q + p q ^ ^ ^ ^ ^ ^ ^ w = p q + p q + 2p q ^ ^ ^ R = [p | q | w ] ^ ^ ^ R = [p | q | w ] p = p p/ p T q=q q q T 2 2 p = p p+ | p | - p T ( ) p q = q q+ | q | -q T 2 ( 2 ) q The position vector for the spacecraft expressed in the inertial frame is below. rS / C ijk = r1ijk + S / C ijk Differentiate twice with respect to the inertial frame. i rS / C ijk = i r1ijk + i S / C ijk In order to express both sides in the tuv frame, multiply by RT/p. 1 T i 1 R ( rS / C )ijk = R T ( i r1 )ijk + ( i S / C )tuv p p Now applying Newton's gravitational law, assuming that the 2-body force due to the central body at the origin of the ijk frame is the only force acting, yields: 1 T rS / C ijk 1 T rE ijk i ) = R (- 3 ) + ( S / C )tuv R (- 3 p rS / C p rE Now filling in for the expression derived in Eq. (A1) and rearranging yields the equations of motion for a body expressed in the tuv frame: ( t S / C )tuv = 1 T rE ijk 1 T rS / C ijk T 2p p 2p ) - 2R R + R ( 3) - R ( I 3 x 3 ( t )tuv - RT R + RT R + I 3 x 3 tuv 3 p rE p rS / C p p p ijk ijk tuv where: rS / C = r1 + pR S / C Therefore everything is known to integrate directly in this frame if desirable. Below, Figure A2 is an example of such an integration. It shows the trajectory of a 194 spacecraft that performs a Hohman transfer from an inner planet to an outer planet, then waits at the outer planet for proper alignment, and returns to the inner planet via Hohman transfer. Both planets have circular co-planar orbits. The results have been verified by integrating in the inertial frame. Figure A2: Example Trajectory in the TRP Frame 195 Appendix B: Combinatorics and Generating Algorithms This section provides generating algorithms created to solve the sub-problems discussed in the Combinatorics section. n-Tuples algorithm The algorithm used to find the n-tuples of a set of m elements is recursive. Begin with all 1-tuples, or simply a row vector of all m elements, or X=[1 2 3 4 ... m]. Then, use the following technique to find all k-tuples based on a complete matrix of all (k-1)-tuples. 1. The columns of the current X represent all (k-1)-tuples of the set of m elements 2. For each column of X, add m-j+1 columns to X, the matrix that contains all the k-tuples, where j is the last entry, or the (k-1)th entry, of the active column in X. 3. The first k-1 entries of the m-j+1 new columns in X are set equal to the active column in X. 4. The last entry, or the kth entry, of the ith new column in X is set equal to i-1+j. (for i=1 m-j+1) The pattern is illustrated in the following example that shows the evolution of X (and a selected few X) when calculating all the 4-tuples (n=4) of a set with 3 elements (m=3) is shown below. 196 X = [1 2 3] 1 1 1 2 2 3 X= 1 2 3 2 3 3 1 1 1 X ' = 1 1 1 1 1 1 X ' = 1 1 1 1 2 3 1 1 1 1 1 X ' = 1 1 1 2 2 1 2 3 1 1 1 1 1 X ' = 1 1 1 2 2 1 2 3 2 3 X ' = ... 1 1 1 1 1 1 2 2 2 3 X = 1 1 1 2 2 3 2 2 3 3 1 2 3 2 3 3 2 3 3 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 3 X= 1 1 1 2 2 3 2 2 3 3 1 2 3 2 3 3 2 3 3 3 step 2 and 3 step 4 step 2 and 3 step 4 2 2 2 2 2 2 2 3 2 2 3 3 2 3 3 3 3 3 3 3 Beaded Necklace Problem Below is the algorithm used to generate all the unique combinations to form a necklace composed of n beads, and each bead is one of q colors. A vector x is given with q positive integers. The ith entry of x represents the number of beads in the necklace associated with the ith color. Thus, the necklace has x(1) beads of the 1st color, x(2) beads of 2nd color, ..., and x(q) beads of qth color. 1. Obtain a matrix Y composed of all unique necklaces if every bead were a different color. Each column of Y contains a unique cyclic permutation of the integers 1n, each integer representing a different bead. When n = 1,2, or 3, there is only one unique necklace for each case. Thus, when n=1, Y=[1], when n=2, Y=[1 2]T, and when n=3, Y=[1 2 3]T. For n>3, the following steps, a-f, 197 give a recursive algorithm to find all the unique necklaces with k beads of k colors using all the unique necklaces of k-1 beads of k-1 colors. a. The columns of the current Y represent all unique necklaces of k-1 beads of k-1 colors. b. For each column of Y, add k-1 columns to Y, the matrix that contains all the unique necklaces with k beads of k colors. c. Fill the first entry of the new k-1 columns of Y with the first entry of the active column of Y. d. For the ith new column of Y, fill the (k-i+1)th entry with k. (for i=1k-1) e. For the ith new column of Y, sequentially fill the empty locations with the 2nd through the (k-1)th entry of the active column of Y. (for i=1k-1) 2. Then, assign each value of the integers 1n a color value according to the vector x., and rewrite the matrix, Y, as X with the newly assigned color values. 3. Now, each column of X needs to be checked against all other columns of X to eliminate repeated necklace patterns. Store the unique patterns in X. The pattern is illustrated in the following example that shows the evolution of Y (and a selected few Y), X, and X for a beaded necklace problem with 2 red beads, 2 blue beads, and 1 yellow bead. Thus, x=[2 2 1]T, n=5, q=3. 198 1 Y = 2 3 1 2 Y= 3 4 1 1 1 1 1 1 1 1 4 Y ' = 2 2 Y '= Y '= 3 4 4 4 4 3 step c step d step e t 1 4 2 3 1 1 1 1 1 1 1 1 2 2 2 5 4 4 4 5 4 4 5 2 2 2 5 4 3 5 4 4 3 5 2 2 5 3 3 3 5 3 3 3 assign {1 red, 2 red , 3 blue, 4 blue , 5 yellow} i r r r r r r r r r r r r r r r y r r r y b b b y X '= b b y r b b y r r r y b b y b b b y b b b y r r y b b b y b b b y b b b r r r r r r y b X = b b r r step 3 b y b b y b b y 1 1 2 4 4 2 3 3 1 1 1 1 2 2 2 5 Y = 3 3 5 2 4 5 3 3 5 4 4 4 step 2 step 2 Partitions of an integer The algorithm used to generate the partitions is given in Chapter 9 of Ref. 47. Table A1 lists the useful partitions for this study. 199 Table B1: The partitions of the integers 1 through 11a 1 1 2 2 11 3 3 21 111 4 4 31 221 211 1111 5 5 41 32 311 221 2111 11111 6 6 51 42 411 33 321 3111 222 2211 21111 111111 7 7 61 52 511 43 421 4111 331 322 3211 31111 2221 22111 211111 1111111 8 8 71 62 611 53 521 5111 44 431 422 4211 41111 332 3311 3221 32111 311111 2222 22211 221111 2111111 11111111 9 9 81 72 711 63 621 6111 54 531 522 5211 51111 441 432 4311 4221 42111 411111 333 3321 33111 3222 32211 321111 3111111 22221 222111 2211111 21111111 111111111 10 10 91 82 811 73 721 7111 64 631 622 6211 61111 55 541 532 5311 5221 52111 511111 442 4411 433 4321 43111 4222 42211 421111 4111111 3331 3322 33211 331111 32221 322111 3211111 31111111 22222 222211 2221111 22111111 211111111 1111111111 11 11 10 1 92 911 83 821 8111 74 731 722 7211 71111 65 641 632 6311 6221 62111 611111 551 542 5411 533 5321 53111 5222 52211 521111 5111111 443 4421 44111 4331 4322 43211 431111 42221 422111 4211111 41111111 3332 33311 33221 332111 3311111 32222 322211 3221111 32111111 311111111 222221 2222111 22211111 221111111 2111111111 11111111111 a Generated using the algorithm in Ref 47. 200 Appendix C: Cycler Trajectories Using a Ephemeris Model This appendix documents the full trajectories that are summarized in Table 5.5. The solutions are found using the method described in Section 5.4. As Section 5.5 states, optimized solutions were obtained for twenty-one launch windows for each of the 203 cyclers, totaling 4263 trajectories. documenting all the trajectories. Clearly, space and interest prohibit The 77 presented here represent the lowest v solutions of the twenty-one launch date periods for a selected set of the parent cyclers. The criteria for the selected cyclers is described in detail in Section 5.5. For an electronic version of this file or a similar file including all 4263 solutions, please contact the author. The format for the solutions is discussed below. Each trajectory begins with a separator line annotated with the shorthand descriptor of the circular-coplanar parent cycler. The "Cycler number" is uniquely associated with the shorthand descriptor, and is primarily used for debugging purposes. Its value ranges from 1-203, indicating one of the 203 circular-coplanar parent cyclers evaluated in Chapter 5. The approximate search space indicates which launch window was used to initiate the search. Namely, a value of i, indicates that the search was initiated with an epoch date between i and i+1 synodic periods following J2000. The total v, given in meters per second, includes the necessary maneuvers at the sphere-ofinfluence of every flyby and intermediate maneuvers. The transit leg information is simply a summary of important solution characteristics. This information, along with 201 the total v is an artifact of the trajectory. The "Epoch time" and the detailed leg information are the only values required to reproduce the solution. described as follows. The "E/M" column indicates the planet where the ith leg is initiated. The "time start" column gives the time (relative to the epoch) of the beginning of the leg. The planet's velocity at this time is found using the DE4056 ephemeris. The spacecraft's velocity at the beginning of the leg is found by adding the planet's velocity to the given v. The spacecraft trajectory is propagated using Kepler's equation until the given "time dv," when the given v is applied. The trajectory is propagated further until the beginning time of the next leg, where it encounters the next planet. The DE4056 ephemeris gives positions and velocities referenced to the Earth's mean equator and equinox of J2000. These positions and velocities are then rotated by the mean obliquity angle of 0.409092629205 radians as given on page 289 of Ref. 5. Thus, all velocities are presented relative to the ecliptic plane. Each leg is Figure C1: Solution diagram for ith leg 202 Below are the solutions: ================ PARENT CYCLER 6.399G1 ======================= Parent cycler number 1 Approximate search space (synodic periods after J2000) 1 Number of steps to walk eccentricity/inclination 243 / 243 Number of cycles 7 Total delta v over 14.56 years (km/s) 0.000000 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 1 67.5 ****** 10.246 8.479 8.479 3 140.8 6.225 6.225 7.278 7.278 5 167.1 6.226 6.226 7.431 7.431 7 169.9 5.929 5.929 8.663 8.663 9 163.0 5.824 5.824 10.149 10.149 11 150.5 5.881 5.881 11.494 11.494 13 139.4 6.044 6.044 11.806 11.806 AVERAGE 142.6 6.022 6.625 9.329 9.329 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 1222.895303 LEG E/M time start vinfx vinfy vinfz (days) (km/s) (km/s) (km/s) 1 E 0.913719539E+02 0.834261459E+01 -0.587278061E+01 -0.939815362E+00 2 M 0.158835048E+03 0.839462641E+01 -0.677475548E+00 -0.985257408E+00 3 E 0.799506629E+03 0.310668622E+01 0.538933590E+01 0.238389920E+00 4 M 0.940331775E+03 0.462032797E+01 0.557723871E+01 -0.722145508E+00 5 E 0.157503895E+04 -0.131365147E+01 0.587628205E+01 0.158336380E+01 6 M 0.174214937E+04 -0.228236193E+01 0.706956842E+01 -0.180335036E+00 7 E 0.234618571E+04 -0.414579329E+01 0.390989992E+01 0.163792698E+01 8 M 0.251612060E+04 -0.730388434E+01 0.464892090E+01 0.289958457E+00 9 E 0.311108949E+04 -0.564474866E+01 0.908224118E+00 0.110764964E+01 10 M 0.327409445E+04 -0.101269501E+02 0.880345454E-01 0.667119586E+00 11 E 0.387716584E+04 -0.511969238E+01 -0.287678333E+01 0.317199932E+00 12 M 0.402768578E+04 -0.984704095E+01 -0.587342786E+01 0.807700399E+00 13 E 0.464740236E+04 -0.148757112E+01 -0.582758220E+01 -0.592734464E+00 14 M 0.478684900E+04 -0.434960247E+01 -0.109596582E+02 0.581111658E+00 15 E 0.541113136E+04 ================ PARENT CYCLER 4.991gG2 ======================= Parent cycler number 83 Approximate search space (synodic periods after J2000) 11 Number of steps to walk eccentricity/inclination 1 / 1 Number of cycles 7 Total delta v over 30.07 years (km/s) 0.000000 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 2 179.8 5.818 5.818 5.248 5.248 5 125.5 5.764 5.764 7.693 7.693 8 138.4 3.752 3.752 4.657 4.657 11 210.9 5.532 5.532 3.198 3.198 14 154.6 6.946 6.946 6.263 6.263 17 112.2 5.121 5.121 8.046 8.046 20 230.2 4.646 4.646 3.219 3.219 AVERAGE 164.5 5.368 5.368 5.475 5.475 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 9325.742435 LEG E/M time start vinfx vinfy vinfz (days) (km/s) (km/s) (km/s) 1 E -0.174074986E+02 -0.238703220E+00 0.579754982E+01 -0.176632468E-02 2 E 0.519582315E+03 -0.227826290E+01 0.532198991E+01 0.573813715E+00 3 M 0.699393400E+03 -0.499623059E+01 0.707421771E+00 0.144166283E+01 4 E 0.152997105E+04 -0.575383361E+01 0.386418107E+00 0.628610087E-04 5 E 0.206677179E+04 -0.550681663E+01 -0.160954336E+01 -0.550229544E+00 6 M 0.219229367E+04 -0.355470987E+01 -0.672927795E+01 0.112658626E+01 7 E 0.310727040E+04 0.130066966E+01 -0.353985845E+01 0.127838602E-02 8 E 0.363605992E+04 0.372962888E+01 -0.150726402E-01 -0.411042515E+00 9 M 0.377449899E+04 0.463453129E+01 -0.454933848E+00 -0.467039188E-01 10 E 0.468119309E+04 0.398651382E+01 0.380698910E+01 0.599174579E-03 11 E 0.521702941E+04 0.195647577E+01 0.500455176E+01 0.131545295E+01 12 M 0.542793047E+04 -0.106091993E+01 0.286375761E+01 0.950243209E+00 13 E 0.621934578E+04 -0.346922206E+01 0.601287549E+01 -0.294263108E-02 14 E 0.676095373E+04 -0.494762984E+01 0.487023018E+01 0.229687664E+00 15 M 0.691560104E+04 -0.580355995E+01 -0.130028398E+01 0.196236847E+01 16 E 0.777160296E+04 -0.455750356E+01 -0.237566423E+01 -0.883416315E-03 17 E 0.830586489E+04 -0.289917665E+01 -0.412851216E+01 -0.880976268E+00 18 M 0.841803274E+04 -0.474724840E+00 -0.793519915E+01 0.124113571E+01 19 E 0.933942338E+04 0.322773002E+01 -0.335738552E+01 0.125353647E-02 20 E 0.987176523E+04 0.254872658E+01 0.229406465E+01 0.313437961E+01 21 M 0.101019833E+05 0.212149789E+01 0.424849528E+00 -0.238347620E+01 22 E 0.109670202E+05 ================ PARENT CYCLER 8.049gGf2 ======================= Parent cycler number 188 Approximate search space (synodic periods after J2000) 19 Number of steps to walk eccentricity/inclination 243 / 243 Number of cycles 7 Total delta v over 29.95 years (km/s) 0.436091 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 2 96.2 11.685 11.685 11.244 11.244 6 76.7 8.689 8.689 11.705 11.705 10 102.1 6.089 6.089 8.522 8.522 14 108.1 9.502 9.502 10.069 10.069 18 91.4 11.088 11.088 11.612 11.612 22 75.5 7.081 7.081 10.765 10.765 time dv (days) 0.101491418E+03 0.254935785E+03 0.820630401E+03 0.103553785E+04 0.160010552E+04 0.184483555E+04 0.237167595E+04 0.260536593E+04 0.313554023E+04 0.336455515E+04 0.389974383E+04 0.412064327E+04 0.466831935E+04 0.488049135E+04 dvx dvy dxz (km/s) (km/s) (km/s) -0.757778282E-10 -0.438397685E-09 -0.134807511E-10 -0.102013702E-11 0.108792571E-10 0.210204338E-11 -0.138102413E-08 -0.568221282E-09 -0.345867992E-10 -0.224826957E-10 0.324616860E-10 0.149618202E-11 0.383772324E-08 0.765410296E-09 0.134760597E-09 -0.848185239E-12 0.299427577E-11 0.341604332E-11 0.226096755E-09 0.591934094E-09 0.195646822E-10 0.214195707E-11 0.294632767E-11 0.358691007E-11 0.591580270E-10 0.193025797E-09 -0.251334812E-11 0.774692386E-11 0.195595154E-10 0.133198047E-11 -0.295492526E-10 0.141486557E-09 -0.409057172E-11 -0.294632767E-11 0.102708123E-10 0.283413017E-12 -0.138087202E-09 0.681854956E-10 0.328330770E-11 -0.398928137E-10 0.333966738E-10 0.973196459E-12 time dv (days) 0.240347612E+03 0.564535086E+03 0.111468222E+04 0.178763540E+04 0.208560007E+04 0.244848716E+04 0.336108937E+04 0.365682578E+04 0.426411381E+04 0.493839453E+04 0.524866457E+04 0.557829938E+04 0.647390151E+04 0.678415083E+04 0.750624237E+04 0.802804869E+04 0.832269007E+04 0.890636978E+04 0.960027089E+04 0.990629794E+04 0.105258514E+05 dvx (km/s) -0.246767343E-11 -0.727488314E-13 -0.968220811E-11 0.276478627E-10 0.410319943E-10 0.240567159E-11 -0.279214975E-11 -0.370357687E-12 -0.104973257E-10 0.215667218E-10 0.177176472E-10 0.652424747E-11 0.694255325E-11 0.306801663E-10 -0.180880049E-11 0.972188929E-11 0.425084649E-11 0.642835129E-11 0.866703124E-11 0.155550229E-10 -0.207499508E-11 dvy (km/s) 0.902746863E-12 0.229820172E-12 0.177011134E-10 -0.757910553E-11 -0.107006918E-10 -0.249330086E-11 -0.115736777E-12 -0.279917663E-11 0.235441673E-11 0.108875240E-10 0.241691459E-10 0.728315006E-11 -0.189742179E-10 -0.483515188E-10 -0.783703320E-12 0.213121008E-11 -0.139214809E-11 0.118613663E-10 -0.120333181E-10 -0.473859434E-11 -0.348202361E-11 dxz (km/s) 0.301369564E-10 -0.412405272E-11 0.106068623E-10 0.224923555E-11 0.369882340E-11 0.698061980E-11 -0.298533208E-10 0.124055358E-11 -0.311497269E-11 0.209391049E-12 -0.128901836E-11 -0.625360942E-11 -0.347896187E-11 -0.203554640E-11 -0.101848364E-11 -0.530653406E-11 0.707337715E-12 -0.129874748E-10 -0.195906858E-10 -0.870795245E-11 0.729177865E-11 203 26 112.5 6.439 6.439 8.425 8.425 AVERAGE 94.7 8.653 8.653 10.335 10.335 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 15567.931545 LEG E/M time start vinfx vinfy vinfz time dv dvx dvy dxz (days) (km/s) (km/s) (km/s) (days) (km/s) (km/s) (km/s) 1 E -0.215814313E+02 -0.686220461E+01 0.943299852E+01 -0.133901350E-02 0.247738521E+03 -0.314952839E-10 0.508415129E-10 0.113031458E-10 2 E 0.539501803E+03 -0.977613095E+01 0.637258366E+01 0.593332504E+00 0.553930798E+03 0.191703091E-09 0.314400609E-09 0.258981706E-10 3 M 0.635695107E+03 -0.110669417E+02 -0.192220229E+01 -0.510376717E+00 0.726121716E+03 -0.137230750E-11 -0.327766553E-10 -0.209752242E-10 4 E 0.116761634E+04 -0.640028065E+01 -0.587067019E+01 -0.271176934E+00 0.140137157E+04 -0.169094176E-07 0.553258335E-08 0.346887151E-05 5 E 0.153285889E+04 -0.716169990E+01 -0.493035001E+01 -0.261389264E-02 0.177429683E+04 -0.325418750E-10 -0.607452743E-11 -0.733573053E-12 6 E 0.208158148E+04 -0.442527052E+01 -0.742368055E+01 -0.901218506E+00 0.209309238E+04 -0.154419315E-09 -0.295062647E-10 0.379244618E-12 7 M 0.215832084E+04 -0.294368226E+00 -0.116847671E+02 0.617971791E+00 0.224696412E+04 -0.508348994E-10 0.183095582E-10 0.221232917E-11 8 E 0.274927606E+04 0.527829816E+01 -0.178596893E+01 0.246550763E+01 0.300131850E+04 0.854999355E-05 -0.361705396E-04 -0.837469343E-03 9 E 0.311455495E+04 0.540096030E+01 -0.283063892E+01 0.148055420E-02 0.338901420E+04 0.103733220E-10 -0.595548388E-11 0.194776334E-11 10 E 0.365271034E+04 0.447580351E+01 0.412847266E+01 0.407652954E-01 0.366802570E+04 0.133871077E-09 -0.115650801E-09 -0.471398736E-11 11 M 0.375481270E+04 0.664118892E+01 0.527066640E+01 -0.858435743E+00 0.383815060E+04 0.620349126E-11 -0.385304265E-10 -0.632201812E-11 12 E 0.431039870E+04 0.120341614E+01 0.935217422E+01 0.119325811E+01 0.454416535E+04 0.374633848E-05 0.168821458E-05 -0.187880800E-03 13 E 0.467565909E+04 0.258295956E+01 0.913055053E+01 0.377177379E-02 0.491855809E+04 -0.377434165E-10 -0.557057643E-10 0.855138246E-12 14 E 0.522770227E+04 -0.199822502E+01 0.922154981E+01 0.112556115E+01 0.524391356E+04 0.386560836E-10 -0.571319721E-09 0.414833677E-11 15 M 0.533577755E+04 -0.719639155E+01 0.653951981E+01 0.261396054E+01 0.561687807E+04 0.352545758E-02 0.260356785E-01 0.342452071E+00 16 E 0.584686941E+04 -0.102058981E+02 0.425279807E+01 0.742192448E+00 0.606967870E+04 0.248194632E-06 -0.183087497E-05 0.133166221E-03 17 E 0.621213054E+04 -0.949165378E+01 0.569381648E+01 -0.111286407E-02 0.644675101E+04 0.455080315E-09 -0.552752235E-09 -0.188194168E-11 18 E 0.677075071E+04 -0.109020320E+02 0.201198205E+01 0.196851251E+00 0.678445900E+04 -0.545979980E-09 -0.431994135E-08 -0.346580057E-09 19 M 0.686213934E+04 -0.983689346E+01 -0.613987917E+01 0.612910911E+00 0.698918729E+04 0.361098745E-11 0.902085510E-11 -0.407589795E-11 20 E 0.741452172E+04 -0.140224597E+01 -0.691615025E+01 -0.651351913E+00 0.765925617E+04 0.126995358E-06 0.459286355E-07 -0.116222772E-04 21 E 0.777979701E+04 -0.240009873E+01 -0.666911450E+01 0.909572774E-03 0.800751231E+04 0.117231435E-09 0.101195278E-09 0.798102240E-11 22 E 0.832197628E+04 0.193311293E+01 -0.670258718E+01 -0.121590198E+01 0.833330383E+04 -0.554736294E-09 -0.174608769E-09 -0.306892599E-10 23 M 0.839749327E+04 0.568618079E+01 -0.913833239E+01 0.198941193E+00 0.848728773E+04 -0.705663665E-11 0.249660762E-11 -0.489153222E-11 24 E 0.899612303E+04 0.610880025E+01 0.199544212E+01 -0.335866964E+00 0.924448848E+04 -0.582932951E-06 0.466007410E-06 -0.124238527E-03 25 E 0.936136634E+04 0.633042698E+01 0.111749571E+01 -0.184004786E-02 0.962033609E+04 0.499751404E-10 -0.354402546E-11 -0.136285712E-11 26 E 0.990088665E+04 0.276463368E+01 0.576484300E+01 0.761426327E+00 0.991776828E+04 0.793466544E-10 0.142521574E-10 -0.299324240E-11 27 M 0.100134309E+05 0.174125530E+01 0.792187044E+01 -0.227966635E+01 0.101966079E+05 0.504281672E-11 -0.710293136E-11 -0.293888745E-11 28 E 0.105521869E+05 -0.385930219E+01 0.834630649E+01 0.326150615E+01 0.107859567E+05 -0.291378851E-05 -0.429281669E-05 0.871638207E-04 29 E 0.109174522E+05 ================ PARENT CYCLER 8.165Gfh-f2 ======================= Parent cycler number 192 Approximate search space (synodic periods after J2000) 20 Number of steps to walk eccentricity/inclination 9 / 9 Number of cycles 7 Total delta v over 30.36 years (km/s) 1.677496 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 1 178.5 ****** 8.180 10.607 10.607 6 81.4 7.626 7.626 11.856 11.856 11 99.1 6.231 6.231 8.359 8.359 16 111.3 9.471 9.472 9.556 9.556 21 95.3 9.574 9.835 11.667 11.667 26 75.1 7.123 7.123 10.874 10.874 31 121.3 6.646 6.646 6.795 6.795 AVERAGE 108.9 7.779 7.873 9.959 9.959 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 16114.706442 LEG E/M time start vinfx vinfy vinfz time dv dvx dvy dxz (days) (km/s) (km/s) (km/s) (days) (km/s) (km/s) (km/s) 1 E -0.118341070E+03 -0.776652940E+01 0.104360865E+01 0.234580458E+01 -0.915653486E+02 -0.368386855E-09 -0.414311209E-09 -0.196223439E-10 2 M 0.601637364E+02 -0.105977858E+02 0.361455507E+00 -0.249654378E+00 0.422470588E+03 0.248575481E-03 -0.166932604E-03 0.793744065E-02 3 E 0.617558893E+03 -0.523317048E+01 -0.428226388E+01 -0.362177109E+01 0.862272817E+03 0.426024218E-04 -0.158347365E-04 -0.703260362E-03 4 E 0.982803556E+03 0.486605302E+00 -0.863697137E+00 -0.762202740E+01 0.100964107E+04 -0.473680869E-09 0.101938639E-08 -0.241711299E-09 5 E 0.116172031E+04 -0.584834758E+01 -0.213454766E+01 0.437989883E+01 0.128225622E+04 -0.307913680E-04 0.857852898E-04 0.245676729E-03 6 E 0.152698063E+04 -0.358320783E+01 -0.667744936E+01 -0.849147109E+00 0.153918699E+04 0.445089255E-08 -0.211834313E-07 0.527855911E-09 7 M 0.160835635E+04 -0.702656511E+00 -0.118152358E+02 0.680191080E+00 0.169662079E+04 0.318970558E-10 -0.144654438E-10 0.111603321E-11 8 E 0.219678597E+04 0.388150757E+01 -0.152027835E+01 -0.483733297E+01 0.239402190E+04 0.111616730E-02 -0.485230585E-03 0.272941312E-03 9 E 0.256203770E+04 0.326771043E+00 0.586216647E+00 -0.633912387E+01 0.258985618E+04 -0.109000955E-07 -0.171172809E-07 -0.435492444E-08 10 E 0.274749425E+04 0.488325769E+01 -0.350324377E+01 0.163170248E+01 0.286437963E+04 -0.616969723E-05 -0.856922449E-05 0.108924551E-03 11 E 0.311276105E+04 0.491306178E+01 0.383178059E+01 0.843339928E-01 0.312761920E+04 -0.424420684E-07 0.385607033E-07 -0.361801309E-08 12 M 0.321181536E+04 0.633697974E+01 0.543153340E+01 -0.463629268E+00 0.355582567E+04 0.100165795E-02 0.109132921E-03 0.129168049E+00 13 E 0.376667070E+04 0.253221474E+00 0.708980874E+01 -0.582734839E+01 0.397852027E+04 -0.101385847E-03 0.623486707E-03 0.140859196E-04 14 E 0.413192857E+04 -0.138645922E+01 0.323812987E+00 -0.906772276E+01 0.415958333E+04 0.159349702E-09 0.985435830E-09 -0.786216462E-10 15 E 0.431629360E+04 0.299990121E+01 0.644120252E+01 0.624389354E+01 0.446604913E+04 0.129088877E-02 -0.644338573E-04 -0.182838676E-03 16 E 0.468155099E+04 -0.176404172E+01 0.924029643E+01 0.110275877E+01 0.469824385E+04 0.234353416E-07 -0.109730748E-06 -0.419092378E-09 17 M 0.479283672E+04 -0.737412512E+01 0.599758478E+01 0.984229653E+00 0.508620660E+04 0.214869079E-02 0.817308165E-02 0.514513444E+00 18 E 0.530752072E+04 -0.690675347E+01 0.175047953E+01 -0.608787555E+01 0.549745662E+04 0.713764026E-04 0.751925203E-04 0.847517346E-04 19 E 0.567278208E+04 -0.683369119E+00 -0.131973701E+01 -0.926457865E+01 0.575722568E+04 -0.435264916E-06 0.251917616E-06 0.363727897E-05 20 E 0.585244932E+04 -0.548136907E+01 0.460268341E+01 0.633708866E+01 0.599124626E+04 0.559253754E-03 0.119821051E-02 -0.121839659E-03 21 E 0.621770442E+04 -0.968537660E+01 0.169660829E+01 0.216873051E+00 0.623200621E+04 -0.193007848E-06 0.173599081E-07 0.601720793E-08 22 M 0.631304973E+04 -0.100187302E+02 -0.595162193E+01 0.573052933E+00 0.641225116E+04 -0.959656290E-10 0.182850881E-09 -0.254171397E-11 23 E 0.686416880E+04 -0.109267726E+01 -0.514154702E+01 -0.511950094E+01 0.710889837E+04 -0.877215913E-05 -0.630908647E-03 0.106915796E-02 24 E 0.722943682E+04 0.820045321E+00 -0.329009544E+00 -0.727422106E+01 0.725649324E+04 -0.919804149E-08 0.102894872E-08 -0.227598274E-08 25 E 0.740981297E+04 -0.294474881E+01 -0.475821023E+01 0.442060633E+01 0.752669655E+04 -0.312221031E-04 0.163839136E-04 0.221006722E-03 26 E 0.777507418E+04 0.197467747E+01 -0.673431904E+01 -0.121948337E+01 0.778633563E+04 -0.121673942E-07 -0.927388876E-09 -0.996284913E-09 27 M 0.785015052E+04 0.566090058E+01 -0.928037069E+01 0.251557026E+00 0.793953229E+04 -0.198405904E-12 -0.288349914E-11 0.556838570E-11 28 E 0.844602899E+04 0.430964624E+01 0.140038597E+01 -0.489399160E+01 0.868710077E+04 -0.756865752E-03 -0.309597639E-03 -0.850720436E-04 29 E 0.881128926E+04 -0.117736144E+00 0.747899649E+00 -0.663404891E+01 0.886164404E+04 -0.391271455E-07 0.196666049E-07 0.588314013E-08 30 E 0.899778842E+04 0.563010684E+01 0.295576619E+00 0.355243776E+01 0.911101522E+04 -0.618254890E-05 0.767596609E-04 -0.577820269E-03 31 E 0.936303618E+04 0.390936035E+01 0.530666105E+01 0.851636561E+00 0.938123708E+04 0.476513758E-07 0.444872464E-07 -0.532394549E-09 32 M 0.948437553E+04 -0.459285568E-01 0.676530292E+01 -0.627611643E+00 0.959995466E+04 -0.173357159E-10 -0.148820962E-10 0.588707518E-12 33 E 0.100622712E+05 -0.504453923E+01 0.134946798E+01 -0.230144516E+00 0.103106583E+05 -0.261243213E-06 0.821613821E-06 0.201531072E-03 34 E 0.104275464E+05 -0.170260265E+00 -0.425378002E+00 -0.522569766E+01 0.104598411E+05 -0.391091104E-10 -0.282232398E-10 0.573715472E-10 35 E 0.106069617E+05 -0.463455451E+01 0.234814522E+01 0.125165921E+01 0.107201927E+05 0.110210816E-06 0.229946431E-06 0.582576560E-05 36 E 0.109722230E+05 ================ PARENT CYCLER 8.165Gfh+f2 ======================= Parent cycler number 193 Approximate search space (synodic periods after J2000) 19 Number of steps to walk eccentricity/inclination 27 / 27 Number of cycles 7 Total delta v over 30.34 years (km/s) 2.611810 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 1 185.4 ****** 9.455 9.593 9.593 6 91.6 9.264 9.276 11.903 11.903 11 76.6 6.612 6.612 10.100 10.100 204 16 112.7 7.553 7.553 8.480 8.480 21 102.0 9.818 11.036 10.763 10.763 26 82.5 9.035 9.035 11.905 11.905 31 96.1 5.838 5.836 7.296 7.296 AVERAGE 106.7 8.020 8.400 10.006 10.006 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 15334.742414 LEG E/M time start vinfx vinfy vinfz time dv dvx dvy dxz (days) (km/s) (km/s) (km/s) (days) (km/s) (km/s) (km/s) 1 E -0.109293925E+03 -0.601422557E+01 0.560015566E+01 0.467494502E+01 -0.684983498E+02 -0.108015911E-06 0.123471374E-06 0.135229409E-07 2 M 0.761405074E+02 -0.741184072E+01 0.578377782E+01 -0.190640136E+01 0.218452378E+03 -0.391045554E-08 0.750886984E-09 0.155337335E-08 3 E 0.623493856E+03 -0.618013355E+01 -0.143201123E+00 0.668894724E+01 0.835342347E+03 -0.106468262E-02 -0.171272529E-03 -0.776183382E-05 4 E 0.988749875E+03 -0.311892453E+00 -0.135911560E+01 0.901166369E+01 0.101561944E+04 -0.279896805E-05 -0.126121335E-05 -0.590127165E-06 5 E 0.116788034E+04 -0.644283246E+01 0.301415602E+01 -0.591684642E+01 0.130667884E+04 0.437281876E-04 0.217085882E-03 0.345238400E-04 6 E 0.153313954E+04 -0.921098244E+01 -0.109781835E+01 -0.598169926E-01 0.154687753E+04 -0.218677509E-05 0.127620153E-06 -0.107993663E-06 7 M 0.162472611E+04 -0.841981587E+01 -0.837617152E+01 0.791925164E+00 0.170918215E+04 0.809383658E-09 -0.162733514E-08 -0.256050963E-09 8 E 0.218776638E+04 0.823072086E+00 -0.502943691E+01 0.451057342E+01 0.243980567E+04 -0.516327101E-04 -0.415469871E-04 -0.818866197E-03 9 E 0.255304071E+04 0.756617959E+00 0.302442841E-02 0.677225511E+01 0.258030716E+04 0.107222836E-06 -0.160041847E-06 -0.243637449E-07 10 E 0.273481709E+04 -0.727807167E+00 -0.453906366E+01 -0.473327360E+01 0.290283265E+04 -0.694023234E-03 -0.423968289E-03 0.117503032E-03 11 E 0.310006831E+04 0.514700447E+01 -0.398173654E+01 -0.116915906E+01 0.311155429E+04 0.941115985E-07 -0.147131785E-06 0.126739350E-07 12 M 0.317664151E+04 0.845599483E+01 -0.552250671E+01 0.302345236E-01 0.326513623E+04 0.277183795E-09 0.797062651E-10 -0.352178746E-10 13 E 0.376660630E+04 0.516343056E+01 0.379378963E+01 0.391928826E+01 0.401497107E+04 -0.640577289E-04 0.264442735E-04 0.981185432E-03 14 E 0.413184860E+04 -0.464144868E+00 0.844331441E+00 0.744058547E+01 0.415980265E+04 0.121646628E-08 0.157819000E-08 -0.127816390E-09 15 E 0.431820894E+04 0.621972758E+01 0.245136853E+01 -0.355235848E+01 0.443509587E+04 0.297008181E-04 -0.681263561E-04 -0.586433198E-03 16 E 0.468348062E+04 0.243486745E+01 0.707723857E+01 0.101387970E+01 0.470038079E+04 -0.264144393E-09 0.171266291E-08 -0.784771819E-10 17 M 0.479614844E+04 -0.138657238E+01 0.823444662E+01 -0.147594926E+01 0.510658008E+04 -0.826304234E-02 -0.147686272E-01 0.335206140E+00 18 E 0.532230376E+04 -0.467399761E+01 0.485734107E+01 0.707301068E+01 0.554511137E+04 -0.228694623E-04 0.706329058E-04 -0.254284145E-04 19 E 0.568756214E+04 -0.142233944E+01 -0.797606327E+00 0.964995959E+01 0.571479126E+04 0.113237127E-07 0.160943562E-08 0.367534206E-08 20 E 0.586908962E+04 -0.211870256E+01 0.700818414E+01 -0.652096779E+01 0.598962044E+04 -0.126827103E-03 -0.560228335E-04 0.572042260E-03 21 E 0.623433453E+04 -0.754240698E+01 0.801692031E+01 0.789535064E+00 0.624963096E+04 0.856173061E-08 0.443333859E-08 -0.523943388E-09 22 M 0.633631074E+04 -0.107037063E+02 0.675832476E+00 -0.898825796E+00 0.665232937E+04 0.471185244E-02 -0.480283990E-02 0.111843048E+00 23 E 0.686300846E+04 -0.621191332E+01 -0.341877457E+01 0.560214783E+01 0.710406962E+04 0.859942639E-04 -0.575121838E-04 0.100259683E-02 24 E 0.722825264E+04 0.392470127E+00 -0.132336126E+01 0.893764052E+01 0.728545553E+04 0.238810274E-07 -0.289131385E-08 -0.300073553E-08 25 E 0.740701167E+04 -0.699603210E+01 -0.557313796E+00 -0.567264990E+01 0.753119457E+04 0.314025036E-04 -0.162985431E-03 0.778703447E-03 26 E 0.777225551E+04 -0.727954000E+01 -0.532188091E+01 -0.570507111E+00 0.778462765E+04 0.192864477E-07 -0.329323973E-07 -0.223208936E-08 27 M 0.785473645E+04 -0.401434262E+01 -0.111814454E+02 0.769990505E+00 0.808234484E+04 0.173489429E-10 -0.422505372E-10 -0.767128161E-11 28 E 0.843834770E+04 0.426715316E+01 -0.384336056E+01 0.184607415E+01 0.869037737E+04 -0.207765296E-05 -0.121441329E-04 -0.375020076E-03 29 E 0.880360809E+04 0.435514641E+00 0.412783304E+00 0.598193279E+01 0.883128677E+04 -0.378566732E-08 -0.143685556E-08 0.811697567E-09 30 E 0.898813258E+04 0.124847557E+01 -0.218087470E+01 -0.526884947E+01 0.924380875E+04 0.157672846E-03 -0.925898399E-03 -0.392229492E-03 31 E 0.935338424E+04 0.577305740E+01 0.787593732E+00 -0.338070444E+00 0.936780317E+04 -0.176617629E-09 -0.355898196E-08 0.253006279E-10 32 M 0.944951040E+04 0.668456638E+01 0.280943472E+01 -0.811307796E+00 0.954131736E+04 0.341258155E-11 -0.139843095E-10 0.445274000E-11 33 E 0.100615568E+05 -0.228456343E+01 0.462909777E+01 0.127041740E+00 0.103135772E+05 -0.672493852E-07 -0.116182337E-06 0.623569675E-04 34 E 0.104268037E+05 -0.420477294E+00 -0.172145988E+00 0.512387138E+01 0.104541199E+05 0.635361840E-10 0.908699040E-11 0.495626215E-10 35 E 0.106089115E+05 -0.142997816E+01 0.502537239E+01 -0.977461477E+00 0.107221410E+05 -0.375297786E-05 -0.109049248E-05 0.117835491E-03 36 E 0.109741680E+05 ================ PARENT CYCLER 9.353Gg2 ======================= Parent cycler number 195 Approximate search space (synodic periods after J2000) 20 Number of steps to walk eccentricity/inclination 243 / 243 Number of cycles 7 Total delta v over 30.26 years (km/s) 0.427585 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 1 183.4 ****** 9.655 10.837 10.837 4 75.9 8.755 8.755 11.945 11.945 7 92.4 6.720 6.720 9.207 9.207 10 99.7 10.966 10.966 10.663 10.663 13 85.1 12.710 12.710 12.065 11.968 16 68.9 8.222 8.222 11.116 11.116 19 101.1 7.373 7.373 9.283 9.283 AVERAGE 100.9 9.124 9.200 10.731 10.717 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 16120.957043 LEG E/M time start vinfx vinfy vinfz time dv dvx dvy dxz (days) (km/s) (km/s) (km/s) (days) (km/s) (km/s) (km/s) 1 E -0.136594758E+03 -0.836055914E+01 0.181599155E+01 0.447419678E+01 -0.109083705E+03 0.188376485E-08 0.337497702E-08 0.387093225E-09 2 M 0.468122614E+02 -0.107424210E+02 0.111622996E+01 -0.892888110E+00 0.129837412E+03 0.107950172E-09 0.861610705E-10 -0.316081272E-10 3 E 0.600313262E+03 -0.800327045E+01 -0.353476720E+01 -0.591648461E-03 0.102716030E+04 0.783544596E-09 -0.286585755E-09 0.204641679E-09 4 E 0.152824159E+04 -0.439021834E+01 -0.752067823E+01 -0.898538333E+00 0.153961955E+04 -0.465052857E-08 -0.269768539E-08 -0.376590319E-09 5 M 0.160409464E+04 -0.525730043E+00 -0.119113586E+02 0.725724390E+00 0.169165603E+04 0.390528954E-11 -0.110297149E-10 0.876974763E-11 6 E 0.218783726E+04 0.571315979E+01 -0.355992807E+01 -0.281201028E-03 0.245455513E+04 -0.105911717E-08 0.607937184E-09 0.217063022E-10 7 E 0.310755403E+04 0.520840769E+01 0.424654083E+01 0.225994610E-01 0.312141186E+04 -0.884844037E-09 0.461353248E-09 -0.585934453E-10 8 M 0.319993955E+04 0.742092484E+01 0.538020071E+01 -0.869352728E+00 0.328161217E+04 -0.186071670E-10 0.198042160E-10 0.270063245E-11 9 E 0.374442373E+04 0.438820487E+01 0.100418402E+02 -0.122803526E-02 0.395986922E+04 0.334479286E-09 -0.313645013E-09 0.212250829E-10 10 E 0.468114326E+04 -0.233339153E+01 0.106512412E+02 0.116454531E+01 0.469610097E+04 0.654957729E-09 0.558388616E-10 -0.543088627E-10 11 M 0.478086137E+04 -0.737016870E+01 0.731479940E+01 0.242284081E+01 0.485576212E+04 -0.311530337E-10 0.505273702E-11 -0.839856325E-11 12 E 0.528019971E+04 -0.984782286E+01 0.805348870E+01 -0.104775189E-02 0.596940427E+04 0.167381834E-09 0.235324282E-09 -0.182055303E-10 13 E 0.622431555E+04 -0.125252827E+02 0.215238608E+01 0.199765584E+00 0.623708386E+04 -0.480818521E-09 -0.459144329E-10 0.189028125E-10 14 M 0.630943767E+04 -0.102169731E+02 -0.621153962E+01 0.505420088E+00 0.639067110E+04 -0.838264944E-12 -0.545798108E-10 0.373395778E-11 15 E 0.685099387E+04 -0.357937243E+01 -0.740809731E+01 -0.757964036E-03 0.711945970E+04 0.477167852E-09 0.112861545E-08 0.709217469E-11 16 E 0.777673812E+04 0.190717112E+01 -0.789322351E+01 -0.129133617E+01 0.778708012E+04 -0.118243305E-09 0.163294176E-08 0.413990452E-10 17 M 0.784568481E+04 0.587156391E+01 -0.943062596E+01 0.385041742E+00 0.793375610E+04 0.578849225E-10 0.162362165E-10 -0.181665406E-11 18 E 0.843282675E+04 0.734876816E+01 0.523102414E+00 -0.779464562E-03 0.870029923E+04 0.142093018E-08 0.157803789E-09 -0.419706918E-11 19 E 0.935514563E+04 0.339857682E+01 0.650056965E+01 0.744284421E+00 0.937030986E+04 0.132909470E-08 0.398200649E-09 0.177941162E-10 20 M 0.945624056E+04 0.274223842E+01 0.884792365E+01 0.605212222E+00 0.973108224E+04 0.687713657E-02 0.221291448E-02 0.213076543E+00 21 E 0.997480976E+04 0.376568919E+00 0.123413751E+02 -0.152824543E-02 0.104365948E+05 0.412485874E-10 0.765383842E-10 -0.389888026E-10 22 E 0.109172282E+05 ================ PARENT CYCLER 3.406gGff3 ======================= Parent cycler number 19 Approximate search space (synodic periods after J2000) 5 Number of steps to walk eccentricity/inclination 3 / 3 Number of cycles 7 Total delta v over 44.74 years (km/s) 0.144267 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 2 122.2 9.523 9.523 7.172 7.172 7 188.7 3.844 3.844 3.251 3.251 12 210.2 4.348 4.348 4.383 4.383 17 126.9 3.675 3.675 7.110 7.110 22 365.7 3.569 3.569 2.681 2.681 27 139.6 5.300 5.300 7.531 7.531 32 183.1 4.175 4.105 3.690 3.690 205 AVERAGE 190.9 4.919 4.909 5.117 5.117 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 4625.516933 LEG E/M time start vinfx vinfy vinfz (days) (km/s) (km/s) (km/s) 1 E 0.979584568E+00 -0.896942727E+01 0.316785360E+01 -0.111238286E-03 2 E 0.553092746E+03 -0.935321005E+01 0.177802478E+01 -0.229883058E+00 3 M 0.675280765E+03 -0.523735987E+01 -0.483397774E+01 0.801328645E+00 4 E 0.160526929E+04 0.946454084E+00 -0.104787982E+01 -0.357962098E+01 5 E 0.197052034E+04 0.131648341E+01 -0.164272616E+01 -0.321357478E+01 6 E 0.233578442E+04 0.175183389E+01 -0.342413162E+01 0.128615002E-02 7 E 0.286483766E+04 0.318676645E+01 0.203513630E+01 0.694202474E+00 8 M 0.305355627E+04 0.271582917E+01 0.791927356E+00 -0.160145156E+01 9 E 0.394258864E+04 -0.121131091E-02 0.436064394E+01 -0.376263831E+00 10 E 0.430786484E+04 -0.181838850E+00 0.177736443E+01 -0.398106542E+01 11 E 0.467312509E+04 0.661419949E+00 0.431938668E+01 0.620625166E-03 12 E 0.520426777E+04 -0.171298622E+01 0.394915218E+01 0.612462823E+00 13 M 0.541450133E+04 -0.436463465E+01 0.204434850E+00 -0.339852539E+00 14 E 0.627329639E+04 -0.255832739E+01 -0.258939956E+01 0.679218706E+00 15 E 0.663855235E+04 0.847284556E-01 -0.223640468E+00 -0.367428066E+01 16 E 0.700381397E+04 -0.295437405E+01 -0.219960979E+01 -0.214366745E-03 17 E 0.753220680E+04 0.228338875E+00 -0.350848998E+01 -0.107101534E+01 18 M 0.765913170E+04 0.215649907E+01 -0.677138052E+01 0.219340801E+00 19 E 0.862148636E+04 0.330417094E+01 0.126942767E+01 0.484248838E-01 20 E 0.898671748E+04 -0.127880494E+01 -0.183479024E+00 -0.331060999E+01 21 E 0.935196700E+04 0.346854086E+01 0.793700180E+00 0.511298663E-03 22 E 0.987995960E+04 -0.598941066E+00 0.350633866E+01 -0.289664756E+00 23 M 0.102456189E+05 -0.145942147E+01 0.210280611E+01 0.798403058E+00 24 E 0.109558983E+05 -0.399861435E+01 0.120949782E+01 -0.320767059E+01 25 E 0.113211279E+05 -0.564338963E+00 -0.280620583E+00 -0.524069408E+01 26 E 0.116863777E+05 -0.472750707E+01 0.234416655E+01 -0.506951581E-03 27 E 0.122212566E+05 -0.528675383E+01 0.200725110E+00 -0.314509992E+00 28 M 0.123608447E+05 -0.509900179E+01 -0.548890318E+01 0.764404225E+00 29 E 0.133054277E+05 0.505252252E+00 -0.828504343E-01 0.417509647E+01 30 E 0.136706881E+05 0.310866560E+01 -0.282823869E+01 -0.216467907E-01 31 E 0.140363814E+05 0.269796006E+01 -0.321039939E+01 0.162304063E-02 32 E 0.145668295E+05 0.283248197E+01 0.283255814E+01 0.896303531E+00 33 M 0.147499654E+05 0.302015515E+01 0.182724292E+01 0.107548302E+01 34 E 0.156121668E+05 0.184506501E+01 0.668436997E+01 -0.128277127E+01 35 E 0.159774326E+05 0.537014818E+00 0.347018090E+01 -0.612290981E+01 36 E 0.163426881E+05 ================ PARENT CYCLER 3.406gfGf3 ======================= Parent cycler number 20 Approximate search space (synodic periods after J2000) 5 Number of steps to walk eccentricity/inclination 27 / 27 Number of cycles 7 Total delta v over 45.02 years (km/s) 0.681385 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 3 126.3 8.523 8.522 7.161 7.161 8 189.6 3.863 3.864 3.248 3.248 13 209.9 4.354 4.354 4.397 4.397 18 123.0 3.823 3.822 7.678 7.678 23 240.4 4.675 4.676 2.771 2.771 28 147.5 4.507 4.507 7.499 7.499 33 187.3 4.137 4.137 3.594 3.594 AVERAGE 174.9 4.840 4.841 5.192 5.192 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 4260.260034 LEG E/M time start vinfx vinfy vinfz (days) (km/s) (km/s) (km/s) 1 E -0.941051403E+00 -0.788520691E+01 0.321297848E+01 -0.792694239E-03 2 E 0.547098303E+03 -0.742667728E+01 0.406391302E+01 -0.888998294E+00 3 E 0.912360307E+03 -0.832816807E+01 0.179517948E+01 -0.213977003E+00 4 M 0.103870311E+04 -0.530636255E+01 -0.474157474E+01 0.800414134E+00 5 E 0.196939267E+04 0.104601481E+01 -0.127132003E+01 -0.349216985E+01 6 E 0.233464367E+04 0.170623431E+01 -0.345612775E+01 -0.338315204E-03 7 E 0.286385608E+04 0.219221679E+00 -0.230254485E+01 0.309949383E+01 8 E 0.322911608E+04 0.319176542E+01 0.205919999E+01 0.709710768E+00 9 M 0.341872876E+04 0.271611618E+01 0.790838076E+00 -0.159486215E+01 10 E 0.430788971E+04 -0.455055023E-02 0.434696072E+01 -0.449798899E+00 11 E 0.467316611E+04 0.659787027E+00 0.430676270E+01 -0.157848808E-02 12 E 0.520434461E+04 0.189741759E+01 0.393491561E+01 0.248444319E+00 13 E 0.556960452E+04 -0.171769952E+01 0.395391080E+01 0.613473328E+00 14 M 0.577946312E+04 -0.436303922E+01 0.225226423E+00 -0.499773164E+00 15 E 0.663686544E+04 -0.240087181E+01 -0.233441593E+01 -0.185580627E+01 16 E 0.700210972E+04 -0.310877447E+01 -0.219813028E+01 0.105109792E-02 17 E 0.753112739E+04 -0.254545064E+01 -0.808919996E+00 0.274046122E+01 18 E 0.789635053E+04 0.413197344E+00 -0.365000148E+01 -0.105737437E+01 19 M 0.801933985E+04 0.203228672E+01 -0.740392201E+01 -0.662142798E-01 20 E 0.900733094E+04 -0.118936378E+01 -0.450560222E+00 0.447500419E+01 21 E 0.937258759E+04 0.379356373E+01 0.255594421E+01 0.838563363E+00 22 E 0.990462362E+04 0.418449885E+01 0.119064534E+01 -0.172723470E+01 23 E 0.102698899E+05 0.136175892E+01 0.401794001E+01 0.196725440E+01 24 M 0.105103073E+05 0.980475764E+00 0.193699138E+01 0.172222094E+01 25 E 0.113159289E+05 -0.410360213E+01 0.181272587E+01 0.359149119E+00 26 E 0.116811831E+05 -0.381817536E+01 0.241964937E+01 0.867382892E-03 27 E 0.122129382E+05 -0.801580962E+00 0.119385767E+01 0.426351600E+01 28 E 0.125781966E+05 -0.449525896E+01 -0.152227879E-01 -0.323722597E+00 29 M 0.127257181E+05 -0.509697313E+01 -0.544593206E+01 0.770229466E+00 30 E 0.136692506E+05 -0.642218787E-01 0.510284918E+00 0.410038130E+01 31 E 0.140345090E+05 0.254192272E+01 -0.325581112E+01 -0.234704157E-03 32 E 0.145647805E+05 0.751023897E+00 -0.235307800E+01 0.333563602E+01 33 E 0.149300438E+05 0.281920397E+01 0.285997892E+01 0.995619625E+00 34 M 0.151173521E+05 0.283473007E+01 0.210729061E+01 -0.661008647E+00 35 E 0.160760832E+05 0.486329823E+01 -0.214909859E+01 0.341637260E+01 36 E 0.164413530E+05 ================ PARENT CYCLER 3.406gGh+fh-3 ======================= Parent cycler number 23 Approximate search space (synodic periods after J2000) 8 Number of steps to walk eccentricity/inclination 1 / 1 Number of cycles 7 Total delta v over 44.78 years (km/s) 1.819780 time dv (days) 0.210782586E+03 0.571420949E+03 0.972877094E+03 0.174041218E+04 0.221889992E+04 0.260031104E+04 0.289314545E+04 0.352474343E+04 0.418732370E+04 0.455624181E+04 0.493338500E+04 0.523580281E+04 0.558626034E+04 0.653262812E+04 0.687962502E+04 0.726272646E+04 0.755124554E+04 0.827503868E+04 0.867627103E+04 0.908168235E+04 0.961068337E+04 0.993480850E+04 0.105652446E+05 0.112006021E+05 0.114635753E+05 0.119431195E+05 0.122421948E+05 0.129653778E+05 0.135647626E+05 0.138315931E+05 0.142909965E+05 0.145942999E+05 0.149137837E+05 0.156815673E+05 0.161783231E+05 dvx (km/s) -0.166555130E-04 -0.147744020E-03 0.716980407E-07 0.830544802E-04 0.491318422E-03 0.982912946E-05 -0.196095449E-04 0.275317291E-08 0.109324326E-03 -0.932176268E-04 -0.107158915E-05 -0.292444781E-05 -0.598405433E-09 -0.318298815E-05 -0.248695185E-03 -0.251162229E-06 -0.622902614E-06 -0.380365612E-08 -0.479176728E-04 0.161858191E-03 0.116613864E-03 -0.917410616E-04 0.217235418E-06 0.470380360E-03 -0.258768171E-03 -0.558873738E-05 0.295345453E-06 -0.142355393E-05 0.676317767E-03 -0.121954629E-01 0.235062715E-03 -0.600995399E-03 0.141697376E-04 -0.528302203E-04 -0.647703512E-04 dvy (km/s) -0.170341832E-04 -0.178199997E-03 -0.118317059E-06 0.307791776E-03 -0.876519261E-03 -0.236963183E-04 0.372037227E-04 -0.824775327E-08 -0.658331076E-04 0.465825805E-03 -0.277094085E-05 0.338116743E-05 0.315820864E-09 0.355093630E-05 -0.473936523E-04 -0.134335367E-06 -0.485234530E-06 -0.104750963E-06 0.730671090E-04 -0.452064422E-03 0.696391971E-05 -0.925511448E-04 0.678989206E-07 0.269189661E-03 0.806785507E-03 0.427955625E-05 0.556543057E-06 0.747676776E-08 -0.498305539E-03 0.202689474E-01 -0.488717939E-03 -0.375372536E-04 -0.726933127E-05 -0.443442594E-05 -0.318381408E-03 dxz (km/s) -0.571179534E-06 0.847658335E-05 0.624081487E-08 0.795200739E-04 0.158970381E-03 0.469728784E-06 0.514891752E-07 -0.239180694E-09 0.265090660E-03 0.908319058E-05 0.140391671E-07 0.356910207E-07 0.122487988E-08 -0.187230291E-03 0.755427794E-04 -0.314720594E-08 0.365532580E-08 0.549702622E-07 -0.509782324E-04 -0.681744800E-04 0.183309988E-05 -0.201652501E-04 -0.196678396E-07 -0.157896244E-02 0.462336523E-04 0.534729782E-07 0.569139392E-07 0.380740694E-06 -0.236927618E-03 0.139498863E-03 0.168683633E-05 -0.254599095E-05 0.138051279E-03 -0.190046822E-03 0.425637902E-04 time dv (days) 0.300480594E+03 0.601887604E+03 0.931311728E+03 0.134583067E+04 0.210453554E+04 0.259395775E+04 0.303918088E+04 0.325755798E+04 0.388998406E+04 0.454531937E+04 0.492813179E+04 0.531026999E+04 0.560108331E+04 0.593379554E+04 0.689618888E+04 0.726132838E+04 0.763704210E+04 0.791479892E+04 0.839477646E+04 0.913882333E+04 0.962796488E+04 0.100105508E+05 0.103564401E+05 0.109695116E+05 0.115679543E+05 0.119417431E+05 0.123809571E+05 0.126003248E+05 0.129899072E+05 0.139139737E+05 0.142890393E+05 0.146889700E+05 0.149581400E+05 0.154337334E+05 0.161893169E+05 dvx (km/s) 0.739143975E-04 0.279737662E-03 -0.692878169E-03 0.141385429E-06 0.162260244E-03 -0.211520418E-04 0.588939029E-03 -0.172364221E-04 0.964202182E-08 0.158858889E-03 -0.119957385E-06 0.488319530E-05 0.465956126E-06 -0.339526402E-08 0.256891376E-04 -0.609354616E-05 0.723943039E-04 -0.215340604E-05 0.746068366E-08 0.765790466E-03 -0.288864650E-02 -0.794669334E-06 0.463155105E-08 0.462163560E-01 0.107222895E-05 0.163730256E-06 -0.927712495E-04 -0.104895325E-05 0.136029221E-07 0.446968149E-04 -0.150808136E-04 0.298559832E-03 -0.861964567E-04 -0.475822754E-05 -0.231974715E-04 dvy (km/s) -0.130386052E-04 0.503266503E-03 0.189773770E-03 -0.116404764E-06 0.582613268E-03 0.736324597E-04 0.110200921E-02 0.687950121E-04 -0.302766450E-07 -0.111221262E-03 -0.260235589E-06 -0.132205562E-06 -0.524805741E-06 -0.671767670E-09 -0.415343314E-04 -0.251481249E-05 -0.414524481E-03 -0.105893117E-05 0.114833898E-07 -0.417388456E-03 0.653384173E-02 0.357141864E-05 0.602144793E-07 -0.152646994E+00 0.763204636E-06 -0.144856379E-06 0.910433012E-03 0.449372395E-06 -0.478011970E-07 -0.464374213E-03 0.442255812E-04 0.247585592E-03 0.229107880E-04 0.286940753E-05 -0.853441850E-04 dxz (km/s) 0.643996888E-06 0.889931652E-06 -0.383852540E-04 0.254585305E-07 0.150081708E-03 -0.878281654E-06 0.180826098E-03 -0.490051103E-06 -0.727237104E-09 0.292253461E-03 -0.101175601E-08 0.441616600E-04 -0.506623575E-08 0.985056792E-09 -0.471615214E-03 -0.647594518E-07 -0.157072822E-02 -0.224221333E-07 0.743490112E-09 -0.469635300E-04 0.233674135E+00 0.338882457E-04 0.360839485E-05 -0.194669017E-01 0.128038950E-04 -0.132263397E-08 -0.247855066E-04 -0.199294248E-07 -0.706692815E-08 -0.172490289E-03 -0.393309147E-06 0.638761665E-04 0.357663583E-06 0.771116197E-06 0.559824451E-03 206 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 2 185.6 3.775 3.775 3.257 3.257 8 210.1 4.352 4.352 4.389 4.389 14 124.2 3.773 3.773 7.478 7.478 20 243.5 4.142 4.339 2.793 2.516 26 155.8 3.885 3.885 7.531 7.531 32 192.0 4.189 4.083 3.421 3.421 38 197.8 4.642 4.642 4.819 4.819 AVERAGE 187.0 4.108 4.121 4.813 4.773 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 6965.408456 LEG E/M time start vinfx vinfy vinfz (days) (km/s) (km/s) (km/s) 1 E -0.212288115E+00 0.195654968E+01 -0.324707920E+01 0.409347137E-03 2 E 0.528606341E+03 0.317460075E+01 0.193588011E+01 0.650292102E+00 3 M 0.714251299E+03 0.270769537E+01 0.793154684E+00 -0.162670245E+01 4 E 0.160276252E+04 -0.340302350E+00 0.281217485E-01 0.437812405E+01 5 E 0.178653677E+04 0.640514479E+00 0.446220766E+01 0.462880981E+00 6 E 0.215178079E+04 0.324460838E+00 -0.275651956E-01 -0.450231058E+01 7 E 0.233326802E+04 0.659090664E+00 0.432366794E+01 0.623973900E-03 8 E 0.286442641E+04 -0.171510160E+01 0.395240281E+01 0.612714611E+00 9 M 0.307447913E+04 -0.436361281E+01 0.214340128E+00 -0.421592800E+00 10 E 0.393249558E+04 0.149380541E+00 -0.194570965E+00 0.375700187E+01 11 E 0.411165708E+04 -0.297313591E+01 -0.208999413E+01 0.743113509E+00 12 E 0.447689546E+04 -0.157985005E+00 0.169966035E+00 -0.371562395E+01 13 E 0.466299809E+04 -0.306345147E+01 -0.221104017E+01 -0.358934143E-03 14 E 0.519177311E+04 0.340389035E+00 -0.360413748E+01 -0.106145749E+01 15 M 0.531595030E+04 0.208568439E+01 -0.718152349E+01 -0.124142854E-01 16 E 0.629820582E+04 -0.174043059E+00 0.254890188E+00 0.416917806E+01 17 E 0.648442918E+04 0.352788759E+01 0.218691495E+01 0.833369962E+00 18 E 0.684968855E+04 0.158279257E+00 -0.241863276E+00 -0.420404831E+01 19 E 0.702872077E+04 0.361675886E+01 0.205958612E+01 -0.150727835E-02 20 E 0.755902332E+04 0.103625951E+01 0.379439866E+01 0.183202282E+01 21 M 0.780253949E+04 0.177556489E+01 0.166921654E+01 0.626729815E+00 22 E 0.860385038E+04 -0.155777951E+00 -0.221052483E+00 0.389404001E+01 23 E 0.878391199E+04 0.907882864E+00 -0.325492421E+00 -0.388420015E+01 24 E 0.914916948E+04 0.148167765E+00 0.204624248E+00 -0.399809137E+01 25 E 0.933436494E+04 -0.301055842E+01 0.248978368E+01 -0.243643714E-03 26 E 0.986366353E+04 -0.385145754E+01 -0.383588529E+00 -0.337755506E+00 27 M 0.100194654E+05 -0.517041890E+01 -0.542160407E+01 0.765407148E+00 28 E 0.109648269E+05 0.225598116E+00 0.213391265E+00 0.421965544E+01 29 E 0.111494215E+05 0.262798112E+01 -0.312167639E+01 0.480981323E+00 30 E 0.115146736E+05 -0.205385933E+00 -0.184775325E+00 -0.407868238E+01 31 E 0.116953311E+05 0.264022654E+01 -0.327650913E+01 0.165202401E-02 32 E 0.122258314E+05 0.278300988E+01 0.276467598E+01 0.113194786E+01 33 M 0.124178120E+05 0.237103362E+01 0.941184011E+00 -0.227873174E+01 34 E 0.133018922E+05 -0.378687546E+00 -0.503727029E-01 0.465904409E+01 35 E 0.134849801E+05 -0.151564462E+00 0.470026035E+01 -0.110165166E+01 36 E 0.138502531E+05 0.377423840E+00 0.302496251E-01 -0.479595322E+01 37 E 0.140324199E+05 -0.196208692E+00 0.465210787E+01 -0.312649249E-03 38 E 0.145647676E+05 -0.230444521E+01 0.400826285E+01 0.417832849E+00 39 M 0.147625221E+05 -0.480178800E+01 -0.384182195E+00 -0.145876139E+00 40 E 0.156254167E+05 0.150093504E+00 -0.167253609E+00 0.361195573E+01 41 E 0.158047360E+05 -0.207577708E+01 -0.167550739E+01 -0.235230701E+01 42 E 0.161699782E+05 -0.159990685E+00 0.144373981E+00 -0.356481363E+01 43 E 0.163559146E+05 ================ PARENT CYCLER 3.418gGff3 ======================= Parent cycler number 33 Approximate search space (synodic periods after J2000) 1 Number of steps to walk eccentricity/inclination 243 / 243 Number of cycles 7 Total delta v over 45.14 years (km/s) 0.009457 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 2 204.9 8.732 8.732 5.444 5.444 7 266.0 7.017 7.017 5.034 5.034 12 153.2 4.157 4.157 6.448 6.448 17 314.1 3.688 3.688 2.540 2.540 22 152.7 3.658 3.658 8.451 8.451 27 279.9 5.540 5.540 2.652 2.652 32 185.2 4.967 4.967 8.374 8.374 AVERAGE 222.3 5.394 5.394 5.563 5.563 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 1517.738795 LEG E/M time start vinfx vinfy vinfz (days) (km/s) (km/s) (km/s) 1 E -0.863680370E+02 0.519280943E+01 0.701954361E+01 0.738304498E-03 2 E 0.393823194E+03 0.868547522E+01 -0.633510236E+00 0.634419430E+00 3 M 0.598678428E+03 0.542513190E+01 0.324328863E+00 0.316446375E+00 4 E 0.152733339E+04 0.672331891E+00 0.176798102E+01 -0.676787767E+01 5 E 0.189259176E+04 -0.250070377E+01 -0.117162924E+01 -0.645687971E+01 6 E 0.225785412E+04 -0.627084891E+01 -0.314141745E+01 0.127475175E-02 7 E 0.274381166E+04 -0.533083499E+01 0.449891731E+01 -0.763199793E+00 8 M 0.300982421E+04 -0.500115666E+01 0.460470904E+00 0.345077566E+00 9 E 0.392316570E+04 0.217933744E+01 0.353326538E+01 0.161131808E+00 10 E 0.428838466E+04 0.215005878E+01 0.352672341E+01 -0.417437404E+00 11 E 0.465365558E+04 0.267737300E+01 0.315535107E+01 0.744885215E-03 12 E 0.515141314E+04 0.299579082E+01 -0.248752251E+01 -0.145423175E+01 13 M 0.530460273E+04 0.385340546E+01 -0.516616384E+01 -0.182910867E+00 14 E 0.628043083E+04 -0.267323223E+01 -0.253214664E+01 0.340697239E-01 15 E 0.664564360E+04 -0.267307450E+01 -0.254449664E+01 0.241288893E+00 16 E 0.701089086E+04 -0.302173382E+01 -0.209773026E+01 -0.428639977E-03 17 E 0.751004889E+04 -0.101480572E+01 0.337828947E+01 -0.107705121E+01 18 M 0.782416497E+04 -0.119364508E+01 0.184179194E+01 0.127812516E+01 19 E 0.860644485E+04 0.138185195E+01 -0.267960519E+00 -0.338671849E+01 20 E 0.897169734E+04 0.364899444E+01 -0.366941401E+00 0.216750026E+00 21 E 0.933693119E+04 0.356358406E+01 -0.899220991E+00 0.488140169E-03 22 E 0.983654050E+04 -0.276461167E+01 -0.232793603E+01 -0.563260578E+00 23 M 0.998923123E+04 -0.459383965E+01 -0.706747387E+01 0.608543776E+00 24 E 0.109638848E+05 -0.508038546E+01 0.204076473E+01 -0.709343007E+00 25 E 0.113291536E+05 -0.503170689E+01 0.204684257E+01 -0.946800336E+00 time dv (days) 0.258908840E+03 0.556453085E+03 0.131843893E+04 0.163032866E+04 0.189610998E+04 0.217900388E+04 0.263071672E+04 0.289593432E+04 0.326324275E+04 0.395936981E+04 0.422122860E+04 0.450481085E+04 0.492209785E+04 0.521039969E+04 0.568920740E+04 0.632613933E+04 0.659035440E+04 0.687654338E+04 0.728326600E+04 0.764912431E+04 0.847564064E+04 0.863085962E+04 0.892270983E+04 0.917694880E+04 0.958842826E+04 0.988703382E+04 0.108230226E+05 0.109925161E+05 0.112589971E+05 0.115417722E+05 0.119818012E+05 0.122546285E+05 0.129040561E+05 0.133293554E+05 0.135945620E+05 0.138775781E+05 0.142826233E+05 0.146122286E+05 0.151767115E+05 0.156523146E+05 0.159106562E+05 0.161978687E+05 dvx (km/s) 0.573074654E-07 -0.155900789E-06 -0.106110447E-06 0.168219438E-08 0.373493527E-06 -0.139456203E-09 -0.114278793E-06 0.727371255E-08 0.195373137E-07 -0.110109887E-07 0.439230580E-05 -0.731028206E-08 0.116192946E-07 -0.108696939E-07 -0.149199586E-09 -0.976269643E-07 -0.510778910E-06 -0.212405954E-07 0.175088644E-06 0.347320764E-02 0.139561158E+00 0.529344050E-07 0.108323301E-03 0.536647669E-07 -0.390936513E-07 0.533281870E-07 0.152737891E-07 -0.149819605E-09 -0.333539851E-06 -0.237604297E-09 0.107635864E-08 0.123618022E-07 0.161436270E-10 -0.576801345E-08 0.238063714E-06 -0.525970083E-08 -0.100992573E-08 -0.225831552E-08 -0.886543714E-11 -0.723913040E-08 0.194537317E-04 -0.616625378E-08 dvy (km/s) -0.188186806E-06 0.932167482E-07 -0.225698709E-06 -0.935993740E-08 -0.135321467E-06 0.212720890E-09 -0.156448677E-08 -0.829541367E-08 -0.252143812E-07 0.388198677E-08 -0.616509416E-05 0.118202301E-07 0.651985940E-08 -0.686128620E-08 -0.245986946E-09 -0.837796954E-07 0.655538645E-06 -0.185629093E-06 0.566224012E-07 -0.364614884E-02 -0.503667634E-02 -0.160744495E-08 -0.154187442E-02 -0.739481621E-08 0.472850275E-07 0.162974412E-07 -0.421527000E-07 0.694255325E-10 -0.271277172E-06 0.325385682E-11 0.936899132E-09 -0.138110680E-08 0.138884133E-12 0.429442966E-08 0.104268519E-07 0.245904277E-08 -0.355909108E-08 -0.288259474E-08 -0.813464206E-11 0.246789333E-08 -0.226077143E-04 0.111844913E-07 dxz (km/s) 0.326152437E-08 0.378085680E-08 -0.115024950E-07 0.296881368E-09 0.218091673E-04 -0.398961205E-11 0.585780282E-10 -0.846730263E-10 0.517458204E-06 0.124866838E-08 -0.300315916E-03 0.172911407E-08 0.878245473E-10 -0.233027734E-10 -0.122623116E-10 0.179648692E-08 -0.229000380E-04 0.198184086E-07 -0.816692882E-09 0.166848957E+00 0.437631174E+00 0.397130332E-08 0.141162405E-03 0.333943921E-08 0.870268462E-09 0.341266225E-08 0.114222028E-08 0.252281374E-10 0.267550261E-04 -0.335140639E-11 0.147150127E-10 0.237503647E-09 -0.147820665E-10 -0.778423243E-09 -0.630692940E-05 -0.819595775E-09 0.733534108E-10 0.962444823E-10 -0.609560805E-11 0.845308354E-09 0.382155814E-03 0.148823194E-08 time dv (days) 0.168133316E+03 0.424551479E+03 0.812269068E+03 0.177570908E+04 0.213731754E+04 0.251055204E+04 0.278371354E+04 0.323815958E+04 0.402542701E+04 0.439066052E+04 0.491248951E+04 0.517439158E+04 0.592913271E+04 0.638634253E+04 0.675521778E+04 0.726546145E+04 0.755716630E+04 0.818401372E+04 0.877811352E+04 0.907031048E+04 0.960672022E+04 0.985944411E+04 0.103498531E+05 0.110771182E+05 0.114460368E+05 dvx (km/s) 0.992575518E-05 0.573153630E-04 0.485906773E-07 0.600488087E-03 0.273036549E-03 -0.343595679E-04 -0.748485892E-04 0.609286873E-07 0.321120585E-04 0.893242851E-05 0.578663384E-05 -0.636398947E-05 -0.206299516E-07 -0.753633457E-05 -0.635982402E-05 -0.106663108E-05 0.144258312E-05 -0.126379931E-08 0.268790562E-03 0.446293927E-05 0.136489387E-05 -0.207904280E-05 0.211888246E-07 0.324526343E-06 0.283035105E-05 dvy (km/s) 0.410841215E-04 0.564617939E-04 -0.284506213E-07 0.187724423E-03 -0.208835041E-03 -0.167250525E-04 -0.706563905E-04 0.578363163E-07 -0.353419878E-05 -0.777823191E-06 0.739791746E-05 -0.163373301E-04 0.371128164E-08 0.289865367E-05 0.126282624E-05 -0.521111106E-06 0.145141763E-05 -0.328348544E-09 -0.205674496E-03 -0.143371110E-04 0.242306666E-07 -0.395738765E-05 -0.319867849E-08 -0.260050416E-05 -0.452357679E-05 dxz (km/s) -0.931556421E-05 -0.147230782E-05 -0.107106201E-06 0.403267292E-03 0.209016859E-03 -0.765851110E-06 0.304424588E-04 -0.237522594E-08 -0.103937572E-03 0.441939987E-04 -0.342900332E-06 0.359971422E-06 0.120364259E-08 0.280767026E-04 -0.235060501E-03 0.977702083E-07 -0.783788839E-07 -0.748398152E-09 0.130029514E-04 0.339026996E-04 0.396094857E-06 0.123902421E-07 0.815873443E-09 -0.133887132E-04 -0.159246865E-03 207 26 E 0.116944135E+05 -0.475153610E+01 0.283408544E+01 0.155543944E-03 27 E 0.121864432E+05 0.192740972E+01 0.436971009E+01 -0.280666122E+01 28 M 0.124663557E+05 0.223175000E+01 0.604357467E+00 0.129875547E+01 29 E 0.132800062E+05 0.276303985E+00 -0.267917595E+01 -0.419720083E+01 30 E 0.136452568E+05 0.862492983E+00 -0.477613919E+01 -0.118446398E+01 31 E 0.140105132E+05 0.117046364E+00 -0.497272312E+01 0.132122065E-02 32 E 0.145047222E+05 -0.482735414E+01 0.511135840E+00 0.105121687E+01 33 M 0.146899292E+05 -0.833261468E+01 0.347382655E-01 0.827548148E+00 34 E 0.156717385E+05 -0.382809712E+01 0.870335805E+01 0.547721112E+00 35 E 0.160369813E+05 -0.385378002E+01 0.871695347E+01 0.299952588E+00 36 E 0.164022401E+05 ================ PARENT CYCLER 3.418gfGf3 ======================= Parent cycler number 34 Approximate search space (synodic periods after J2000) 20 Number of steps to walk eccentricity/inclination 243 / 243 Number of cycles 7 Total delta v over 45.20 years (km/s) 0.019571 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 3 243.0 13.286 13.287 7.188 7.186 8 224.9 6.925 6.925 3.611 3.611 13 251.8 4.502 4.502 4.996 4.994 18 138.8 3.684 3.683 6.945 6.945 23 305.6 3.950 3.950 2.704 2.704 28 131.4 5.119 5.117 8.212 8.212 33 293.5 5.317 5.316 2.485 2.485 AVERAGE 227.0 6.112 6.111 5.163 5.162 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 15971.800569 LEG E/M time start vinfx vinfy vinfz (days) (km/s) (km/s) (km/s) 1 E -0.168979542E+03 -0.131804165E+02 0.130659162E+01 0.123871562E-02 2 E 0.292649929E+03 -0.738118050E+01 0.110304109E+02 0.552188828E+00 3 E 0.657901402E+03 -0.535710719E+01 0.121446702E+02 -0.583209261E+00 4 M 0.900905719E+03 -0.709362880E+01 0.948097281E+00 0.643312013E+00 5 E 0.191809980E+04 0.452582725E+00 0.685384955E+01 -0.629504681E+00 6 E 0.228337445E+04 0.152315805E+01 0.673803942E+01 0.272428454E-03 7 E 0.277040610E+04 0.525848506E+01 0.448302477E+01 0.202294687E+00 8 E 0.313565021E+04 0.665324799E+01 0.188679557E+01 -0.357706467E+00 9 M 0.336058331E+04 0.334672800E+01 0.909741127E+00 -0.100455141E+01 10 E 0.427733005E+04 -0.520642442E+00 -0.894742655E+00 0.438358431E+01 11 E 0.464259273E+04 -0.190204749E+01 -0.408515121E+01 0.823888056E-03 12 E 0.513850831E+04 -0.368332454E+01 -0.217243807E+01 0.144967908E+01 13 E 0.550374752E+04 -0.321941323E+01 0.182215913E+01 -0.256576134E+01 14 M 0.575552727E+04 -0.495322351E+01 -0.448974853E+00 0.453535116E+00 15 E 0.663984844E+04 0.167634289E+01 0.326910600E+01 0.854661926E-01 16 E 0.700513645E+04 0.216781357E+01 0.296516768E+01 -0.983657048E-03 17 E 0.750465537E+04 0.766154353E+00 0.528908848E+00 0.357102406E+01 18 E 0.786991050E+04 0.193065475E+01 -0.289274959E+01 -0.121194092E+01 19 M 0.800867198E+04 0.322634457E+01 -0.614869406E+01 -0.137066534E+00 20 E 0.898918391E+04 -0.307743520E+01 -0.249821423E+01 -0.820029470E-01 21 E 0.935440988E+04 -0.341418762E+01 -0.200764433E+01 0.129345522E-02 22 E 0.985245039E+04 -0.716842664E-01 -0.251163814E+00 -0.394029604E+01 23 E 0.102177099E+05 -0.749090561E+00 0.348690578E+01 -0.169684716E+01 24 M 0.105232971E+05 -0.115799877E+01 0.205042800E+01 -0.132976745E+01 25 E 0.113469920E+05 0.479804893E+01 0.175737472E+01 -0.259580716E+00 26 E 0.117122288E+05 0.502919865E+01 0.102376345E+01 -0.108924595E-02 27 E 0.122062316E+05 0.224507603E+00 0.389414024E+00 -0.510081174E+01 28 E 0.125714861E+05 -0.500268100E+01 -0.996330456E+00 -0.407696431E+00 29 M 0.127028951E+05 -0.493175627E+01 -0.653325684E+01 0.659588273E+00 30 E 0.136690410E+05 -0.166682936E+01 0.125993451E+01 0.487148379E+01 31 E 0.140343023E+05 -0.415162522E+01 0.330361649E+01 0.662571132E-03 32 E 0.145271876E+05 -0.139745646E+01 0.490158090E+01 -0.153394316E+01 33 E 0.148924370E+05 0.241268494E+01 0.419236688E+01 -0.220594879E+01 34 M 0.151859811E+05 0.192544590E+01 0.143702275E+01 0.635196326E+00 35 E 0.159762138E+05 -0.585476020E+00 0.598515882E+01 -0.224447355E+01 36 E 0.163414560E+05 ================ PARENT CYCLER 3.639gGf3 ======================= Parent cycler number 49 Approximate search space (synodic periods after J2000) 14 Number of steps to walk eccentricity/inclination 27 / 27 Number of cycles 7 Total delta v over 44.79 years (km/s) 0.007701 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 2 93.9 7.919 7.919 6.740 6.740 6 327.7 3.694 3.694 2.595 2.595 10 138.2 5.490 5.490 7.509 7.509 14 224.7 4.170 4.170 2.808 2.808 18 185.8 5.810 5.811 4.864 4.864 22 118.5 3.895 3.895 7.429 7.429 26 372.3 5.045 5.044 4.145 4.145 AVERAGE 208.7 5.146 5.146 5.156 5.156 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 11272.659420 LEG E/M time start vinfx vinfy vinfz (days) (km/s) (km/s) (km/s) 1 E 0.933126310E+01 -0.368752656E+01 -0.700518883E+01 -0.256897984E-03 2 E 0.933828124E+03 -0.155222565E+01 -0.769151016E+01 -0.106812246E+01 3 M 0.102774233E+04 0.293362637E+01 -0.605891550E+01 0.339969780E+00 4 E 0.196948396E+04 0.299771401E+01 0.926792030E+00 -0.191602820E+01 5 E 0.233474144E+04 0.361903246E+01 0.735546658E+00 -0.137595212E-02 6 E 0.324239089E+04 0.261339839E+00 0.346236703E+01 -0.126164887E+01 7 M 0.357008616E+04 -0.635685966E+00 0.238153839E+01 0.810324389E+00 8 E 0.429589024E+04 -0.470939905E+01 0.273483010E+01 0.610520952E+00 9 E 0.466115403E+04 -0.446393977E+01 0.316677412E+01 -0.348160684E-03 10 E 0.557592344E+04 -0.547579929E+01 0.243670694E+00 -0.316111435E+00 11 M 0.571408008E+04 -0.506659129E+01 -0.548793438E+01 0.768046751E+00 12 E 0.665760318E+04 -0.376070741E+00 0.836779673E+00 -0.404354805E+01 13 E 0.702286012E+04 0.278652443E+01 -0.307660433E+01 0.127368444E-02 14 E 0.793239242E+04 0.278197392E+01 0.206116480E+01 0.232322059E+01 15 M 0.815712096E+04 0.218232219E+01 0.895743319E+00 0.152289252E+01 0.119502689E+05 0.122284300E+05 0.125884033E+05 0.135502916E+05 0.137365709E+05 0.142625598E+05 0.145417636E+05 0.154753767E+05 0.157922686E+05 0.161575167E+05 0.671516937E-05 0.159312657E-05 -0.845478966E-07 -0.443384800E-03 -0.445452751E-04 -0.389334109E-06 0.211673707E-04 -0.322520371E-07 0.236968381E-04 0.324930871E-04 -0.460503181E-05 0.914950358E-06 0.164789945E-05 0.877952788E-07 -0.266625327E-07 0.215642307E-06 0.949013730E-03 0.103679903E-03 -0.218860717E-04 -0.624138094E-04 -0.297496617E-04 0.191714272E-05 0.413533481E-05 0.123603748E-06 -0.582683948E-07 0.181747630E-07 0.228150633E-04 0.720472408E-04 0.328473743E-04 -0.110027103E-03 time dv (days) 0.202885412E+02 0.347437650E+03 0.696782093E+03 0.120606394E+04 0.197289100E+04 0.251714964E+04 0.301511965E+04 0.316939017E+04 0.384645908E+04 0.443074037E+04 0.490542799E+04 0.521155615E+04 0.554151449E+04 0.629496318E+04 0.672021180E+04 0.727487667E+04 0.763614722E+04 0.789072472E+04 0.818516413E+04 0.904396781E+04 0.972794026E+04 0.100496905E+05 0.102635480E+05 0.109269076E+05 0.114017775E+05 0.119691103E+05 0.123523334E+05 0.125911974E+05 0.132439368E+05 0.138151455E+05 0.143546778E+05 0.145965850E+05 0.150098546E+05 0.155415858E+05 0.160456098E+05 dvx (km/s) 0.475363383E-04 -0.179593518E-04 0.154035466E-04 0.417773134E-06 0.119734440E-03 0.243627082E-04 -0.799842854E-06 -0.736250821E-04 0.159664793E-06 -0.141091360E-02 0.610300618E-05 0.997128374E-04 0.581331403E-04 0.117879065E-06 0.195287894E-04 -0.377174310E-05 -0.174480843E-04 -0.123335588E-04 0.736285136E-08 0.111751144E-04 0.177655459E-05 -0.159885375E-03 -0.158885729E-04 -0.353618809E-07 -0.187456329E-05 -0.948689655E-05 0.469884913E-03 -0.904108952E-05 -0.545938315E-08 -0.134416944E-03 -0.396220757E-07 -0.545556066E-04 -0.678083126E-06 0.184969142E-06 -0.349508523E-04 dvy (km/s) 0.392021388E-04 -0.254943514E-04 -0.126966506E-03 0.534625230E-06 -0.386861258E-05 0.282598223E-04 0.263104212E-05 0.733255422E-04 -0.910354985E-06 -0.808731500E-03 0.403488203E-04 -0.622214557E-04 0.137714240E-03 -0.139148433E-06 -0.414156669E-05 -0.800647913E-05 -0.375213095E-04 -0.385461986E-04 -0.547954119E-08 -0.120018045E-04 -0.330749834E-06 0.439685465E-04 -0.240479683E-04 0.609695721E-08 0.426478420E-05 -0.284626690E-05 -0.191419421E-02 0.107137058E-04 -0.107624357E-07 0.137289432E-02 -0.362340237E-07 -0.444725059E-04 0.446359119E-07 -0.688890982E-06 -0.294738611E-04 dxz (km/s) -0.798060279E-05 0.135404416E-04 -0.458261269E-04 0.213470698E-07 -0.145973257E-04 0.528625893E-05 0.346449809E-04 0.932106828E-05 0.319790398E-06 -0.847869196E-04 -0.125592941E-04 -0.403208429E-03 -0.462500008E-05 0.102998360E-06 -0.250426875E-04 -0.188267711E-05 0.126339145E-04 -0.183077667E-05 -0.356232324E-09 0.250207469E-05 0.307828689E-06 -0.178877373E-04 0.113952116E-05 -0.367996188E-06 0.148322285E-05 0.664726847E-06 0.888258948E-04 0.973097717E-06 0.570475514E-09 -0.349681887E-04 -0.174814725E-07 0.220416035E-03 0.674330540E-06 -0.194270759E-06 0.385881822E-03 time dv (days) 0.268190384E+03 0.947915254E+03 0.149861314E+04 0.204984061E+04 0.261611277E+04 0.329154518E+04 0.388943996E+04 0.436529036E+04 0.492643716E+04 0.559664693E+04 0.630849963E+04 0.690232533E+04 0.729571981E+04 0.796610170E+04 0.884664682E+04 dvx (km/s) 0.518010799E-04 -0.120890525E-03 0.455347393E-06 -0.596666003E-04 -0.444926615E-04 0.183122261E-04 0.245919237E-06 -0.449888807E-05 0.700564617E-06 -0.547744354E-07 -0.993018805E-07 0.482074426E-04 0.457939803E-04 -0.105766909E-03 0.467167646E-05 dvy (km/s) 0.100185542E-03 0.124105821E-03 -0.328764585E-07 0.548183399E-04 -0.225548702E-04 -0.373686107E-04 -0.405113855E-09 -0.418538855E-04 -0.440650670E-06 0.337029179E-06 -0.277019012E-06 -0.494237102E-03 -0.478152206E-04 -0.128305020E-04 0.985802894E-06 dxz (km/s) -0.112732753E-05 -0.115654994E-04 -0.173357748E-08 0.301882623E-03 -0.567689370E-06 0.482587590E-05 -0.308352318E-07 -0.174187816E-03 0.407853032E-08 0.191519398E-07 0.418693684E-07 0.111796219E-03 -0.855125241E-05 -0.402551638E-05 0.563865264E-04 208 16 E 0.898787500E+04 -0.550747285E+00 0.570006150E+01 -0.963463420E+00 17 E 0.935313628E+04 -0.369352777E-01 0.578705014E+01 0.848218607E-02 18 E 0.102691898E+05 -0.278431384E+01 0.508734664E+01 0.358553624E+00 19 M 0.104549977E+05 -0.482673815E+01 -0.360327985E+00 -0.484248535E+00 20 E 0.113148201E+05 -0.272577005E+01 -0.279699326E+01 -0.264149009E+00 21 E 0.116800534E+05 -0.293888936E+01 -0.255977302E+01 0.125515979E-01 22 E 0.125886064E+05 0.155579697E+01 -0.340256266E+01 -0.108192551E+01 23 M 0.127071421E+05 0.345214463E+01 -0.656914178E+01 -0.345074563E+00 24 E 0.137031735E+05 -0.311615912E+00 0.294994648E+00 -0.503278504E+01 25 E 0.140684317E+05 0.270226365E+01 0.427243648E+01 -0.667623658E-02 26 E 0.149815271E+05 -0.545283493E+00 0.482938142E+01 0.135052265E+01 27 M 0.153538375E+05 -0.214891346E+01 0.346093877E+01 0.763690062E+00 28 E 0.160058311E+05 -0.590947490E+01 0.910194006E+00 0.239387152E+00 29 E 0.163710741E+05 ================ PARENT CYCLER 3.768Gh-3 ======================= Parent cycler number 54 Approximate search space (synodic periods after J2000) 11 Number of steps to walk eccentricity/inclination 3 / 3 Number of cycles 7 Total delta v over 44.66 years (km/s) 0.000000 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 1 221.5 ****** 10.192 3.768 3.768 4 122.9 5.862 5.862 7.953 7.953 7 214.8 4.302 4.302 2.737 2.737 10 174.2 5.469 5.469 5.632 5.632 13 137.9 3.665 3.665 5.696 5.696 16 238.0 4.266 4.266 3.260 3.260 19 130.1 4.655 4.655 7.668 7.668 AVERAGE 177.1 4.703 5.487 5.245 5.245 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 9063.079462 LEG E/M time start vinfx vinfy vinfz (days) (km/s) (km/s) (km/s) 1 E 0.544261030E+02 -0.216961725E+00 0.101766180E+02 0.520259079E+00 2 M 0.275969105E+03 -0.331606979E+01 0.147205622E+01 0.101730044E+01 3 E 0.104257188E+04 -0.558781423E+01 0.179105897E+01 -0.123686602E-02 4 E 0.232989346E+04 -0.558158446E+01 -0.171080289E+01 -0.532483706E+00 5 M 0.245278108E+04 -0.378926514E+01 -0.696224123E+01 0.649795404E+00 6 E 0.341140003E+04 0.108172368E+00 0.301430875E+00 -0.437683444E+01 7 E 0.469293767E+04 0.278535914E+01 0.272299986E+01 0.182686197E+01 8 M 0.490774697E+04 0.202000374E+01 0.905123229E+00 0.160983392E+01 9 E 0.573339551E+04 -0.114078098E+01 0.342833115E+01 -0.396517512E+01 10 E 0.701172184E+04 -0.374761365E+01 0.397924906E+01 0.181604274E+00 11 M 0.718595048E+04 -0.539688339E+01 -0.159303206E+01 0.236592535E+00 12 E 0.805578842E+04 -0.225945646E+01 -0.290071324E+01 -0.114011969E-02 13 E 0.933441735E+04 0.319559223E+01 -0.148475273E+01 -0.100892024E+01 14 M 0.947231413E+04 0.464739208E+01 -0.329327263E+01 0.697784563E-01 15 E 0.104108388E+05 0.346414288E+01 0.248281331E+01 -0.340554113E-02 16 E 0.116918305E+05 -0.100889523E+00 0.394713246E+01 0.161408808E+01 17 M 0.119297899E+05 -0.224760068E+01 0.186281871E+01 0.145168657E+01 18 E 0.127363257E+05 -0.452106189E+01 0.108721153E+01 -0.112686885E-03 19 E 0.140188225E+05 -0.396080770E+01 -0.234503487E+01 -0.696408166E+00 20 M 0.141489070E+05 -0.248618346E+01 -0.721205930E+01 0.774444667E+00 21 E 0.150894886E+05 0.281758080E+01 -0.191298279E+01 0.528412709E-02 22 E 0.163670285E+05 ================ PARENT CYCLER 3.768Gh+3 ======================= Parent cycler number 55 Approximate search space (synodic periods after J2000) 8 Number of steps to walk eccentricity/inclination 243 / 243 Number of cycles 7 Total delta v over 44.69 years (km/s) 0.000000 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 1 93.5 ****** 7.712 5.921 5.921 4 262.8 4.725 4.725 2.843 2.843 7 132.5 4.695 4.695 8.061 8.061 10 235.7 4.561 4.561 2.574 2.574 13 176.6 5.181 5.181 5.651 5.651 16 141.6 3.772 3.772 5.745 5.745 19 239.5 4.250 4.250 3.235 3.235 AVERAGE 183.2 4.531 4.985 4.862 4.862 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 6723.187401 LEG E/M time start vinfx vinfy vinfz (days) (km/s) (km/s) (km/s) 1 E 0.348860951E+02 0.265146145E+01 -0.718053080E+01 -0.942743041E+00 2 M 0.128389070E+03 0.461370874E+01 -0.369597641E+01 0.340655974E+00 3 E 0.105123584E+04 0.427532396E+01 0.200822703E+01 0.217826774E-01 4 E 0.233402492E+04 0.361893209E+00 0.371920668E+01 0.289230956E+01 5 M 0.259686992E+04 -0.172550850E+01 0.191822126E+01 0.119480642E+01 6 E 0.338272796E+04 -0.269443969E+01 0.754803621E+00 0.368852697E+01 7 E 0.465789197E+04 -0.437010002E+01 -0.163218604E+01 -0.530993610E+00 8 M 0.479035926E+04 -0.398354089E+01 -0.697890559E+01 0.637354886E+00 9 E 0.575306392E+04 0.112089299E+00 0.335930882E+00 -0.463325605E+01 10 E 0.703467090E+04 0.282503669E+01 0.262276886E+01 0.243766149E+01 11 M 0.727033030E+04 0.231770977E+01 0.636924412E+00 0.921306198E+00 12 E 0.807036648E+04 -0.924157399E+00 0.354483591E+01 0.354917021E+01 13 E 0.934888801E+04 -0.357183831E+01 0.374844153E+01 0.190403312E+00 14 M 0.952544366E+04 -0.542516918E+01 -0.157902170E+01 0.103833402E+00 15 E 0.103935284E+05 -0.176767805E+01 -0.220464620E+01 0.255451953E+01 16 E 0.116687982E+05 0.332769663E+01 -0.142397640E+01 -0.106160687E+01 17 M 0.118103820E+05 0.463558871E+01 -0.339318828E+01 0.764179252E-01 18 E 0.127501290E+05 0.347506176E+01 0.243757967E+01 -0.206852868E-02 19 E 0.140310541E+05 -0.116402153E+00 0.390707321E+01 0.166724314E+01 20 M 0.142705628E+05 -0.188059349E+01 0.184991953E+01 0.187239071E+01 21 E 0.150848050E+05 -0.268710654E+01 0.151003205E+00 0.336163201E+01 22 E 0.163595656E+05 ================ PARENT CYCLER 3.768Gh-f3 ======================= Parent cycler number 56 Approximate search space (synodic periods after J2000) 17 Number of steps to walk eccentricity/inclination 1 / 1 0.904266419E+04 0.962795233E+04 0.102970610E+05 0.106699533E+05 0.113696051E+05 0.119707903E+05 0.126063867E+05 0.130557531E+05 0.138237087E+05 0.143240984E+05 0.150373737E+05 0.156798343E+05 0.160715749E+05 -0.645266555E-05 0.107264354E-05 -0.105033185E-04 -0.771035600E-08 -0.210621505E-04 -0.759813090E-05 0.978576323E-05 0.156465078E-06 -0.927365084E-03 0.327568083E-04 0.868426463E-04 0.102124347E-07 0.186317032E-05 -0.221873641E-05 0.482134596E-05 -0.974746733E-05 -0.158018236E-07 -0.333489776E-05 0.528463086E-06 0.128169222E-07 0.278562623E-07 0.130030099E-04 -0.691790485E-06 -0.112232630E-04 0.276974114E-07 -0.152696741E-05 0.605751250E-06 0.164052530E-06 0.953531055E-08 0.151014926E-03 0.300874006E-04 0.385235548E-04 0.282653894E-08 0.837150447E-05 -0.283800166E-05 0.254151250E-07 0.175892920E-08 -0.852924153E-05 0.244584415E-04 time dv (days) 0.876575533E+02 0.590276242E+03 0.213679522E+04 0.234832660E+04 0.277871152E+04 0.369333831E+04 0.472515906E+04 0.527928881E+04 0.598906077E+04 0.703785613E+04 0.761217107E+04 0.909147786E+04 0.935510187E+04 0.995096172E+04 0.107054669E+05 0.117275244E+05 0.120507703E+05 0.134288739E+05 0.140383351E+05 0.144498931E+05 0.161370713E+05 dvx (km/s) -0.335570519E-10 -0.705332988E-11 -0.498660172E-11 -0.235124223E-09 0.354716689E-10 -0.587612152E-11 -0.584553394E-10 0.236599040E-11 0.709301107E-12 0.582652005E-11 -0.351839803E-11 -0.272312103E-11 0.526635404E-10 0.343109943E-10 -0.895141303E-11 -0.179193599E-10 0.740715375E-12 0.385899483E-11 0.131397617E-09 -0.206507478E-10 -0.575377121E-11 dvy (km/s) 0.375842784E-10 0.420620516E-11 -0.109850735E-10 0.741905810E-10 -0.162626706E-10 0.687972472E-11 0.251975498E-11 0.112099336E-11 0.148804428E-12 -0.148076940E-10 -0.595879065E-11 0.153433899E-11 0.704936177E-10 0.136999277E-10 -0.166826298E-10 -0.335702789E-10 -0.233953628E-12 0.195429815E-11 -0.320557805E-10 0.117555498E-10 -0.911013776E-12 dxz (km/s) 0.104457608E-11 0.239358123E-11 0.237865234E-11 0.332825904E-11 0.141315122E-11 -0.261348110E-10 0.524246267E-11 0.154425929E-11 0.253380873E-12 0.114186731E-12 0.885928862E-12 -0.228998486E-11 0.847358548E-13 -0.189002291E-12 -0.509620633E-11 0.112182005E-11 -0.389040910E-11 0.790497084E-12 -0.206548813E-11 0.294147086E-12 -0.517853901E-11 time dv (days) 0.489115413E+02 0.571355522E+03 0.137193311E+04 0.237345167E+04 0.292693030E+04 0.368876732E+04 0.467776206E+04 0.511767884E+04 0.606064959E+04 0.709122916E+04 0.764634730E+04 0.836442643E+04 0.937537136E+04 0.965565638E+04 0.106995932E+05 0.116900358E+05 0.122802555E+05 0.130319325E+05 0.140669804E+05 0.146288293E+05 0.153779999E+05 dvx (km/s) -0.543506520E-09 0.131609250E-10 0.711814248E-10 0.121394652E-09 -0.309513210E-11 0.700802720E-10 -0.246597045E-09 0.391190307E-10 -0.150920758E-10 0.281008895E-10 0.361098745E-11 -0.276445559E-11 -0.139545486E-11 0.545946912E-11 0.519162115E-12 -0.434045982E-10 0.131080167E-10 -0.392380743E-10 -0.308951060E-10 -0.628946715E-11 0.278277507E-09 dvy (km/s) 0.360056287E-09 -0.106312497E-11 0.117238049E-09 0.237276927E-09 -0.156476123E-10 0.242757890E-10 -0.191775840E-10 -0.843158956E-10 -0.540490750E-11 -0.597532447E-11 0.254951587E-11 0.545946912E-11 -0.122995127E-10 -0.273469471E-11 -0.106312497E-10 -0.253777685E-10 0.252140836E-11 -0.542474809E-10 -0.898282730E-10 0.144174957E-11 0.667139852E-10 dxz (km/s) 0.296393620E-10 0.543136162E-11 0.490426521E-11 -0.572070356E-12 -0.525563806E-11 0.970204870E-11 -0.178151968E-11 0.229703919E-11 -0.793747619E-11 -0.591869612E-11 0.206424809E-11 -0.111954665E-11 -0.411692251E-12 -0.327483412E-11 -0.558285279E-11 0.269232678E-11 0.431972021E-11 0.689400811E-10 0.679126875E-12 0.153289228E-11 0.254350169E-10 209 Number of cycles 7 Total delta v over 45.04 years (km/s) 0.033288 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 1 337.6 ****** 7.656 3.320 3.319 5 177.6 4.889 4.889 5.782 5.782 9 137.1 3.644 3.643 5.677 5.678 13 227.2 4.308 4.307 3.581 3.581 17 124.9 4.839 4.840 8.174 8.174 21 225.8 4.597 4.597 2.597 2.597 25 169.6 5.067 5.068 6.086 6.086 AVERAGE 200.0 4.557 5.000 5.031 5.031 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 13742.863583 LEG E/M time start vinfx vinfy vinfz time dv dvx dvy dxz (days) (km/s) (km/s) (km/s) (days) (km/s) (km/s) (km/s) 1 E -0.775507826E+02 0.472168137E+01 0.592253739E+01 -0.111608164E+01 -0.269149535E+02 0.212016453E-02 -0.175613305E-02 0.142890532E-03 2 M 0.260021412E+03 0.323384636E+01 -0.283950129E+00 0.690279611E+00 0.378138072E+03 -0.290140581E-04 -0.210686946E-05 0.635870460E-04 3 E 0.104746581E+04 -0.373499791E+00 -0.422499335E-01 -0.470990988E+01 0.107490955E+04 0.441292835E-03 -0.345252607E-05 -0.734802780E-04 4 E 0.123042409E+04 -0.348805520E+01 0.272372930E+01 0.207027355E+01 0.139478738E+04 0.441242018E-04 0.139040514E-03 -0.290563666E-03 5 E 0.232617935E+04 -0.343652358E+01 0.347009675E+01 0.215909010E+00 0.235281781E+04 0.136391329E-03 0.113706982E-03 -0.788769195E-04 6 M 0.250376905E+04 -0.552624513E+01 -0.157446680E+01 0.643251763E+00 0.279266757E+04 0.612396279E-04 -0.179225841E-03 -0.492076867E-04 7 E 0.340657691E+04 0.220471844E+00 -0.336719500E-01 -0.376692623E+01 0.343376498E+04 -0.341546678E-03 0.271633431E-03 -0.770356922E-04 8 E 0.358783069E+04 0.309831066E+01 -0.194391636E+01 0.386263976E-03 0.414323823E+04 0.484591827E-03 0.686924740E-04 0.149228479E-04 9 E 0.465592212E+04 0.316317443E+01 -0.150780542E+01 -0.997629632E+00 0.467648956E+04 -0.793490103E-03 -0.657135710E-03 0.516851366E-04 10 M 0.479303835E+04 0.464596783E+01 -0.326290016E+01 0.629794704E-01 0.529039967E+04 -0.265597074E-03 -0.237593667E-03 -0.991347246E-04 11 E 0.573145592E+04 -0.189066864E+00 0.236214892E+00 -0.424188488E+01 0.575938532E+04 -0.864616685E-04 -0.342317854E-03 -0.371374406E-04 12 E 0.591765189E+04 -0.399970817E+01 0.991467443E+00 0.126236456E+01 0.636690926E+04 0.120538189E-03 0.178906128E-03 0.218432455E-03 13 E 0.701340158E+04 -0.149937222E+00 0.404950018E+01 0.146011798E+01 0.704748282E+04 0.240915987E-04 0.271605734E-03 -0.100222844E-03 14 M 0.724060986E+04 -0.159168776E+01 0.200872273E+01 0.250106740E+01 0.756224545E+04 -0.120385081E-04 -0.161792540E-03 -0.692554380E-04 15 E 0.806531651E+04 -0.350836529E-01 -0.379337542E+00 -0.476518039E+01 0.809216458E+04 0.284797737E-03 0.125212822E-03 0.691145583E-05 16 E 0.824430362E+04 0.117538620E+01 -0.331888418E+01 0.331387957E+01 0.840866970E+04 0.320639472E-03 -0.513002150E-03 0.139749611E-03 17 E 0.934007748E+04 -0.405200157E+01 -0.256229202E+01 -0.661092841E+00 0.935881103E+04 -0.171417004E-02 0.103763833E-03 -0.547275413E-04 18 M 0.946496785E+04 -0.291134168E+01 -0.761968376E+01 0.530537928E+00 0.980413037E+04 0.567166422E-03 -0.394434187E-04 0.879055743E-04 19 E 0.104340036E+05 0.550576574E-01 0.344747622E+00 -0.465322046E+01 0.104619378E+05 -0.400077745E-03 -0.361858198E-03 -0.970999385E-04 20 E 0.106202312E+05 -0.284906942E+01 0.361194576E+01 -0.212043168E+00 0.110804541E+05 0.204349496E-04 0.360297895E-04 0.607696565E-04 21 E 0.117160001E+05 0.277071975E+01 0.308577300E+01 0.198218570E+01 0.117611572E+05 0.150109035E-03 -0.188854474E-03 -0.758858378E-04 22 M 0.119417858E+05 0.221094828E+01 0.809884685E+00 0.109551411E+01 0.120867371E+05 -0.812258673E-04 -0.198985639E-04 0.120747104E-03 23 E 0.127470707E+05 -0.397373515E+00 -0.126575872E+00 -0.487838867E+01 0.127744056E+05 -0.260808093E-03 0.163846045E-03 0.565315373E-04 24 E 0.129293033E+05 -0.370128062E+01 0.145079313E+01 0.313205988E+01 0.134223947E+05 0.604827694E-03 -0.185261218E-03 0.302034425E-03 25 E 0.140250618E+05 -0.403172924E+01 0.307040631E+01 0.759196047E-01 0.140505043E+05 -0.480366583E-03 0.106829471E-02 -0.108072325E-03 26 M 0.141946782E+05 -0.556079202E+01 -0.236848223E+01 0.714992638E+00 0.144927821E+05 0.162325681E-03 -0.394731613E-03 -0.702088915E-04 27 E 0.150980234E+05 0.189721446E+00 -0.120700202E-01 -0.350215336E+01 0.151252430E+05 -0.389868524E-03 0.723086261E-03 -0.757232891E-04 28 E 0.152794871E+05 0.337372605E+01 -0.409512558E+00 -0.556613306E-02 0.158372541E+05 0.840875400E-03 0.191289139E-04 -0.917325114E-05 29 E 0.163731479E+05 ================ PARENT CYCLER 3.768Gh-fff3 ======================= Parent cycler number 62 Approximate search space (synodic periods after J2000) 20 Number of steps to walk eccentricity/inclination 243 / 243 Number of cycles 7 Total delta v over 44.81 years (km/s) 0.037802 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 1 144.8 ****** 10.669 5.789 5.789 7 140.2 3.730 3.730 5.728 5.728 13 239.5 4.286 4.286 3.195 3.195 19 128.9 4.484 4.484 8.154 8.154 25 227.2 4.571 4.571 2.603 2.603 31 168.7 5.070 5.070 6.161 6.161 37 162.9 4.060 4.063 4.712 4.712 AVERAGE 173.2 4.367 5.268 5.191 5.191 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 16082.755644 LEG E/M time start vinfx vinfy vinfz time dv dvx dvy dxz (days) (km/s) (km/s) (km/s) (days) (km/s) (km/s) (km/s) 1 E 0.283206092E+02 -0.824958794E+01 0.676434518E+01 0.904481095E-01 0.500461126E+02 -0.244775319E-04 0.518583379E-04 0.412821783E-05 2 M 0.173157298E+03 -0.543642826E+01 -0.196953336E+01 0.279801337E+00 0.604547243E+03 0.546883355E-07 -0.210438991E-07 0.130643159E-07 3 E 0.103593719E+04 0.179928724E+00 -0.145415725E+00 -0.382196486E+01 0.106287373E+04 0.194750616E-05 0.723310698E-05 -0.384720357E-06 4 E 0.121551413E+04 0.217894437E+01 0.300570954E+01 -0.540387180E+00 0.147481969E+04 -0.208557056E-04 0.337208810E-04 -0.332316664E-03 5 E 0.158073323E+04 -0.839017794E+00 -0.624423991E+00 -0.359761151E+01 0.171952830E+04 0.129656333E-02 -0.221153967E-03 0.580949343E-04 6 E 0.194598343E+04 -0.210160337E+01 -0.218695745E+01 -0.220138409E+01 0.204825634E+04 -0.129601863E-04 0.113136200E-05 0.885116958E-04 7 E 0.231124383E+04 0.328022792E+01 -0.143909320E+01 -0.104125391E+01 0.233227928E+04 0.324598054E-05 -0.843942572E-05 0.419236911E-06 8 M 0.245148016E+04 0.464056613E+01 -0.335683102E+01 0.744793056E-01 0.288354831E+04 -0.523478105E-08 0.135137403E-08 0.799782052E-09 9 E 0.339075874E+04 -0.187483209E+00 0.240135762E+00 -0.424143925E+01 0.341869121E+04 -0.150537656E-05 0.345002043E-05 0.532821828E-06 10 E 0.357697522E+04 -0.326869150E+01 -0.277730322E+01 0.388850402E+00 0.382897923E+04 0.184539441E-05 -0.714073349E-05 0.538631054E-04 11 E 0.394219842E+04 -0.431137820E-01 -0.432355635E+00 -0.428446431E+01 0.418692069E+04 -0.537426529E-03 -0.144608769E-03 0.144099456E-03 12 E 0.430745554E+04 0.348259337E+01 0.218815117E+01 -0.127412030E+01 0.440607668E+04 0.412696015E-05 -0.312161773E-05 -0.127795764E-04 13 E 0.467271900E+04 -0.633670900E-01 0.397398625E+01 0.160368963E+01 0.470864455E+04 -0.265120418E-07 0.489157805E-05 0.128497602E-06 14 M 0.491222261E+04 -0.209363277E+01 0.182585651E+01 0.157751699E+01 0.533276329E+04 -0.352993864E-09 -0.277733544E-09 -0.181320262E-10 15 E 0.572095468E+04 -0.563761438E-01 -0.328073400E+00 -0.438212936E+01 0.574781937E+04 -0.204002045E-06 -0.172984959E-06 0.313718531E-07 16 E 0.590005263E+04 0.438059022E+01 -0.607376326E+00 -0.541048673E+00 0.615208159E+04 -0.477699054E-05 -0.653666851E-05 0.212008811E-04 17 E 0.626531199E+04 -0.162392548E+00 0.365050808E+00 -0.445509911E+01 0.642237140E+04 -0.234331293E-03 0.339592213E-03 0.869604065E-05 18 E 0.663056643E+04 -0.423595953E+01 0.127158521E+01 -0.668038581E+00 0.674013630E+04 -0.188461803E-05 -0.661633629E-05 0.302720011E-03 19 E 0.699579934E+04 -0.367649166E+01 -0.247993145E+01 -0.666381812E+00 0.701513750E+04 -0.418719226E-07 0.184342113E-06 0.141486111E-07 20 M 0.712472043E+04 -0.293005605E+01 -0.759006085E+01 0.538741770E+00 0.775421263E+04 -0.225983663E-08 -0.111675408E-08 0.128289154E-09 21 E 0.809316997E+04 0.567717737E-01 0.336556974E+00 -0.460536413E+01 0.812110047E+04 -0.328815381E-05 -0.334530353E-06 -0.138022186E-06 22 E 0.827937329E+04 -0.121971926E+01 -0.168339294E+00 0.438859010E+01 0.850217913E+04 0.449919890E-03 0.104242153E-04 -0.957234561E-05 23 E 0.864462877E+04 -0.453787285E+01 0.346483781E+00 -0.186825663E+00 0.869942133E+04 0.361839090E-06 -0.248977726E-05 0.936101441E-06 24 E 0.900991250E+04 0.268984896E+00 -0.384939018E+00 0.454411714E+01 0.921810970E+04 0.966254811E-03 -0.203571829E-03 -0.932390245E-04 25 E 0.937517075E+04 0.270330793E+01 0.306783938E+01 0.204406067E+01 0.941833287E+04 -0.252679008E-05 0.437340823E-06 -0.136997074E-06 26 M 0.960233980E+04 0.222659481E+01 0.798681495E+00 0.108531866E+01 0.999667775E+04 -0.684433241E-08 0.486958853E-08 -0.186195643E-08 27 E 0.104071111E+05 -0.396551080E+00 -0.125543925E+00 -0.487221931E+01 0.104344467E+05 -0.180903106E-05 -0.156517570E-05 0.374743440E-06 28 E 0.105893482E+05 0.203627947E+01 -0.462436005E+01 0.107063268E+00 0.108267728E+05 0.175177971E-05 -0.356878890E-06 0.323639367E-05 29 E 0.109546168E+05 0.438384864E+00 -0.144361757E-01 -0.502762112E+01 0.112066462E+05 -0.191632195E-05 0.315368715E-04 -0.364834584E-05 30 E 0.113198768E+05 -0.122278985E+01 0.489501113E+01 -0.184364139E+00 0.114331067E+05 0.551841520E-05 0.450915531E-05 -0.114993891E-03 31 E 0.116851346E+05 -0.404843884E+01 0.305027880E+01 0.855077551E-01 0.117104367E+05 -0.399336387E-05 0.168203646E-05 0.203316619E-06 32 M 0.118538155E+05 -0.569222538E+01 -0.233268767E+01 0.330912485E+00 0.120451137E+05 0.128544985E-03 -0.113631926E-03 0.714262309E-02 33 E 0.127233528E+05 0.222571428E+00 -0.168483422E+00 -0.410515161E+01 0.127502838E+05 0.756345584E-05 -0.859386389E-05 0.155509843E-05 34 E 0.129028928E+05 0.239406548E+01 0.324275669E+01 0.733894759E-01 0.131038467E+05 -0.244158452E-02 -0.324643534E-03 0.441457196E-04 35 E 0.132682635E+05 -0.352474695E+00 0.143439261E-01 -0.404825194E+01 0.134034072E+05 0.433679199E-03 -0.707566311E-04 0.251679114E-04 36 E 0.136335167E+05 -0.192138339E+01 -0.182876569E+01 -0.308195766E+01 0.138818931E+05 0.798722610E-03 -0.621808238E-04 0.354946855E-03 37 E 0.139987761E+05 0.391929817E+01 -0.316330712E+00 -0.102155340E+01 0.140232064E+05 -0.242124390E-05 -0.449342467E-05 -0.272555590E-06 210 38 M 0.141616448E+05 0.438604888E+01 -0.165937303E+01 -0.459490637E+00 0.146191409E+05 0.203512541E-07 -0.130357573E-07 0.622468463E-08 39 E 0.151147616E+05 -0.198158653E+00 0.584076909E-01 -0.348638762E+01 0.151424735E+05 -0.184618964E-04 0.125798022E-04 0.258266171E-05 40 E 0.152995076E+05 -0.999622277E+00 -0.343894095E+01 -0.239244707E+00 0.155551693E+05 0.235052162E-04 -0.229454166E-04 -0.121482294E-03 41 E 0.156647386E+05 0.718210383E+00 0.136024071E+01 -0.324752148E+01 0.158144924E+05 -0.128042346E-02 -0.421396814E-03 -0.584630807E-05 42 E 0.160299917E+05 0.138602174E+01 0.328898869E+01 -0.479103718E+00 0.160847787E+05 0.254040477E-04 -0.798308378E-05 0.346938151E-05 43 E 0.163952383E+05 ================ PARENT CYCLER 3.768Gfh-ff3 ======================= Parent cycler number 64 Approximate search space (synodic periods after J2000) 17 Number of steps to walk eccentricity/inclination 9 / 9 Number of cycles 7 Total delta v over 44.93 years (km/s) 0.007746 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 1 285.3 ****** 5.459 2.547 2.547 7 175.6 5.274 5.274 5.655 5.655 13 141.9 3.775 3.775 5.737 5.737 19 239.3 4.292 4.292 3.195 3.195 25 128.8 4.484 4.484 8.174 8.174 31 226.6 4.584 4.584 2.598 2.598 37 170.3 5.057 5.057 6.045 6.045 AVERAGE 195.4 4.578 4.703 4.850 4.850 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 13742.863583 LEG E/M time start vinfx vinfy vinfz time dv dvx dvy dxz (days) (km/s) (km/s) (km/s) (days) (km/s) (km/s) (km/s) 1 E -0.392759792E+02 0.195735001E+01 0.442682612E+01 -0.252392845E+01 0.352531936E+01 0.333951548E-04 0.204725684E-04 -0.208609726E-05 2 M 0.246066011E+03 0.222344574E+01 0.698000684E+00 0.102724298E+01 0.673014049E+03 0.157641199E-06 -0.689178151E-07 -0.424822731E-07 3 E 0.105162835E+04 -0.148463873E+01 0.480329429E+01 -0.848721118E+00 0.129270061E+04 0.658072555E-04 -0.147443057E-04 0.149682786E-03 4 E 0.141688935E+04 -0.433932385E+00 -0.103292845E+00 -0.506051038E+01 0.144429268E+04 0.691556049E-05 -0.290283958E-05 -0.107545368E-05 5 E 0.159957826E+04 0.148780638E+01 -0.480838484E+01 0.149856657E+01 0.185161182E+04 0.167072421E-04 -0.826530550E-05 -0.207008530E-03 6 E 0.196484430E+04 0.348126201E+00 0.564276760E+00 0.522742982E+01 0.212190487E+04 0.105454774E-03 0.186119735E-03 -0.494811046E-04 7 E 0.233010144E+04 -0.362928712E+01 0.382165008E+01 0.188921995E+00 0.235644804E+04 -0.552065971E-05 0.492901229E-05 0.202223984E-06 8 M 0.250574543E+04 -0.542992075E+01 -0.157885796E+01 0.803307827E-01 0.322602291E+04 -0.569040037E-08 0.416924214E-08 0.178281361E-07 9 E 0.337354962E+04 -0.223084817E+01 -0.299859768E+01 -0.846238283E+00 0.361096581E+04 0.327357393E-05 -0.133040685E-04 0.237094797E-04 10 E 0.373880530E+04 0.182083208E+00 -0.173179808E+00 -0.383867726E+01 0.376572810E+04 0.434195147E-05 -0.412254072E-05 0.770530479E-06 11 E 0.391829066E+04 -0.374272202E+00 -0.409739707E-01 0.375294877E+01 0.405708849E+04 0.869492458E-03 -0.132431563E-03 -0.685446328E-04 12 E 0.428354810E+04 0.143480056E-01 0.378818121E+00 0.375320427E+01 0.441504012E+04 0.649218857E-03 -0.119628579E-03 -0.381711051E-04 13 E 0.464880370E+04 0.333092604E+01 -0.142132711E+01 -0.106484713E+01 0.467009224E+04 -0.228958468E-05 0.834571962E-05 -0.661906860E-06 14 M 0.479072728E+04 0.462904898E+01 -0.338797822E+01 0.683216939E-01 0.520437444E+04 0.351687030E-08 -0.282959556E-08 -0.114913021E-08 15 E 0.573083446E+04 0.323800480E+01 0.261753992E+01 -0.707470104E+00 0.598652662E+04 0.513417703E-05 -0.179508155E-04 0.132695109E-03 16 E 0.609610897E+04 -0.166337389E+00 0.244515769E+00 -0.423298193E+01 0.612404051E+04 0.283319296E-06 0.306829092E-05 0.335615203E-06 17 E 0.628231922E+04 0.361064924E+01 0.227934295E+01 0.505027822E+00 0.638823714E+04 0.880223110E-05 -0.364840638E-05 -0.194822819E-03 18 E 0.664755343E+04 0.360631525E+01 0.229779253E+01 0.554915705E+00 0.675348008E+04 -0.129039583E-05 -0.485439824E-07 0.853638396E-04 19 E 0.701281775E+04 -0.566012541E-01 0.398617588E+01 0.159060039E+01 0.704870602E+04 0.853266778E-07 -0.324792706E-05 -0.867711584E-07 20 M 0.725207288E+04 -0.209698262E+01 0.182647850E+01 0.157398044E+01 0.767258274E+04 -0.616443837E-09 -0.388336569E-09 -0.139611931E-09 21 E 0.806074568E+04 -0.437119381E+01 0.474844151E+00 0.150998147E+00 0.812649749E+04 -0.236321874E-06 0.109681854E-05 -0.194849678E-05 22 E 0.842603351E+04 -0.648238311E-01 -0.309657846E+00 -0.439985351E+01 0.845289902E+04 -0.207751724E-05 -0.661963363E-06 0.159340999E-06 23 E 0.860513694E+04 0.232457930E+01 -0.175237918E+00 0.381729499E+01 0.879141755E+04 -0.393592027E-03 0.145376778E-03 0.172920523E-04 24 E 0.897039303E+04 -0.382753187E+01 0.118821562E+01 0.198904830E+01 0.907996228E+04 -0.181841548E-04 -0.754748093E-05 -0.629265579E-03 25 E 0.933562384E+04 -0.367075985E+01 -0.248734863E+01 -0.665028491E+00 0.935494566E+04 0.163007890E-06 -0.812449942E-06 -0.625923617E-07 26 M 0.946443595E+04 -0.294897920E+01 -0.760502540E+01 0.534810290E+00 0.101040404E+05 -0.273280985E-09 -0.193105160E-09 -0.100913170E-11 27 E 0.104335336E+05 -0.443534675E+01 0.115106784E+01 -0.866617280E+00 0.104883229E+05 -0.747988108E-08 -0.404887788E-07 -0.119837765E-08 28 E 0.107987956E+05 0.694208202E-01 0.364242709E+00 -0.464148818E+01 0.108416257E+05 0.857920389E-07 -0.123685999E-06 -0.169672761E-07 29 E 0.109850134E+05 0.447952295E+01 -0.102606708E+01 0.169944433E+00 0.110397979E+05 -0.385045445E-07 -0.238561295E-06 -0.138132203E-08 30 E 0.113502437E+05 0.445717233E+01 -0.100682885E+01 0.329406266E+00 0.114050349E+05 0.262529766E-07 0.164030910E-06 0.186643412E-08 31 E 0.117155183E+05 0.273579894E+01 0.307555301E+01 0.201664595E+01 0.117495129E+05 -0.186013716E-06 0.192485869E-07 -0.983936997E-08 32 M 0.119421485E+05 0.221867664E+01 0.806839543E+00 0.108435862E+01 0.123364990E+05 -0.202636248E-08 0.121142677E-08 -0.133061953E-09 33 E 0.127469454E+05 -0.197745926E+01 0.447629263E+01 -0.192372195E+00 0.128017357E+05 -0.587606643E-06 -0.432596839E-06 0.467675897E-08 34 E 0.131122139E+05 -0.382390305E+00 -0.145012803E+00 -0.485873811E+01 0.131395505E+05 0.972084459E-06 -0.938180835E-07 -0.149689119E-06 35 E 0.132944579E+05 0.136978553E+01 -0.275204873E+01 0.399461471E+01 0.135391777E+05 0.235768137E-04 -0.573374118E-04 -0.676055055E-05 36 E 0.136597113E+05 0.105479712E-01 0.128257032E+01 0.488652037E+01 0.139226945E+05 -0.109651573E-03 0.202414243E-03 0.307609463E-04 37 E 0.140249657E+05 -0.401442851E+01 0.307450862E+01 0.703675914E-01 0.140505041E+05 -0.793710716E-06 0.343153669E-06 0.396505717E-07 38 M 0.141952214E+05 -0.551532755E+01 -0.236893915E+01 0.711111603E+00 0.147167667E+05 -0.316644910E-08 0.652991858E-08 0.170998957E-10 39 E 0.150944374E+05 -0.176916126E+00 -0.272646834E+01 -0.201546988E+01 0.153574225E+05 0.805646055E-05 0.109369617E-05 -0.170338876E-03 40 E 0.154596945E+05 0.187694226E+00 -0.379605637E-01 -0.340840552E+01 0.154868789E+05 0.100421557E-04 -0.539343488E-05 0.121169208E-05 41 E 0.156409239E+05 0.211660518E+00 0.280353916E+01 0.174631262E+01 0.158966218E+05 0.254219158E-03 -0.153389587E-03 -0.736183775E-03 42 E 0.160062066E+05 -0.323942256E+00 -0.938090041E+00 0.313750689E+01 0.161742262E+05 0.714574995E-03 0.673924328E-03 0.140299676E-04 43 E 0.163714665E+05 ================ PARENT CYCLER 3.768Gfh+ff3 ======================= Parent cycler number 65 Approximate search space (synodic periods after J2000) 17 Number of steps to walk eccentricity/inclination 9 / 9 Number of cycles 7 Total delta v over 44.91 years (km/s) 0.008819 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 1 280.4 ****** 5.619 2.562 2.562 7 174.9 5.360 5.360 5.652 5.652 13 141.7 3.768 3.768 5.735 5.735 19 239.2 4.293 4.293 3.195 3.195 25 128.8 4.484 4.484 8.172 8.172 31 226.7 4.581 4.582 2.599 2.599 37 170.9 5.057 5.057 5.995 5.995 AVERAGE 194.7 4.591 4.737 4.844 4.844 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 13742.863583 LEG E/M time start vinfx vinfy vinfz time dv dvx dvy dxz (days) (km/s) (km/s) (km/s) (days) (km/s) (km/s) (km/s) 1 E -0.431215636E+02 0.213621323E+01 0.454823049E+01 -0.251530349E+01 -0.106308548E+01 0.380487335E-04 -0.289983326E-05 -0.248122972E-05 2 M 0.237268290E+03 0.208242694E+01 0.825703577E+00 0.124395767E+01 0.653034855E+03 0.376433349E-06 -0.659865200E-07 -0.382909244E-06 3 E 0.105249685E+04 -0.153849925E+01 0.463148746E+01 0.175294351E+01 0.130087683E+04 0.109288744E-03 -0.116788958E-04 -0.297189274E-03 4 E 0.141776152E+04 -0.445215222E+00 -0.112485419E+00 0.514682145E+01 0.144515689E+04 0.181999395E-04 -0.550608917E-05 0.294583291E-05 5 E 0.160039731E+04 0.199668846E+00 0.118441475E+01 -0.520445776E+01 0.176111155E+04 -0.118997833E-03 -0.206053189E-03 -0.568036023E-04 6 E 0.196565695E+04 -0.702448078E+00 0.518350871E+01 -0.107868206E+01 0.203870839E+04 0.193202530E-04 0.614159388E-05 -0.199688770E-04 7 E 0.233091416E+04 -0.368219306E+01 0.389002466E+01 0.186648867E+00 0.235715099E+04 -0.166150278E-04 0.119507497E-04 0.428029162E-06 8 M 0.250582640E+04 -0.542528978E+01 -0.158211450E+01 0.105299070E+00 0.289644703E+04 -0.680889381E-09 -0.260781413E-09 0.369127984E-11 9 E 0.337387226E+04 -0.205082218E+01 -0.281046800E+01 0.155068247E+01 0.363320263E+04 0.495791686E-05 -0.154312302E-05 0.160312221E-03 10 E 0.373912630E+04 0.181594851E+00 -0.169777380E+00 0.381404371E+01 0.376605109E+04 0.328391555E-06 -0.253165233E-06 -0.529167287E-07 11 E 0.391862492E+04 -0.261185196E+01 -0.258116055E+01 -0.752818592E+00 0.397341048E+04 -0.164817577E-06 0.140562800E-06 0.555735795E-08 12 E 0.428386196E+04 0.343270052E+00 0.750396374E+00 -0.367393613E+01 0.441900807E+04 0.528872252E-03 -0.468271746E-04 0.444769085E-04 13 E 0.464912172E+04 0.332376452E+01 -0.142313870E+01 -0.106149326E+01 0.467037754E+04 -0.117313575E-07 0.657355164E-06 -0.175454536E-07 211 14 M 0.479082720E+04 0.463011020E+01 -0.338270914E+01 0.683981826E-01 15 E 0.573084677E+04 0.316748433E+01 0.256998707E+01 0.110306540E+01 16 E 0.609612161E+04 -0.166470222E+00 0.245774587E+00 0.423408604E+01 17 E 0.628233512E+04 0.351248936E+01 0.220756487E+01 -0.113195432E+01 18 E 0.664756867E+04 0.351932637E+01 0.223436873E+01 -0.110527510E+01 19 E 0.701283317E+04 -0.560537557E-01 0.398707315E+01 0.158969227E+01 20 M 0.725207506E+04 -0.209653394E+01 0.182644627E+01 0.157441901E+01 21 E 0.806075645E+04 -0.437066708E+01 0.474027791E+00 0.153467152E+00 22 E 0.842604426E+04 -0.646325712E-01 -0.308061565E+00 0.439948921E+01 23 E 0.860514657E+04 -0.304697632E+00 0.396072102E+00 -0.444428116E+01 24 E 0.897040103E+04 -0.427527195E+01 0.128462755E+01 0.271357228E+00 25 E 0.933563427E+04 -0.367164183E+01 -0.248641278E+01 -0.665201178E+00 26 M 0.946445891E+04 -0.294652039E+01 -0.760310224E+01 0.535308459E+00 27 E 0.104334737E+05 -0.119844588E+01 0.571180705E+00 0.446443473E+01 28 E 0.107987227E+05 0.701101026E-01 0.365424435E+00 0.464322981E+01 29 E 0.109849392E+05 0.447601555E+01 -0.103129106E+01 0.173536795E+00 30 E 0.113501692E+05 0.445373129E+01 -0.101176271E+01 0.334761203E+00 31 E 0.117154439E+05 0.273050317E+01 0.307432806E+01 0.202068358E+01 32 M 0.119421466E+05 0.221990883E+01 0.805306230E+00 0.108514519E+01 33 E 0.127469639E+05 -0.197824717E+01 0.447469443E+01 -0.181334173E+00 34 E 0.131122325E+05 -0.381993634E+00 -0.143460349E+00 0.485912088E+01 35 E 0.132944751E+05 -0.220028555E-01 0.133205518E+01 -0.485888572E+01 36 E 0.136597352E+05 -0.116805064E+01 0.473706910E+01 0.125200477E+01 37 E 0.140249933E+05 -0.400374071E+01 0.308865122E+01 0.636275600E-01 38 M 0.141958860E+05 -0.546599844E+01 -0.236434370E+01 0.684530742E+00 39 E 0.150828645E+05 -0.897966949E+00 -0.317168902E+01 -0.134589358E+00 40 E 0.154481463E+05 0.167308665E+00 -0.466092071E-01 0.328034221E+01 41 E 0.156286683E+05 -0.114271505E+01 -0.264516779E+01 0.137658645E+01 42 E 0.159939235E+05 -0.125759592E+01 -0.290886604E+01 -0.287753037E+00 43 E 0.163591664E+05 ================ PARENT CYCLER 3.784Gg3 ======================= Parent cycler number 74 Approximate search space (synodic periods after J2000) 1 Number of steps to walk eccentricity/inclination 81 / 81 Number of cycles 7 Total delta v over 44.71 years (km/s) 0.074448 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 1 140.9 ****** 8.288 3.978 3.978 4 210.0 5.740 5.740 3.635 3.635 7 122.0 4.596 4.596 8.338 8.338 10 226.5 4.947 4.947 2.553 2.553 13 163.3 5.078 5.078 6.408 6.408 16 146.2 3.608 3.608 4.242 4.242 19 220.8 4.784 4.785 3.642 3.642 AVERAGE 175.7 4.792 5.292 4.685 4.685 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 1263.763258 LEG E/M time start vinfx vinfy vinfz (days) (km/s) (km/s) (km/s) 1 E 0.595220741E+02 0.723888443E+01 -0.403514150E+01 0.600537830E-01 2 M 0.200400713E+03 0.359373791E+01 0.386901167E+00 -0.166008621E+01 3 E 0.106367473E+04 0.311715368E+01 0.481813053E+01 0.190282228E-02 4 E 0.235040691E+04 -0.129513504E+00 0.567554096E+01 0.850704466E+00 5 M 0.256037830E+04 -0.276623683E+01 0.166853129E+01 0.166688099E+01 6 E 0.337650377E+04 -0.456916584E+01 0.466685232E+00 -0.146537869E-02 7 E 0.465875053E+04 -0.322727161E+01 -0.318629468E+01 -0.747954155E+00 8 M 0.478077046E+04 -0.207785926E+01 -0.806512224E+01 0.405587510E+00 9 E 0.575874883E+04 -0.135121965E-01 0.404720720E+00 -0.495735013E+01 10 E 0.704097730E+04 0.292580171E+01 0.356233368E+01 0.179607693E+01 11 M 0.726750159E+04 0.211078970E+01 0.112829167E+01 0.887711489E+00 12 E 0.806023032E+04 -0.185805442E+01 0.472668314E+01 0.218124233E-02 13 E 0.934437287E+04 -0.438826837E+01 0.255441600E+01 -0.312941994E-02 14 M 0.950763912E+04 -0.560794854E+01 -0.300485976E+01 0.763344217E+00 15 E 0.104185025E+05 0.268608974E+00 -0.345639813E+01 0.105485104E+01 16 E 0.116962898E+05 0.358990468E+01 0.613823592E-03 -0.356670175E+00 17 M 0.118425368E+05 0.421604436E+01 -0.307156903E+00 -0.350584025E+00 18 E 0.127533646E+05 0.256942697E+01 0.404041005E+01 0.940190646E-02 19 E 0.140363740E+05 -0.702865988E+00 0.466090181E+01 0.820619454E+00 20 M 0.142571712E+05 -0.346398852E+01 0.103039740E+01 -0.453238938E+00 21 E 0.151094790E+05 -0.309943986E+01 -0.237503837E+01 -0.192925111E-02 22 E 0.163890467E+05 ================ PARENT CYCLER 5.125gGgf3 ======================= Parent cycler number 89 Approximate search space (synodic periods after J2000) 11 Number of steps to walk eccentricity/inclination 27 / 27 Number of cycles 7 Total delta v over 44.85 years (km/s) 0.210006 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 2 106.4 11.701 11.701 9.683 9.683 7 91.5 5.592 5.592 10.981 10.981 12 153.8 5.251 5.251 7.553 7.553 17 115.8 5.621 5.621 11.605 11.605 22 122.2 5.198 5.195 6.958 6.958 27 138.5 5.525 5.525 9.856 9.856 32 94.5 5.402 5.402 10.810 10.810 AVERAGE 117.5 6.327 6.327 9.635 9.635 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 9307.533494 LEG E/M time start vinfx vinfy vinfz (days) (km/s) (km/s) (km/s) 1 E 0.865407746E+01 -0.319344687E+01 0.112355763E+02 -0.978238049E-03 2 E 0.569779474E+03 -0.659075706E+01 0.963317158E+01 0.826983469E+00 3 M 0.676131170E+03 -0.962965974E+01 0.832907604E+00 0.583221128E+00 4 E 0.126236079E+04 -0.420043090E+01 0.364000440E+01 -0.212428877E-03 5 E 0.195987598E+04 -0.353919300E+01 -0.425045097E+01 0.823007648E+00 6 E 0.232513372E+04 -0.416690955E+01 -0.375577077E+01 0.117009236E-02 7 E 0.286127900E+04 0.757376611E+00 -0.544530625E+01 -0.102519965E+01 8 M 0.295275743E+04 0.318408958E+01 -0.105093468E+02 -0.909320078E-02 9 E 0.358378635E+04 0.353568787E+01 -0.388996862E+01 -0.315586692E+00 10 E 0.431416623E+04 0.423053317E+01 0.269679415E+01 0.155412701E+01 0.520443581E+04 0.599019191E+04 0.612405364E+04 0.638825285E+04 0.675349537E+04 0.704871945E+04 0.767258938E+04 0.811554963E+04 0.845290961E+04 0.876951108E+04 0.902518602E+04 0.935495797E+04 0.100846284E+05 0.105978357E+05 0.108378282E+05 0.110397237E+05 0.114049604E+05 0.117494493E+05 0.120709174E+05 0.128017542E+05 0.131395689E+05 0.133492641E+05 0.137145239E+05 0.140506272E+05 0.147280731E+05 0.151376568E+05 0.154752246E+05 0.157199821E+05 0.160487099E+05 0.619558809E-09 0.919262923E-05 0.133687062E-06 -0.950222883E-06 -0.114763160E-06 0.492899886E-07 -0.215452278E-09 -0.482511023E-07 -0.831168358E-06 -0.252761702E-03 0.182482309E-07 0.113127092E-06 -0.838257008E-08 -0.374214410E-03 0.364198287E-05 -0.170297778E-05 0.114630895E-05 -0.822615479E-05 0.355189341E-06 -0.477953296E-04 0.763059502E-04 -0.378726610E-04 0.663715556E-04 -0.759464399E-04 -0.506034628E-06 0.626810010E-04 -0.149510984E-03 0.653832254E-04 -0.177099147E-03 -0.453932868E-09 -0.106183059E-04 0.159625075E-05 0.770995971E-05 -0.242687377E-06 -0.176857919E-05 -0.139631462E-09 0.439645305E-06 -0.259850417E-06 0.317044054E-03 0.841252309E-07 -0.549985861E-06 -0.429549443E-08 0.520047896E-03 -0.580807355E-05 -0.105282452E-04 0.712857528E-05 0.105767812E-05 0.338444759E-08 -0.352753240E-04 -0.757073412E-05 -0.123934047E-04 0.219805918E-04 0.324830071E-04 -0.151680103E-06 -0.447363411E-05 0.434067388E-04 -0.523407957E-05 0.592707846E-04 -0.106123185E-09 -0.394246233E-03 -0.176222057E-06 0.353070578E-03 -0.158713254E-04 -0.473695898E-07 -0.167580653E-10 0.217345573E-08 -0.629389053E-07 -0.373906103E-05 -0.842909758E-08 -0.422810395E-07 0.557788593E-09 -0.239540268E-04 0.806559935E-06 -0.613351813E-07 0.814802259E-07 -0.369966204E-06 0.660169940E-06 0.463318683E-05 0.114503547E-04 -0.674331169E-05 -0.183578637E-04 0.375479733E-05 -0.168345718E-06 0.235121511E-06 0.168884049E-04 -0.102785792E-03 0.118776318E-05 time dv (days) 0.806538699E+02 0.632037723E+03 0.135962313E+04 0.238190262E+04 0.316431115E+04 0.364577559E+04 0.467705352E+04 0.510350332E+04 0.606648366E+04 0.707495594E+04 0.766386596E+04 0.839410739E+04 0.936886281E+04 0.102454385E+05 0.111085077E+05 0.117182269E+05 0.122979507E+05 0.129714762E+05 0.140694936E+05 0.146918482E+05 0.154037796E+05 dvx (km/s) 0.100657194E-06 -0.577493451E-10 -0.170504412E-08 0.213837518E-07 -0.405244059E-11 0.161882684E-08 -0.196251050E-08 -0.295327188E-10 0.178749070E-07 0.300727069E-07 0.856266973E-09 0.777673997E-07 0.866975337E-07 -0.943196005E-06 0.289721059E-05 -0.517513358E-04 -0.217430841E-06 -0.388447527E-06 0.198616065E-05 -0.709474745E-07 -0.676326492E-04 dvy (km/s) 0.182907129E-07 -0.136579317E-09 -0.155470932E-07 -0.325812420E-07 0.903408216E-11 -0.316209409E-13 0.123352919E-08 0.110578224E-10 0.145194234E-07 0.216186909E-07 -0.720874784E-12 -0.161520196E-07 -0.153495583E-06 -0.134309516E-05 -0.173669288E-05 0.305241360E-04 -0.364153432E-06 -0.904055995E-07 0.284131600E-05 0.288326072E-06 -0.887719784E-04 dxz (km/s) -0.370132625E-08 0.187958593E-11 -0.140990536E-09 0.255970587E-09 -0.875951732E-11 -0.258591457E-10 -0.104658701E-09 0.506319983E-11 0.224397615E-08 0.440396418E-09 -0.587934148E-09 -0.202675089E-08 -0.821684853E-09 -0.330874476E-06 0.239545044E-05 -0.214549883E-06 -0.738436967E-07 -0.590413060E-08 0.207281209E-07 -0.779994348E-06 -0.198567826E-06 time dv dvx dvy dxz (days) (km/s) (km/s) (km/s) 0.249937998E+03 -0.287814218E-09 -0.188246265E-07 0.802473483E-10 0.585732228E+03 -0.143157498E-06 -0.101348594E-06 -0.152629259E-07 0.764065613E+03 0.310951653E-10 -0.154280431E-09 0.129785362E-11 0.136698807E+04 0.144077682E-06 0.162486499E-08 0.237435380E-08 0.220825124E+04 0.117606277E-05 0.113053720E-05 0.118549030E-03 0.258248346E+04 0.164156878E-07 0.860874454E-08 0.103285816E-09 0.287500077E+04 0.733888227E-08 0.301276058E-09 0.265227772E-09 0.304741177E+04 0.123458074E-09 -0.107056519E-10 0.378051083E-11 0.416809026E+04 -0.148084331E-05 0.687022958E-05 -0.442617867E-03 0.456986408E+04 0.179441678E-04 -0.767520351E-05 -0.722774031E-03 212 11 E 0.467944888E+04 0.481259813E+01 0.210774931E+01 -0.679316333E-03 12 E 0.521419518E+04 -0.933719563E+00 0.500658141E+01 0.127944210E+01 13 M 0.536797813E+04 -0.403398010E+01 0.636620865E+01 -0.495835882E+00 14 E 0.593096323E+04 0.262712531E+01 0.490302912E+01 -0.106404111E-02 15 E 0.663022106E+04 -0.502237682E+01 0.250133261E+01 -0.240376832E+00 16 E 0.699547028E+04 -0.462734297E+01 0.319516239E+01 -0.177040908E-02 17 E 0.753169669E+04 -0.533536833E+01 -0.176985961E+01 -0.123750698E-01 18 M 0.764744984E+04 -0.838198582E+01 -0.799000270E+01 0.758386081E+00 19 E 0.826158387E+04 -0.143958222E+01 -0.538830847E+01 0.456239760E-03 20 E 0.900298504E+04 0.463601909E+01 -0.234274208E+01 0.119232655E+00 21 E 0.936825181E+04 0.423180453E+01 -0.297816024E+01 -0.621782360E-03 22 E 0.990272572E+04 0.368491046E+01 0.365940992E+01 0.149312484E+00 23 M 0.100249389E+05 0.468237335E+01 0.508251071E+01 -0.809287098E+00 24 E 0.106194153E+05 0.548019913E+01 0.642635048E+00 -0.120905533E-02 25 E 0.113223440E+05 -0.850031029E+00 0.536930682E+01 0.976932093E+00 26 E 0.116875995E+05 -0.607242028E-01 0.554086974E+01 -0.898228398E-03 27 E 0.122234899E+05 -0.444983160E+01 0.317414043E+01 0.809125724E+00 28 M 0.123619855E+05 -0.983524355E+01 0.233181645E+00 0.589293619E+00 29 E 0.129511050E+05 -0.439088738E+01 0.311355849E+01 -0.124787532E-03 30 E 0.136548751E+05 -0.267876193E+01 -0.415384331E+01 0.218011216E+01 31 E 0.140201361E+05 -0.358118305E+01 -0.406746054E+01 -0.268806323E-03 32 E 0.145555107E+05 0.271079607E+01 -0.454261493E+01 -0.109397603E+01 33 M 0.146500423E+05 0.505034138E+01 -0.955134700E+01 -0.355769364E+00 34 E 0.152958843E+05 0.280264444E+01 -0.441760509E+01 0.262501417E+01 35 E 0.160264016E+05 0.918366206E-03 0.913582372E+00 0.579257267E+01 36 E 0.163916542E+05 ================ PARENT CYCLER 5.301gGff3 ======================= Parent cycler number 94 Approximate search space (synodic periods after J2000) 21 Number of steps to walk eccentricity/inclination 9 / 9 Number of cycles 7 Total delta v over 44.78 years (km/s) 0.247075 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 2 74.1 9.551 9.552 11.452 11.455 7 141.8 5.461 5.459 7.648 7.648 12 109.0 7.328 7.328 11.206 11.206 17 105.8 5.346 5.338 7.814 7.814 22 135.3 6.185 6.185 9.306 9.306 27 91.1 5.836 5.836 11.432 11.432 32 139.4 5.640 5.637 8.260 8.260 AVERAGE 113.8 6.478 6.476 9.588 9.589 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 17112.294551 LEG E/M time start vinfx vinfy vinfz (days) (km/s) (km/s) (km/s) 1 E -0.908115517E+01 -0.755310257E+01 -0.586466462E+01 0.625798158E-04 2 E 0.543252922E+03 -0.509902947E+01 -0.802091687E+01 -0.946868226E+00 3 M 0.617391858E+03 0.209510967E+00 -0.114511487E+02 0.194071912E+00 4 E 0.124475605E+04 -0.364379587E+01 -0.422382979E+00 -0.406334274E+01 5 E 0.161001598E+04 0.196936292E+01 -0.466437394E+01 -0.204795840E+01 6 E 0.234053614E+04 0.537242012E+01 0.105702528E+01 0.742374526E-03 7 E 0.287617513E+04 0.227654667E+00 0.534395500E+01 0.109172508E+01 8 M 0.301795888E+04 -0.153884253E+01 0.738874474E+01 -0.124010937E+01 9 E 0.356679127E+04 -0.612485480E+01 0.370477490E+01 -0.161465346E+01 10 E 0.393206222E+04 -0.147356993E+01 0.716856890E+01 -0.550609959E+00 11 E 0.466254693E+04 -0.562943484E+01 0.468694156E+01 -0.501904663E-02 12 E 0.520573757E+04 -0.726105589E+01 0.968704739E+00 0.213337895E+00 13 M 0.531473128E+04 -0.950244586E+01 -0.588846319E+01 0.783975896E+00 14 E 0.591818578E+04 -0.117285104E+01 0.301973675E+01 -0.432521192E+01 15 E 0.628345045E+04 -0.377522951E+01 -0.333835898E+01 -0.194320168E+01 16 E 0.701944420E+04 0.285304875E+01 -0.451179611E+01 0.244881353E-02 17 E 0.755457761E+04 0.496115786E+01 0.191476684E+01 -0.461676558E+00 18 M 0.766039435E+04 0.750470463E+01 0.206881153E+01 -0.681208328E+00 19 E 0.825784312E+04 -0.407705299E+00 0.353101926E+01 -0.506968214E+01 20 E 0.862309045E+04 0.518435273E+01 0.296012602E+01 -0.160213784E+01 21 E 0.935358559E+04 0.697333912E+00 0.613153373E+01 -0.331830570E-03 22 E 0.989207362E+04 -0.356970499E+01 0.496378360E+01 0.933762775E+00 23 M 0.100273358E+05 -0.904844600E+01 0.217212155E+01 -0.106977333E-02 24 E 0.105817852E+05 -0.434890676E+01 -0.374239768E+01 -0.109920066E+01 25 E 0.109470293E+05 -0.526948861E+01 0.227686861E+01 0.105300822E+01 26 E 0.116775395E+05 -0.493849670E+01 -0.314111429E+01 0.774043028E-03 27 E 0.122146737E+05 -0.579114760E+00 -0.573215719E+01 -0.929265985E+00 28 M 0.123058217E+05 0.135449198E+01 -0.113514617E+02 0.626325288E-01 29 E 0.129422433E+05 -0.199746803E+01 -0.293943744E+00 -0.527610937E+01 30 E 0.133074988E+05 0.351666513E+01 -0.389969463E+01 -0.208669537E+01 31 E 0.140385752E+05 0.528073309E+01 0.203175003E+01 0.776729627E-03 32 E 0.145749404E+05 -0.514347480E+00 0.549462618E+01 0.115031655E+01 33 M 0.147142907E+05 -0.261595289E+01 0.760636920E+01 -0.187756214E+01 34 E 0.152531120E+05 -0.722082856E+01 0.451226537E+01 -0.950084473E+00 35 E 0.156183624E+05 -0.276765444E+01 0.805993291E+01 -0.651993496E+00 36 E 0.163488614E+05 ================ PARENT CYCLER 5.301gGff3 ======================= Parent cycler number 96 Approximate search space (synodic periods after J2000) 6 Number of steps to walk eccentricity/inclination 3 / 3 Number of cycles 7 Total delta v over 45.04 years (km/s) 0.004582 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 2 113.9 5.165 5.165 11.872 11.872 7 130.6 5.647 5.647 7.518 7.518 12 112.2 7.393 7.393 10.939 10.939 17 100.9 5.315 5.315 8.269 8.269 22 139.6 5.856 5.856 9.032 9.032 27 92.5 6.078 6.078 11.519 11.519 32 148.5 5.216 5.216 7.080 7.080 AVERAGE 119.8 5.810 5.810 9.461 9.461 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 5412.834470 LEG E/M time start vinfx vinfy vinfz (days) (km/s) (km/s) (km/s) 0.490938979E+04 0.523726263E+04 0.545242590E+04 0.603585191E+04 0.687859053E+04 0.724213442E+04 0.754905966E+04 0.802207160E+04 0.856555835E+04 0.905777506E+04 0.962479929E+04 0.992105770E+04 0.101141104E+05 0.107248546E+05 0.115707177E+05 0.119394680E+05 0.122442642E+05 0.124503534E+05 0.130566705E+05 0.139069052E+05 0.142771159E+05 0.145696904E+05 0.147469186E+05 0.155734809E+05 0.161578926E+05 -0.157567193E-06 -0.386390399E-06 -0.935652482E-09 -0.614937972E-06 0.237005240E-05 0.322986146E-05 0.530680989E-05 -0.115558491E-06 -0.264740443E-01 -0.808836383E-05 0.312534200E-04 -0.169735603E-05 0.208697516E-07 0.194851193E-05 0.340782911E-06 -0.187818586E-06 -0.355721734E-06 -0.494280362E-09 0.193835009E-06 -0.772849822E-05 0.395113271E-06 0.232713343E-06 -0.645888926E-09 0.625553981E-03 0.195454112E-03 0.287975538E-07 -0.418726388E-06 0.762716936E-09 0.217933155E-05 0.135331718E-05 -0.442347401E-05 -0.194269103E-05 -0.529937563E-06 0.989914282E-01 -0.342546759E-04 -0.402895086E-04 -0.167578285E-04 -0.670026856E-07 0.163960106E-05 -0.397179136E-05 -0.776653182E-06 -0.729546501E-06 -0.528844323E-09 -0.311153035E-08 0.299831911E-05 0.255295307E-06 0.669169710E-07 -0.698881490E-09 -0.631102857E-04 -0.259985935E-04 0.906027379E-08 0.746677141E-08 -0.170587536E-08 0.589788528E-07 -0.963061070E-04 -0.468957687E-07 0.157308261E-06 -0.795188306E-07 -0.148540924E-04 -0.221153714E-04 0.286488073E-06 0.136342002E-05 -0.379621605E-08 -0.111540180E-07 0.237346399E-03 0.818254619E-08 0.113167023E-07 0.104672806E-10 -0.415510285E-08 -0.157705965E-03 0.655213458E-11 0.184385981E-07 -0.423483452E-10 -0.648593762E-04 -0.171018139E-04 time dv (days) 0.233945839E+03 0.554373762E+03 0.711496487E+03 0.136529183E+04 0.176342521E+04 0.261371202E+04 0.289744269E+04 0.319907357E+04 0.380787010E+04 0.439957243E+04 0.492327844E+04 0.522208662E+04 0.570094216E+04 0.597662813E+04 0.643064920E+04 0.724420023E+04 0.757045012E+04 0.775001166E+04 0.849160141E+04 0.915635190E+04 0.964975400E+04 0.991236294E+04 0.102657490E+05 0.108191938E+05 0.115022170E+05 0.119299926E+05 0.122283459E+05 0.124458344E+05 0.130810404E+05 0.138850492E+05 0.143013941E+05 0.145958429E+05 0.149352074E+05 0.153078996E+05 0.157279373E+05 dvx (km/s) 0.308345327E-04 0.309800429E-03 -0.136159362E-05 -0.106194600E-03 0.918633065E-04 -0.375878652E-04 0.505519532E-04 0.602561723E-07 0.145603252E-05 -0.664391017E-04 0.261505653E-04 0.208676568E-04 -0.856741495E-06 -0.105513061E-02 0.248386263E-01 0.503383658E-05 0.131458351E-03 0.624176145E-07 -0.257400832E-03 0.736658819E-04 0.338261958E-06 -0.103839258E-04 -0.423669354E-09 0.194874377E-04 0.927156966E-06 0.362149300E-04 0.182265774E-04 0.670963688E-06 -0.121528215E-02 -0.445378607E-02 0.289526360E-03 0.199666781E-03 0.111016668E-04 0.255585473E-03 0.220266153E-03 dvy (km/s) -0.757267514E-04 -0.611169268E-03 -0.855352752E-07 0.364027136E-03 -0.142297038E-03 -0.360112982E-05 0.780592986E-04 -0.199591776E-07 0.169429430E-04 -0.732450760E-05 -0.330912987E-04 -0.355269436E-05 -0.109162568E-04 -0.734942570E-03 0.158675366E-01 0.410559971E-04 -0.717608788E-04 -0.131432437E-06 0.101810803E-02 -0.876175522E-04 -0.215165399E-05 0.999275328E-06 0.905959550E-08 -0.101759569E-03 -0.122019698E-04 0.131865416E-04 0.469561473E-06 -0.614699667E-06 0.103792116E-02 -0.386897766E-02 0.437582298E-04 -0.109871848E-03 0.240658973E-05 0.102122598E-02 -0.133116914E-02 dxz (km/s) -0.163926771E-05 0.307758482E-04 0.152962485E-07 -0.267642945E-03 -0.169457712E-03 -0.830410966E-06 -0.400587072E-05 0.527750512E-07 0.540927956E-03 -0.100997895E-03 0.366049333E-07 -0.663885238E-06 -0.310597843E-06 -0.142376831E-02 -0.124637544E+00 0.204099324E-05 0.911453009E-05 -0.190993018E-07 0.365009484E-04 -0.296305895E-03 -0.891124463E-07 0.663293921E-06 0.888111896E-08 -0.211763301E-03 0.645149923E-04 -0.198883615E-06 -0.396968145E-06 -0.354644333E-07 0.245487151E-03 0.129341501E-01 -0.264633602E-06 0.188805145E-04 0.171180354E-04 -0.700680456E-04 -0.188577603E-04 time dv (days) dvx (km/s) dvy (km/s) dxz (km/s) 213 1 E -0.430216417E+02 -0.508750361E+01 0.101658463E+01 -0.973418317E-03 2 E 0.491413343E+03 -0.180811424E+01 -0.480532715E+01 -0.562291091E+00 3 M 0.605304815E+03 -0.354041493E+01 -0.113216379E+02 0.469803154E+00 4 E 0.123543715E+04 -0.476397632E+01 0.104520122E+01 -0.277925544E+01 5 E 0.160066167E+04 -0.156886745E+01 0.539917853E+01 0.105225783E-02 6 E 0.235218576E+04 0.554293056E+01 0.116398229E+01 -0.125515283E-02 7 E 0.288857771E+04 0.166227762E+01 0.531566669E+01 0.930767686E+00 8 M 0.301919029E+04 -0.294712467E+00 0.740893588E+01 -0.124183106E+01 9 E 0.356929968E+04 -0.565437273E+01 0.470910500E+01 0.606650416E+00 10 E 0.393457375E+04 -0.736590351E+01 -0.312734631E+00 -0.113556749E+00 11 E 0.466506757E+04 -0.492462812E+01 0.550241313E+01 -0.856097748E-03 12 E 0.520848415E+04 -0.707477964E+01 0.211538962E+01 0.350579686E+00 13 M 0.532071587E+04 -0.992080663E+01 -0.454474070E+01 0.769426969E+00 14 E 0.591940547E+04 -0.832146280E+00 0.368751659E+01 -0.379014652E+01 15 E 0.628467295E+04 0.518356893E+01 0.447514364E+00 -0.126278545E+01 16 E 0.701518000E+04 0.152003947E+01 -0.511552589E+01 0.243522211E-02 17 E 0.755023634E+04 0.523640915E+01 0.506579401E+00 -0.759464248E+00 18 M 0.765117804E+04 0.824176899E+01 -0.308634233E+00 -0.597375755E+00 19 E 0.825884635E+04 0.840909138E+00 0.573365929E+01 -0.936254218E+00 20 E 0.862409458E+04 -0.450994840E+01 0.371196223E+01 -0.349162795E+00 21 E 0.935458060E+04 0.169294171E+01 0.560780441E+01 0.342023626E-03 22 E 0.989177443E+04 -0.296795842E+01 0.494216083E+01 0.103036137E+01 23 M 0.100313833E+05 -0.841827599E+01 0.326670626E+01 -0.212918358E+00 24 E 0.105822367E+05 -0.527200470E+01 -0.306140594E+01 0.460098741E-01 25 E 0.109474807E+05 -0.572446598E+00 -0.601088660E+01 0.944741675E+00 26 E 0.116780001E+05 -0.565428831E+01 -0.227220215E+01 -0.980660162E-03 27 E 0.122161182E+05 -0.234643599E+01 -0.555062482E+01 -0.788377564E+00 28 M 0.123086194E+05 -0.694062952E+00 -0.114934397E+02 0.321699021E+00 29 E 0.129333344E+05 -0.500830655E+01 0.155406533E+00 -0.145922368E+01 30 E 0.132985609E+05 -0.140866433E+01 0.501837461E+01 -0.231856483E-03 31 E 0.140402506E+05 0.513206090E+01 0.100972330E+01 0.701668560E-03 32 E 0.145748973E+05 0.373450374E+00 0.507665711E+01 0.113961598E+01 33 M 0.147234006E+05 -0.199313964E+01 0.677946572E+01 -0.435701976E+00 34 E 0.153032410E+05 -0.108330416E+01 -0.403574004E+00 0.508490141E+01 35 E 0.156684944E+05 0.707624374E+00 -0.517336469E+01 0.129930053E-02 36 E 0.164086611E+05 ================ PARENT CYCLER 5.301gGff3 ======================= Parent cycler number 97 Approximate search space (synodic periods after J2000) 9 Number of steps to walk eccentricity/inclination 3 / 3 Number of cycles 7 Total delta v over 44.71 years (km/s) 0.184814 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 2 97.3 12.716 12.716 7.705 7.705 7 109.5 7.880 7.880 10.938 10.938 12 101.0 5.314 5.314 8.267 8.267 17 139.4 5.875 5.875 9.040 9.040 22 92.1 6.110 6.110 11.650 11.650 27 141.9 5.456 5.456 7.642 7.642 32 109.5 7.319 7.319 11.095 11.095 AVERAGE 112.9 7.239 7.239 9.477 9.477 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 7752.726486 LEG E/M time start vinfx vinfy vinfz (days) (km/s) (km/s) (km/s) 1 E 0.329049307E+02 0.985904601E+01 0.804005377E+01 -0.108204922E-03 2 E 0.598299127E+03 0.709215914E+01 0.105047501E+02 0.101918836E+01 3 M 0.695613695E+03 -0.120302138E+01 0.745101927E+01 -0.154859291E+01 4 E 0.123079929E+04 -0.108470083E+01 0.759418783E+01 -0.179350299E+01 5 E 0.196133580E+04 -0.459180764E+01 0.319468100E+01 -0.556869610E+01 6 E 0.232657667E+04 -0.543974601E+01 0.569868102E+01 0.969183597E-02 7 E 0.287201823E+04 -0.749992519E+01 0.239289077E+01 0.345995205E+00 8 M 0.298149778E+04 -0.989411514E+01 -0.459931279E+01 0.768597246E+00 9 E 0.357950726E+04 -0.402041779E+01 -0.349734396E+01 -0.968545215E-01 10 E 0.431001921E+04 0.213949851E+01 -0.459397391E+01 -0.161234555E+01 11 E 0.467526714E+04 0.151770326E+01 -0.511504025E+01 0.243427219E-02 12 E 0.521031888E+04 0.523520902E+01 0.507318723E+00 -0.759385844E+00 13 M 0.531129220E+04 0.823973589E+01 -0.307394305E+00 -0.598285393E+00 14 E 0.591904699E+04 0.558559666E+01 0.169999936E+01 -0.549392155E+00 15 E 0.664957199E+04 -0.138331861E-02 0.235583451E+01 -0.538156921E+01 16 E 0.701483047E+04 0.168257478E+01 0.562942529E+01 0.347681194E-03 17 E 0.755209695E+04 -0.296795752E+01 0.496423122E+01 0.102953210E+01 18 M 0.769146725E+04 -0.842273672E+01 0.327427912E+01 -0.224671836E+00 19 E 0.824192217E+04 -0.523711115E+01 0.301048843E+01 0.103496546E+01 20 E 0.897240941E+04 -0.484757344E+01 -0.284625863E+01 -0.240381526E+01 21 E 0.933767253E+04 -0.569947892E+01 -0.224403517E+01 -0.890478729E-03 22 E 0.987592115E+04 -0.230702729E+01 -0.560280851E+01 -0.784494553E+00 23 M 0.996798777E+04 -0.813497002E+00 -0.116185013E+02 0.263221315E+00 24 E 0.105996989E+05 0.331799700E+01 -0.449491609E+01 0.982720608E-03 25 E 0.113348295E+05 0.347613887E+01 0.140835982E+01 -0.397256785E+01 26 E 0.117000996E+05 0.536812770E+01 0.105525447E+01 0.747753565E-03 27 E 0.122357204E+05 0.228389296E+00 0.534104585E+01 0.109227043E+01 28 M 0.123775849E+05 -0.154352589E+01 0.738254272E+01 -0.123266880E+01 29 E 0.129265581E+05 -0.144344624E+01 0.709132758E+01 -0.109168870E+01 30 E 0.136570747E+05 -0.580910942E+01 0.343503366E+01 -0.286901060E+01 31 E 0.140223252E+05 -0.563769943E+01 0.466209877E+01 -0.484576763E-02 32 E 0.145654779E+05 -0.724450073E+01 0.101999506E+01 0.206134108E+00 33 M 0.146750103E+05 -0.936056786E+01 -0.590712327E+01 0.761386513E+00 34 E 0.152692215E+05 -0.446581593E+01 -0.234990656E+01 0.448846874E-01 35 E 0.159995952E+05 0.198677204E+01 -0.396586958E+01 -0.237140733E+01 36 E 0.163648706E+05 ================ PARENT CYCLER 5.301ggFf3 ======================= Parent cycler number 98 Approximate search space (synodic periods after J2000) 16 Number of steps to walk eccentricity/inclination 1 / 1 Number of cycles 7 Total delta v over 44.75 years (km/s) 0.429957 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 3 165.4 5.622 5.622 7.519 7.519 8 117.7 5.489 5.489 11.417 11.417 0.213507151E+03 0.508497064E+03 0.699824666E+03 0.134865675E+04 0.187872558E+04 0.260428998E+04 0.290816960E+04 0.313471326E+04 0.362409079E+04 0.419755152E+04 0.493134170E+04 0.522531891E+04 0.541051931E+04 0.602533304E+04 0.645268957E+04 0.727200704E+04 0.756537759E+04 0.774232828E+04 0.850721515E+04 0.879941123E+04 0.959631782E+04 0.991271575E+04 0.102076564E+05 0.108269502E+05 0.111081950E+05 0.119524403E+05 0.122299934E+05 0.124023266E+05 0.130465546E+05 0.135952368E+05 0.143075740E+05 0.145971728E+05 0.148161750E+05 0.153580290E+05 0.159423561E+05 -0.440732328E-07 0.106508605E-07 -0.235110996E-10 -0.173636935E-04 0.831218889E-08 0.845093526E-06 0.692774768E-06 -0.189244511E-09 -0.490201467E-09 0.293673144E-08 0.275401283E-08 0.187593708E-07 -0.699149338E-10 0.797327729E-04 0.157332478E-03 -0.142074893E-06 0.142733976E-05 0.594804366E-09 -0.834217306E-06 -0.342249905E-06 -0.691150835E-07 0.239130964E-07 0.579543645E-10 -0.515683894E-07 0.172773532E-05 0.143487715E-06 0.371634355E-06 -0.717171207E-10 -0.386079478E-04 0.762035478E-07 -0.527518354E-05 0.742350940E-05 -0.588080489E-07 -0.798855845E-04 -0.837577177E-04 0.164848125E-07 0.185370636E-07 -0.586206777E-11 -0.957147477E-04 -0.162456705E-07 -0.538345106E-07 -0.153754801E-06 0.364494793E-09 -0.993663062E-09 0.401597689E-08 -0.385525819E-08 -0.214499235E-07 -0.713963645E-10 0.218079785E-04 0.456961755E-04 0.182891212E-05 0.304832425E-06 0.359167264E-08 -0.417937936E-06 0.143774346E-06 -0.919835927E-07 -0.293060632E-06 -0.755314742E-10 0.957862046E-06 -0.535056335E-04 0.513007779E-07 -0.469957273E-06 0.352021675E-10 0.571007063E-04 0.222676039E-05 -0.506459092E-06 0.894894262E-05 0.117770357E-06 -0.330961848E-03 0.225064605E-03 -0.281241744E-09 -0.160221169E-08 -0.157433018E-11 0.102200895E-02 -0.337132224E-09 -0.728853427E-08 -0.459153920E-07 0.698403249E-10 -0.173951601E-10 -0.228870898E-10 -0.365194114E-10 0.694358714E-09 0.467837505E-11 -0.586727821E-03 -0.440205739E-03 0.237665809E-08 -0.812277702E-07 0.516962083E-10 -0.558351136E-04 -0.641307669E-06 0.707322198E-09 -0.158533268E-07 -0.102140806E-10 -0.797610039E-05 -0.343482746E-03 -0.147758750E-08 0.326266099E-07 -0.278913750E-10 0.462965063E-03 0.756976350E-07 -0.114352230E-06 -0.518465291E-06 -0.540025009E-07 -0.565109833E-04 0.195692649E-05 time dv (days) 0.242100783E+03 0.612896312E+03 0.936447212E+03 0.174217485E+04 0.220969959E+04 0.258838862E+04 0.288844017E+04 0.307119921E+04 0.370369429E+04 0.456204028E+04 0.493209197E+04 0.522546488E+04 0.540245542E+04 0.648885649E+04 0.685776932E+04 0.725660038E+04 0.757300249E+04 0.791164922E+04 0.879709247E+04 0.903815677E+04 0.961756181E+04 0.988973114E+04 0.101701353E+05 0.109672642E+05 0.115941713E+05 0.119679100E+05 0.122570001E+05 0.125038488E+05 0.134598352E+05 0.138908351E+05 0.142776070E+05 0.145819077E+05 0.148770421E+05 0.153787775E+05 0.160726503E+05 dvx (km/s) -0.237305587E-06 0.158580323E-06 -0.105588770E-08 0.133147100E-03 -0.422793080E-05 0.327356848E-06 0.153780901E-05 0.814509144E-09 0.314561394E-05 -0.504048495E-04 -0.114181261E-06 0.114908089E-05 0.410782890E-09 0.344839724E-05 -0.169127831E-03 0.143802053E-07 -0.101911655E-07 0.125678566E-09 0.461044983E-05 0.600152020E-06 0.261318260E-06 0.253246078E-06 0.187712652E-08 -0.454545970E-01 0.765687668E-05 0.367739483E-06 -0.150856375E-05 -0.891963502E-09 -0.161628653E-04 0.205351024E-04 0.230353420E-05 0.222362741E-05 -0.612127930E-06 0.181028394E-03 0.194996565E-03 dvy (km/s) -0.148311108E-07 -0.702368780E-05 0.733627654E-08 0.342772345E-04 -0.908854465E-04 -0.510540926E-06 -0.248011044E-06 0.151886994E-08 0.521688965E-05 0.146554634E-04 0.146181892E-05 0.236514913E-06 0.255934357E-08 -0.205151875E-05 -0.162296866E-02 0.185286446E-07 0.875675766E-07 -0.914221338E-10 0.339324699E-05 -0.571665635E-06 0.112191427E-06 -0.385136903E-06 -0.194394963E-08 0.123890761E-01 -0.117524264E-03 0.541051156E-07 -0.235674521E-05 -0.407851443E-09 -0.253676265E-05 0.652875424E-05 -0.262958586E-05 -0.276521245E-06 -0.149386169E-05 0.217269831E-03 0.192334413E-03 dxz (km/s) -0.404790173E-08 -0.444940124E-06 0.118739992E-08 0.389287197E-03 -0.925995686E-03 -0.243472826E-09 0.995382640E-07 -0.569566128E-10 0.970235765E-04 -0.456606735E-03 0.197657447E-08 -0.647797201E-07 0.370761733E-10 0.120634562E-03 0.119667708E-03 -0.159299228E-09 0.488644600E-08 -0.445586593E-12 0.218046126E-03 -0.186295779E-05 -0.355047856E-08 0.226664933E-07 0.178833368E-09 -0.873217227E-05 0.412894847E-03 0.794661680E-08 0.139510001E-06 0.360577723E-09 0.141607678E-03 -0.765809101E-04 0.270729570E-08 -0.650041776E-07 -0.150324558E-06 0.113632483E-05 -0.743110045E-03 214 13 124.8 5.647 5.543 7.388 7.388 18 128.6 6.609 6.609 9.789 9.789 23 101.6 5.210 5.214 10.590 10.590 28 158.6 5.297 5.277 7.799 7.799 33 114.0 5.472 5.472 11.564 11.564 AVERAGE 130.1 5.621 5.604 9.438 9.438 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 13259.108943 LEG E/M time start vinfx vinfy vinfz (days) (km/s) (km/s) (km/s) 1 E 0.120268486E+02 0.437394775E+01 0.348311920E+01 0.110782880E-02 2 E 0.548190863E+03 0.356561922E-01 0.560920601E+01 -0.508812861E-03 3 E 0.124572480E+04 -0.174595286E+01 0.512463565E+01 0.151649381E+01 4 M 0.141111437E+04 -0.376320477E+01 0.649112747E+01 -0.494480115E+00 5 E 0.197844526E+04 -0.546506504E+01 -0.312968469E+00 -0.815920622E+00 6 E 0.234369223E+04 -0.549291235E+01 0.493844135E+00 -0.118156216E-02 7 E 0.287952950E+04 -0.350614697E+01 -0.428214860E+01 -0.134112649E-03 8 E 0.357917940E+04 -0.521151240E+01 -0.172200509E+01 -0.218743087E-01 9 M 0.369683784E+04 -0.815516859E+01 -0.795386975E+01 0.761277728E+00 10 E 0.429213308E+04 0.281051021E+01 -0.413083777E+01 0.225740375E+00 11 E 0.465740865E+04 0.218318090E+01 -0.451177154E+01 0.175714563E-03 12 E 0.519114919E+04 0.490846397E+01 0.169662263E+01 0.633160552E-03 13 E 0.594021268E+04 0.356053209E+01 0.424834679E+01 0.591579298E-01 14 M 0.606498094E+04 0.545947065E+01 0.490896632E+01 -0.820739796E+00 15 E 0.664850008E+04 -0.206228534E+01 0.626980981E+01 0.357587071E+00 16 E 0.701376363E+04 -0.110266299E+01 0.650157955E+01 -0.231975867E-02 17 E 0.755396858E+04 -0.500672420E+01 0.420434498E+01 -0.878323400E+00 18 E 0.828449202E+04 -0.476245791E+01 0.451823339E+01 0.759737448E+00 19 M 0.841306040E+04 -0.978122807E+01 0.411961163E-01 0.379169664E+00 20 E 0.898021731E+04 -0.208516592E+01 -0.438265725E+01 0.193163866E+01 21 E 0.934547822E+04 -0.292825566E+01 -0.430803370E+01 0.262631795E-02 22 E 0.988008376E+04 0.296228326E+01 -0.430617575E+01 -0.124246974E-02 23 E 0.105961751E+05 0.288293926E+01 -0.420732072E+01 -0.108120984E+01 24 M 0.106977299E+05 0.491291870E+01 -0.938171171E+01 -0.509815163E-01 25 E 0.113222493E+05 0.433430218E+01 0.293516253E+01 0.179103548E+00 26 E 0.116875188E+05 0.473816911E+01 0.226977159E+01 0.810916034E-03 27 E 0.122222457E+05 0.840469384E-01 0.523285273E+01 -0.819652829E-02 28 E 0.129541613E+05 -0.180135013E+01 0.477090194E+01 0.135659801E+01 29 M 0.131127464E+05 -0.505815959E+01 0.592388856E+01 -0.388455822E+00 30 E 0.136756089E+05 -0.536934754E+01 -0.115079519E+01 0.611221858E+00 31 E 0.140408755E+05 -0.549578570E+01 -0.344318663E+00 -0.155500059E-02 32 E 0.145766717E+05 -0.273620054E+01 -0.479517478E+01 0.133705150E-02 33 E 0.152770933E+05 -0.478967746E+01 -0.263913133E+01 -0.181347651E+00 34 M 0.153910811E+05 -0.706169289E+01 -0.912785913E+01 0.742548120E+00 35 E 0.159906871E+05 0.346436106E+01 -0.351738207E+01 -0.900971836E+00 36 E 0.163559469E+05 ================ PARENT CYCLER 5.301gfGf3 ======================= Parent cycler number 99 Approximate search space (synodic periods after J2000) 1 Number of steps to walk eccentricity/inclination 3 / 3 Number of cycles 7 Total delta v over 44.79 years (km/s) 0.391556 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 3 123.4 5.417 5.416 7.302 7.303 8 124.4 6.900 6.899 10.049 10.050 13 95.7 5.261 5.262 10.069 10.070 18 153.3 5.362 5.360 8.321 8.323 23 98.6 6.821 6.821 11.625 11.626 28 136.3 5.592 5.584 7.293 7.296 33 121.1 6.989 6.988 10.320 10.322 AVERAGE 121.8 6.049 6.047 9.283 9.284 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 782.500646 LEG E/M time start vinfx vinfy vinfz (days) (km/s) (km/s) (km/s) 1 E 0.427836675E+01 0.485059297E+01 -0.241932503E+01 0.723562696E-03 2 E 0.539702907E+03 0.452544981E+01 0.296756525E+01 0.224130133E+00 3 E 0.127015854E+04 0.314960058E+01 0.439254401E+01 0.336710995E+00 4 M 0.139353021E+04 0.406622966E+01 0.599585706E+01 -0.918055505E+00 5 E 0.196545370E+04 -0.298498197E+01 0.622466840E+01 -0.292056822E+00 6 E 0.233071444E+04 -0.203142422E+01 0.658412863E+01 -0.591031736E-03 7 E 0.287213383E+04 -0.563587397E+01 0.395808929E+01 0.343857248E+00 8 E 0.360262737E+04 -0.536810739E+01 0.428040662E+01 0.680090903E+00 9 M 0.372698609E+04 -0.999993278E+01 -0.897246728E+00 0.436325116E+00 10 E 0.429464186E+04 -0.186779713E+01 -0.486045751E+01 -0.795055051E+00 11 E 0.465988333E+04 -0.256135710E+01 -0.458320061E+01 0.662847792E-03 12 E 0.519467099E+04 0.402780176E+01 -0.312366489E+01 0.135244480E+01 13 E 0.592519653E+04 0.363818191E+01 -0.363237674E+01 -0.112102434E+01 14 M 0.602090347E+04 0.653371056E+01 -0.766029774E+01 -0.195058365E+00 15 E 0.664279291E+04 0.372095460E+01 0.372573853E+01 -0.938244278E+00 16 E 0.700806135E+04 0.431428114E+01 0.319203388E+01 -0.170932493E-02 17 E 0.754324984E+04 -0.698633774E+00 0.521706278E+01 -0.940463938E+00 18 E 0.827370086E+04 -0.219451620E+01 0.471506627E+01 0.129781563E+01 19 M 0.842697670E+04 -0.585152975E+01 0.586005657E+01 -0.828814243E+00 20 E 0.897336019E+04 -0.651241481E+01 -0.223171334E+00 0.197269711E+01 21 E 0.933863077E+04 -0.677066484E+01 0.801705431E+00 -0.180923545E-02 22 E 0.987969171E+04 -0.357488714E+01 -0.413804679E+01 0.405518139E+01 23 E 0.106101927E+05 -0.596728624E+01 -0.328656621E+01 -0.335015576E+00 24 M 0.107087447E+05 -0.573230188E+01 -0.100905556E+02 0.697810558E+00 25 E 0.113113185E+05 0.287040535E+01 -0.179609727E+01 -0.379196991E+01 26 E 0.116765828E+05 0.378453567E+01 -0.347133411E+01 0.154120210E-02 27 E 0.122108371E+05 0.404400295E+01 0.321339406E+01 -0.791814278E-03 28 E 0.129583519E+05 0.196233686E+01 0.519321192E+01 0.599819468E+00 29 M 0.130946880E+05 0.275274862E+01 0.668309214E+01 -0.997530214E+00 30 E 0.136622458E+05 -0.390446495E+01 0.577622527E+01 -0.481196891E+00 31 E 0.140275121E+05 -0.300481952E+01 0.629468192E+01 -0.353581366E-02 32 E 0.145692690E+05 -0.604373972E+01 0.274775941E+01 0.214224425E+01 33 E 0.152997482E+05 -0.597688169E+01 0.357651471E+01 0.564219218E+00 34 M 0.154208439E+05 -0.100660504E+02 -0.220513865E+01 0.594501392E+00 35 E 0.159982515E+05 -0.464567207E+00 -0.481815318E+01 0.150757922E+01 36 E 0.163635236E+05 ================ PARENT CYCLER 5.301gFgf3 ======================= time dv (days) 0.269385576E+03 0.652820953E+03 0.127053323E+04 0.149621401E+04 0.222681320E+04 0.260089412E+04 0.298447698E+04 0.359682816E+04 0.389328527E+04 0.434692441E+04 0.495096595E+04 0.559564347E+04 0.595892792E+04 0.615250881E+04 0.689322666E+04 0.726225790E+04 0.766354709E+04 0.830377728E+04 0.871365356E+04 0.904596427E+04 0.960208888E+04 0.998749746E+04 0.106114083E+05 0.107914078E+05 0.113953032E+05 0.119548823E+05 0.125662460E+05 0.129779490E+05 0.131971758E+05 0.139239902E+05 0.142926997E+05 0.146817350E+05 0.152941915E+05 0.154810220E+05 0.160637391E+05 dvx (km/s) 0.165828071E-06 0.133116030E-06 0.681065523E-06 0.784857382E-09 0.132979843E-05 0.616562507E-06 0.387435145E-07 -0.869820180E-06 -0.103282931E-06 -0.128183618E-03 0.274469025E-04 0.146235810E-01 0.135185078E-03 -0.721179999E-07 0.266688739E-04 -0.978760270E-06 0.145454871E-04 -0.667471878E-05 -0.168021429E-06 -0.921107710E-04 -0.174515458E-03 -0.129574373E-03 -0.555264777E-04 -0.147187652E-05 -0.136160320E-04 -0.698753519E-05 -0.924436441E-02 0.347278281E-05 0.674804603E-08 -0.813511863E-06 -0.411294149E-06 -0.537070794E-07 0.130130843E-06 -0.357858115E-10 -0.821631364E-06 dvy (km/s) 0.732910963E-07 -0.122244202E-06 0.119097522E-06 0.290109113E-09 0.451220562E-05 -0.155567237E-06 -0.755709964E-06 0.461837488E-06 -0.633407275E-06 -0.141316386E-03 -0.365978629E-04 0.951686949E-01 -0.165109891E-03 0.980621040E-07 -0.283369596E-05 -0.849495298E-05 0.546101292E-05 -0.100074136E-04 -0.235333573E-06 0.480900005E-05 -0.149840123E-03 0.453704271E-04 -0.713408128E-03 0.280910350E-06 0.853289942E-05 -0.247575101E-05 0.489013596E-02 0.188804091E-05 -0.148733002E-08 0.106837088E-05 0.723762896E-07 0.141488085E-06 0.397301540E-07 -0.194494001E-09 -0.289213460E-05 dxz (km/s) 0.622603574E-09 -0.534281056E-08 -0.250784164E-07 -0.180739512E-10 0.477807007E-04 -0.312083537E-08 -0.140693348E-07 -0.154805933E-07 -0.509237603E-08 -0.991099107E-04 -0.266160105E-05 -0.169136089E-03 0.123945510E-04 -0.261199017E-08 -0.149441082E-03 0.590260601E-07 0.178145439E-04 0.252250107E-06 0.138448619E-06 0.218329410E-03 0.919819786E-06 -0.100515017E-05 -0.264321949E-04 0.813338196E-07 0.530717735E-04 0.103727877E-07 -0.493475998E-05 0.387998746E-07 -0.288577460E-09 -0.113712688E-03 -0.731385264E-10 0.637476924E-08 -0.417094787E-08 0.219982546E-11 -0.188284393E-04 time dv (days) 0.271990637E+03 0.649271252E+03 0.128866429E+04 0.147931873E+04 0.202024281E+04 0.241192735E+04 0.298170786E+04 0.362128118E+04 0.381213446E+04 0.436769015E+04 0.490053777E+04 0.530424982E+04 0.593955257E+04 0.611418689E+04 0.669758318E+04 0.726495183E+04 0.769664455E+04 0.829669223E+04 0.855810874E+04 0.902815077E+04 0.959292941E+04 0.998926686E+04 0.106249755E+05 0.107991308E+05 0.113989820E+05 0.117567210E+05 0.127415726E+05 0.129788023E+05 0.131798217E+05 0.137170357E+05 0.142767202E+05 0.146788409E+05 0.153179126E+05 0.155074550E+05 0.160530423E+05 dvx (km/s) -0.174818573E-02 -0.513267402E-04 0.198119347E-02 0.100271102E-03 -0.785228167E-03 0.112966413E-03 -0.135019075E-02 -0.107497574E-02 0.136015225E-03 0.191595613E-03 -0.130208647E-02 0.138226156E-03 -0.368492560E-02 -0.975915504E-03 -0.488313093E-03 -0.124248025E-02 -0.264974504E-06 0.129768984E-02 0.177154381E-02 0.468875019E-03 0.153059681E-02 -0.109548796E-02 -0.347122953E-02 -0.116147916E-02 0.354383807E-03 0.915584772E-04 -0.151512552E-01 0.271480474E-02 0.112099505E-02 -0.114882714E-02 0.367272530E-03 -0.116211994E-02 -0.307629352E-02 0.132984230E-02 0.199194560E-02 dvy (km/s) 0.121941160E-02 0.184914170E-02 -0.116343180E-02 -0.219816263E-03 -0.700466107E-03 -0.164254845E-03 -0.132938771E-02 0.194249808E-02 0.257675982E-03 -0.579597754E-04 -0.102770060E-02 -0.845918132E-04 -0.972189439E-03 -0.648897980E-03 0.179970811E-03 -0.447670842E-03 0.134610639E-03 0.255319527E-02 0.239943255E-03 -0.105063409E-02 -0.492157421E-03 0.714131796E-03 0.396280291E-03 0.139581866E-02 -0.238487620E-03 -0.495336544E-04 0.703706609E-01 -0.448486818E-02 -0.133576351E-02 -0.144802579E-02 0.808524762E-04 -0.921537222E-03 0.477882907E-02 0.237437521E-02 0.511509941E-03 dxz (km/s) -0.225684838E-04 -0.190696777E-03 0.185705917E-05 -0.151336391E-03 -0.318429082E-03 -0.922876645E-05 0.153836135E-03 -0.202380630E-03 0.169953130E-03 0.498578298E-03 -0.915027263E-05 0.349155740E-04 0.476552128E-04 -0.199255995E-03 -0.270031884E-03 0.419976304E-04 0.670326377E-03 0.114961971E-04 -0.806798350E-03 0.503945274E-03 0.660725834E-05 -0.592053152E-03 -0.152814150E-03 -0.104865837E-03 0.208106128E-03 0.555863554E-04 0.177071209E-04 0.237224905E-03 -0.336593879E-04 -0.201565147E-03 0.261187465E-04 -0.601076612E-03 -0.403071576E-03 0.339882252E-03 0.664658066E-03 215 Parent cycler number 100 Approximate search space (synodic periods after J2000) 9 Number of steps to walk eccentricity/inclination 27 / 27 Number of cycles 7 Total delta v over 44.62 years (km/s) 0.205926 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 2 99.0 11.744 11.754 7.698 7.698 7 111.0 7.596 7.596 10.944 10.944 12 100.3 5.345 5.345 8.323 8.324 17 139.8 5.800 5.799 9.110 9.110 22 93.7 5.941 5.942 11.681 11.681 27 141.6 5.487 5.466 7.662 7.663 32 110.1 7.104 7.103 11.284 11.284 AVERAGE 113.6 7.002 7.001 9.529 9.529 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 7752.256864 LEG E/M time start vinfx vinfy vinfz (days) (km/s) (km/s) (km/s) 1 E 0.432398662E+02 0.592044953E+01 0.721495840E+01 -0.701476281E+01 2 E 0.594402595E+03 0.658123207E+01 0.968714163E+01 0.100108675E+01 3 M 0.693395937E+03 -0.121693731E+01 0.760108392E+01 -0.928234436E-01 4 E 0.130483056E+04 0.354574355E+01 0.668249603E+01 0.592632833E-03 5 E 0.196101402E+04 -0.586570615E+01 0.475876086E+01 -0.930190453E+00 6 E 0.232625654E+04 -0.513967198E+01 0.558529562E+01 0.649613542E-03 7 E 0.287051651E+04 -0.725226898E+01 0.223072717E+01 0.348904359E+00 8 M 0.298154116E+04 -0.991628978E+01 -0.456583328E+01 0.770410399E+00 9 E 0.358028834E+04 -0.399786070E+01 -0.351142097E+01 -0.651754115E+00 10 E 0.431080720E+04 0.211018034E+01 -0.441828757E+01 -0.215615096E+01 11 E 0.467605356E+04 0.155859299E+01 -0.513500237E+01 0.243441262E-02 12 E 0.521123058E+04 0.526714459E+01 0.498048508E+00 -0.762408924E+00 13 M 0.531150024E+04 0.826607307E+01 -0.280861357E+00 -0.936730894E+00 14 E 0.595520052E+04 0.568874199E+01 -0.103464862E+01 0.107976353E-02 15 E 0.664906482E+04 0.859620836E+00 0.536021475E+01 -0.203423659E+01 16 E 0.701429936E+04 0.176502737E+01 0.553002051E+01 0.268701562E-03 17 E 0.755127268E+04 -0.303177969E+01 0.483324817E+01 0.103854921E+01 18 M 0.769108658E+04 -0.848173974E+01 0.329325150E+01 0.451342874E+00 19 E 0.828323709E+04 -0.284381100E+01 0.518245921E+01 -0.149389015E-02 20 E 0.897117929E+04 -0.519472403E+01 -0.277961340E+01 0.812019776E+00 21 E 0.933644430E+04 -0.560763875E+01 -0.200028186E+01 -0.103733757E-03 22 E 0.987400797E+04 -0.209622360E+01 -0.550537261E+01 -0.774676119E+00 23 M 0.996775214E+04 -0.900137904E+00 -0.116438047E+02 0.261828912E+00 24 E 0.106005061E+05 0.348796954E+01 -0.444004210E+01 -0.636119146E-02 25 E 0.113352340E+05 0.279786763E+01 0.123883041E+01 -0.456631513E+01 26 E 0.117004997E+05 0.539814019E+01 0.105729183E+01 0.791673454E-03 27 E 0.122362438E+05 0.224696678E+00 0.535095475E+01 0.109056533E+01 28 M 0.123778605E+05 -0.166934163E+01 0.747827113E+01 -0.962887993E-01 29 E 0.129916614E+05 0.299277642E+01 0.641384258E+01 -0.188827660E-02 30 E 0.136563622E+05 -0.598477642E+01 0.382795945E+01 0.405489632E+00 31 E 0.140216096E+05 -0.536022192E+01 0.465067496E+01 -0.452918907E-02 32 E 0.145638648E+05 -0.705211134E+01 0.818266276E+00 0.221620080E+00 33 M 0.146739369E+05 -0.961972862E+01 -0.584432476E+01 0.794186125E+00 34 E 0.152815841E+05 -0.566455681E+01 -0.175421311E+00 0.525097858E-03 35 E 0.159758275E+05 0.238234280E+00 -0.519769543E+01 -0.226691099E+01 36 E 0.163410809E+05 ================ PARENT CYCLER 5.301gfGff3 ======================= Parent cycler number 102 Approximate search space (synodic periods after J2000) 9 Number of steps to walk eccentricity/inclination 3 / 3 Number of cycles 7 Total delta v over 44.69 years (km/s) 0.086683 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 3 97.3 12.715 12.715 7.705 7.705 9 109.6 7.868 7.868 10.921 10.921 15 101.8 5.276 5.318 8.279 8.279 21 139.7 5.847 5.847 9.036 9.036 27 92.1 6.110 6.110 11.614 11.614 33 146.0 5.480 5.493 7.623 7.623 39 110.0 7.263 7.263 11.071 11.071 AVERAGE 113.8 7.223 7.231 9.464 9.464 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 7387.469588 LEG E/M time start vinfx vinfy vinfz (days) (km/s) (km/s) (km/s) 1 E 0.328952327E+02 0.984271018E+01 0.805689446E+01 0.400103086E-03 2 E 0.598295700E+03 0.942279392E+01 0.721687133E+01 -0.461469459E+01 3 E 0.963554686E+03 0.709200829E+01 0.105045092E+02 0.101918299E+01 4 M 0.106086922E+04 -0.120292713E+01 0.745107949E+01 -0.154866584E+01 5 E 0.159605438E+04 -0.619952706E+01 0.483792475E+01 0.476737074E+00 6 E 0.196132874E+04 -0.548951659E+01 0.413761309E+01 -0.383806397E+01 7 E 0.232658237E+04 -0.544356885E+01 0.571163105E+01 -0.279495736E-01 8 E 0.287196899E+04 -0.454917022E+01 0.643859072E+01 0.290153916E+00 9 E 0.323722584E+04 -0.748680946E+01 0.239409753E+01 0.344944802E+00 10 M 0.334685297E+04 -0.987310610E+01 -0.460397690E+01 0.763946634E+00 11 E 0.394367631E+04 -0.633497989E+00 0.327514533E+01 -0.408202361E+01 12 E 0.430893353E+04 0.164658933E+01 -0.341556758E+01 -0.365855222E+01 13 E 0.467419853E+04 0.139757937E+01 -0.506649120E+01 -0.355938256E-03 14 E 0.520898446E+04 -0.428104777E+00 -0.393669582E+00 -0.524250414E+01 15 E 0.557423830E+04 0.523257130E+01 0.562836946E+00 -0.764291821E+00 16 M 0.567601717E+04 0.824968363E+01 -0.352203065E+00 -0.596897279E+00 17 E 0.628412360E+04 0.864544212E+00 0.578319257E+01 0.390198339E+00 18 E 0.664937156E+04 -0.228569972E+00 0.149521506E+01 -0.564289897E+01 19 E 0.701462904E+04 0.171940384E+01 0.558647140E+01 -0.274616337E-02 20 E 0.755174510E+04 0.303969245E+01 0.496736342E+01 -0.274910018E+00 21 E 0.791701065E+04 -0.297211650E+01 0.492898882E+01 0.103112350E+01 22 M 0.805669616E+04 -0.841921521E+01 0.327289984E+01 -0.219099643E+00 23 E 0.860737117E+04 -0.421690160E+01 -0.256459918E+01 0.360691648E+01 24 E 0.897261330E+04 -0.398970224E+01 -0.248495865E+01 -0.392690539E+01 25 E 0.933786682E+04 -0.567070144E+01 -0.227027902E+01 0.284541231E-02 26 E 0.987608728E+04 -0.589579660E+01 -0.110623580E+01 0.123657927E+01 27 E 0.102413684E+05 -0.232735515E+01 -0.559418413E+01 -0.785985627E+00 time dv (days) 0.291263094E+03 0.609251596E+03 0.785111130E+03 0.140325808E+04 0.217285468E+04 0.257117353E+04 0.288717021E+04 0.307135324E+04 0.368986617E+04 0.438385647E+04 0.493293853E+04 0.522627103E+04 0.540805528E+04 0.605928017E+04 0.688281493E+04 0.726667682E+04 0.757224476E+04 0.777990915E+04 0.838642842E+04 0.903692699E+04 0.959447486E+04 0.988806959E+04 0.100626652E+05 0.109164391E+05 0.113900238E+05 0.119844441E+05 0.122574863E+05 0.124699307E+05 0.130913665E+05 0.137184542E+05 0.142710470E+05 0.145803756E+05 0.147650840E+05 0.153857206E+05 0.161949795E+05 dvx (km/s) -0.424160986E-03 0.442585864E-03 0.588228860E-04 -0.180480119E-03 -0.457784638E-05 -0.103880927E-03 -0.283506355E-03 0.214749599E-04 -0.788600678E-04 0.724179772E-04 0.602855054E-06 -0.812532963E-04 -0.152501366E-05 -0.284272063E-03 -0.297130565E-03 0.842446194E-04 0.205675370E-03 0.153568766E-03 0.748469666E-04 0.157613352E-03 0.111041505E-03 -0.586802413E-03 -0.291357392E-03 -0.447510030E-01 0.199774945E-03 0.192135451E-03 0.183535198E-03 0.212421615E-03 -0.965613050E-04 -0.249749220E-04 -0.131247932E-03 -0.508332583E-03 -0.137572961E-04 -0.143381390E-04 -0.108789888E-03 dvy (km/s) -0.232987679E-03 0.494248290E-03 -0.160165065E-04 0.352508635E-03 -0.555039627E-05 0.213887091E-03 0.129658728E-03 0.141948952E-03 -0.175814583E-03 0.118817245E-03 0.202913853E-03 -0.470878665E-03 -0.175049093E-03 -0.105472912E-03 0.241547946E-03 0.144392991E-03 0.158382815E-03 0.761843580E-04 -0.485145294E-04 -0.128863425E-03 0.146343081E-04 0.427764406E-04 0.143968109E-03 0.197567025E-01 -0.960553025E-03 0.591201647E-04 -0.238430335E-03 -0.695589042E-04 0.150073414E-03 -0.912762912E-04 0.214005806E-03 0.452052833E-03 0.337418282E-03 -0.532899807E-04 0.116695471E-03 dxz (km/s) 0.406835688E-04 0.162564070E-04 0.131384570E-04 0.145919085E-04 0.151093825E-04 -0.159670732E-06 -0.361198623E-04 0.365042716E-05 0.376182015E-04 0.883217294E-03 -0.333013714E-05 0.362968779E-05 -0.158363525E-04 0.326277541E-05 -0.582773519E-03 -0.396769391E-06 -0.311057050E-04 0.225073652E-04 -0.768411922E-05 0.405171805E-03 0.262946921E-06 -0.713147060E-05 -0.218365935E-04 -0.227992020E-02 -0.439417423E-03 0.680043933E-05 0.186075262E-04 0.271702390E-04 -0.146351073E-04 -0.162954981E-03 0.202200274E-05 -0.505718722E-04 -0.124937073E-05 -0.211239400E-05 -0.216931880E-03 time dv (days) 0.332557481E+03 0.740746705E+03 0.978151866E+03 0.131775810E+04 0.183348271E+04 0.220239614E+04 0.257746022E+04 0.299615632E+04 0.325366991E+04 0.346024940E+04 0.421761923E+04 0.456827168E+04 0.493089577E+04 0.546831469E+04 0.558950513E+04 0.576723313E+04 0.653249221E+04 0.685756832E+04 0.731004287E+04 0.766497742E+04 0.793796347E+04 0.813929742E+04 0.885938824E+04 0.922463823E+04 0.959621264E+04 0.999663006E+04 0.102551835E+05 dvx (km/s) -0.680067815E-08 -0.183496171E-04 -0.173128363E-07 -0.631596261E-10 0.138329069E-06 -0.418487509E-05 -0.620313083E-08 -0.737257150E-06 0.199425512E-07 0.247081486E-10 -0.142474091E-03 -0.158388546E-04 -0.488988649E-08 0.243924398E-03 0.106846506E-06 0.218891314E-10 0.864781397E-06 0.965490085E-04 0.907856642E-09 0.980881882E-06 -0.668149077E-08 -0.149135104E-11 0.384838764E-04 -0.369017181E-05 -0.492248421E-07 -0.425195905E-05 0.168955292E-07 dvy (km/s) -0.751305124E-08 0.218648636E-04 0.739194574E-06 0.946892176E-10 -0.109262389E-06 -0.338629188E-04 0.105559150E-07 -0.577578313E-06 -0.339683804E-08 0.285175419E-10 0.354594410E-03 -0.246220674E-04 -0.130528612E-06 -0.663874270E-03 -0.160062293E-07 0.131205824E-09 -0.284787552E-06 0.570464380E-03 0.312263083E-07 -0.498564163E-06 0.110474431E-06 -0.453126017E-10 -0.203739273E-04 0.173313555E-05 -0.727938365E-08 0.196520891E-04 -0.229263710E-07 dxz (km/s) 0.745124723E-10 0.149653987E-03 0.466733651E-07 0.130910695E-10 0.215279127E-04 -0.479265916E-03 0.185112563E-11 -0.835603606E-04 0.128439839E-08 -0.148402708E-11 -0.396211494E-04 0.460684765E-03 0.121968712E-09 -0.553346980E-04 -0.340277141E-08 0.260914097E-11 -0.430371660E-04 -0.618356547E-04 -0.255690064E-09 -0.967618861E-04 0.586004196E-08 -0.225511044E-10 0.685272493E-03 0.598367837E-04 0.178911368E-09 0.451612399E-03 0.149882630E-08 216 28 M 0.103334692E+05 -0.777851966E+00 -0.115850718E+02 0.275162709E+00 0.104721135E+05 -0.259415719E-11 -0.757083862E-11 0.691416158E-11 29 E 0.109636705E+05 -0.433551945E+01 -0.137428508E+00 -0.336674801E+01 0.112412646E+05 0.730715446E-03 0.278474325E-03 0.254563622E-04 30 E 0.113289258E+05 0.157406411E+01 0.731950304E+00 -0.519193365E+01 0.115809521E+05 -0.952646747E-03 -0.369041070E-03 -0.932190608E-04 31 E 0.116941812E+05 0.545099362E+01 0.532152943E+00 -0.153035650E-02 0.117745311E+05 0.695927226E-08 -0.197243907E-08 0.225172324E-09 32 E 0.122298471E+05 0.561004978E+00 -0.570600682E+00 0.542576772E+01 0.124380434E+05 -0.702684431E-04 0.126022525E-04 0.626536014E-05 33 E 0.125951038E+05 -0.857376838E-01 0.537584667E+01 0.112598303E+01 0.126170028E+05 -0.652888852E-08 -0.686166978E-09 0.123535163E-09 34 M 0.127410966E+05 -0.144574924E+01 0.738670735E+01 -0.120745721E+01 0.128457069E+05 0.330742642E-10 -0.101236612E-10 0.153392564E-11 35 E 0.132916772E+05 -0.610461661E+01 0.370158376E+01 0.136991291E+01 0.135327561E+05 0.228795458E-06 0.560797511E-05 -0.218794008E-03 36 E 0.136569482E+05 -0.503645171E+01 0.287964392E+01 -0.438185902E+01 0.139053118E+05 0.119206110E-05 -0.881264313E-04 -0.113555192E-02 37 E 0.140221887E+05 -0.559745253E+01 0.465621999E+01 -0.305590738E-02 0.142719356E+05 -0.189037111E-07 0.297584783E-07 0.108902212E-10 38 E 0.145651168E+05 -0.474568164E+01 0.550919803E+01 -0.403384184E+00 0.146893034E+05 -0.103463662E-05 -0.103374601E-05 0.906956308E-04 39 E 0.149303717E+05 -0.719060516E+01 0.100555384E+01 0.205319258E+00 0.149468672E+05 0.745138834E-07 -0.138021894E-07 -0.253773588E-08 40 M 0.150403418E+05 -0.934140462E+01 -0.589622136E+01 0.738684183E+00 0.151282083E+05 -0.150027931E-10 0.152789080E-10 0.888628526E-12 41 E 0.156261184E+05 0.151923290E+01 -0.475342691E+01 -0.359515215E+00 0.158781588E+05 -0.462770143E-06 0.106614213E-06 -0.366178140E-05 42 E 0.159913944E+05 0.149713698E+01 -0.474850502E+01 -0.569314879E+00 0.162470614E+05 0.370702963E-07 0.775733779E-06 -0.915794821E-04 43 E 0.163566330E+05 ================ PARENT CYCLER 5.219Ggh-3 ======================= Parent cycler number 111 Approximate search space (synodic periods after J2000) 16 Number of steps to walk eccentricity/inclination 243 / 243 Number of cycles 7 Total delta v over 44.70 years (km/s) 0.021325 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 1 74.0 ****** 10.335 7.534 7.533 5 143.0 5.628 5.628 8.901 8.901 9 100.2 5.458 5.458 11.586 11.586 13 145.1 5.281 5.281 7.197 7.197 17 120.2 5.868 5.868 11.138 11.138 21 105.8 5.244 5.244 7.559 7.559 25 142.5 5.559 5.559 9.188 9.188 AVERAGE 118.7 5.506 6.196 9.015 9.015 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 12965.276998 LEG E/M time start vinfx vinfy vinfz time dv dvx dvy dxz (days) (km/s) (km/s) (km/s) (days) (km/s) (km/s) (km/s) 1 E 0.551162081E+02 0.977440846E+01 -0.331811074E+01 -0.517491084E+00 0.662170895E+02 0.606157756E-03 0.788443446E-03 -0.811506452E-04 2 M 0.129122084E+03 0.722654853E+01 0.200333168E+01 -0.718919845E+00 0.218227051E+03 -0.352774612E-05 -0.860633300E-05 -0.405335698E-06 3 E 0.723155200E+03 0.511290829E+01 -0.243997358E+01 -0.618001875E-03 0.827866265E+03 -0.234697273E-03 -0.793317230E-04 0.677689608E-05 4 E 0.142122896E+04 0.270678170E+01 0.493640236E+01 -0.299848449E-01 0.158599788E+04 -0.526286293E-05 0.138347672E-05 0.216204139E-06 5 E 0.233661182E+04 -0.295542797E+01 0.467541886E+01 0.104115721E+01 0.235806109E+04 -0.330108631E-04 -0.820885071E-04 -0.246791694E-05 6 M 0.247960691E+04 -0.834091168E+01 0.310867863E+01 0.328147271E-01 0.266476603E+04 0.164530113E-07 -0.195271091E-07 -0.346243609E-08 7 E 0.304069513E+04 -0.106133473E+01 0.530158207E+01 -0.130470450E-02 0.314621014E+04 -0.229461058E-05 0.169614450E-05 0.143078251E-07 8 E 0.374412853E+04 -0.404738006E+01 -0.744612608E+00 -0.358158418E+01 0.402600468E+04 0.367107861E-06 0.149818874E-06 -0.124888379E-07 9 E 0.465340641E+04 -0.152839200E+01 -0.518524345E+01 -0.755727285E+00 0.466842911E+04 -0.433229043E-06 0.766504637E-07 -0.324851796E-07 10 M 0.475355776E+04 -0.995860559E+00 -0.115384843E+02 0.327926202E+00 0.494174374E+04 -0.206711175E-07 0.244392920E-07 0.117926827E-08 11 E 0.538084437E+04 0.194837420E+01 -0.490269473E+01 0.437054362E+00 0.549773054E+04 -0.451013527E-04 0.641420976E-04 0.175210066E-03 12 E 0.611138291E+04 0.529184384E+01 0.352634528E+00 -0.411005157E-03 0.638559600E+04 -0.813130862E-05 -0.917624287E-06 -0.218788223E-07 13 E 0.702542653E+04 0.520232022E+00 0.513654276E+01 0.111258059E+01 0.704718959E+04 0.636400242E-06 0.710291706E-05 -0.323313742E-06 14 M 0.717051364E+04 -0.193496902E+01 0.689566426E+01 -0.707685758E+00 0.725507436E+04 0.503626602E-08 -0.562321186E-08 -0.275546533E-10 15 E 0.773425178E+04 0.439300226E+01 0.373044354E+01 0.334033111E-03 0.783831162E+04 -0.816478885E-07 -0.618825362E-06 0.649347884E-08 16 E 0.842798402E+04 -0.285745124E+01 0.372023531E+01 -0.339262919E+01 0.870148983E+04 0.129424507E-05 -0.120433885E-05 -0.419414235E-07 17 E 0.933967005E+04 -0.586104344E+01 0.154508498E+00 0.234352385E+00 0.935769265E+04 -0.165382833E-05 0.350972915E-06 0.537661607E-07 18 M 0.945982070E+04 -0.952830319E+01 -0.571516180E+01 0.774351987E+00 0.954993509E+04 -0.125644341E-08 0.109339245E-07 0.636054646E-10 19 E 0.100605833E+05 -0.383182786E+01 -0.356527343E+01 -0.170631723E+00 0.101993754E+05 0.652663160E-04 0.393932476E-04 0.178523609E-03 20 E 0.107910682E+05 0.222922770E+01 -0.472406912E+01 0.840223200E-01 0.110652096E+05 0.282885582E-04 -0.493910699E-04 -0.940022724E-07 21 E 0.117048729E+05 0.494284681E+01 0.169927214E+01 -0.424508182E+00 0.117207370E+05 -0.524230627E-04 -0.620372095E-05 -0.359012305E-05 22 M 0.118106339E+05 0.716450343E+01 0.229224838E+01 -0.747578355E+00 0.120406253E+05 -0.445624653E-07 -0.781793994E-08 -0.335086057E-08 23 E 0.124158745E+05 0.527676245E+01 -0.184172891E+01 0.764442130E-03 0.125209625E+05 -0.696548236E-05 -0.263942808E-05 -0.264791858E-07 24 E 0.131164611E+05 0.207655196E+01 0.515230414E+01 -0.342393647E-01 0.134458965E+05 -0.618618605E-06 0.142334339E-06 0.295088583E-07 25 E 0.140315594E+05 -0.352601014E+01 0.418754348E+01 0.966344166E+00 0.140529311E+05 -0.156586701E-07 -0.354394167E-05 0.130090786E-06 26 M 0.141740374E+05 -0.894272550E+01 0.209680958E+01 0.226570077E+00 0.145143524E+05 -0.203888601E-05 -0.171030744E-05 0.135023892E-05 27 E 0.147412291E+05 -0.329787504E+01 0.379453032E+01 -0.130479165E+01 0.149384656E+05 -0.192166358E-03 0.163822568E-03 0.245409610E-03 28 E 0.154717346E+05 -0.281851098E+01 -0.320522673E+01 -0.294708277E+01 0.157538052E+05 0.534445348E-04 0.852565470E-04 -0.115145696E-04 29 E 0.163816397E+05 ================ PARENT CYCLER 5.219Ggh+3 ======================= Parent cycler number 112 Approximate search space (synodic periods after J2000) 16 Number of steps to walk eccentricity/inclination 1 / 1 Number of cycles 7 Total delta v over 45.10 years (km/s) 0.000079 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 1 141.2 ****** 7.666 9.240 9.240 5 139.9 5.716 5.716 9.328 9.328 9 86.5 6.726 6.726 12.023 12.023 13 121.8 6.584 6.584 8.102 8.102 17 103.3 8.291 8.291 11.316 11.316 21 96.0 5.720 5.720 7.916 7.916 25 131.0 6.799 6.799 9.211 9.211 AVERAGE 117.1 6.639 6.786 9.591 9.591 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 12965.276998 LEG E/M time start vinfx vinfy vinfz time dv dvx dvy dxz (days) (km/s) (km/s) (km/s) (days) (km/s) (km/s) (km/s) 1 E -0.802122613E+02 0.740718437E+01 -0.119595861E+01 -0.157382091E+01 -0.590392600E+02 -0.431464384E-05 -0.139652470E-04 0.366954882E-06 2 M 0.609410807E+02 0.839594255E+01 -0.383328907E+01 -0.443823044E+00 0.331283956E+03 0.645512286E-08 -0.498352808E-08 0.278125882E-08 3 E 0.689645441E+03 0.567520136E+01 -0.248389072E+00 -0.681175105E+00 0.799219307E+03 0.237668929E-05 0.664298911E-07 0.331491152E-05 4 E 0.142013788E+04 0.283435720E+01 0.496621822E+01 -0.185867463E+00 0.166738319E+04 0.319665002E-06 0.348459975E-06 -0.238297329E-09 5 E 0.233586126E+04 -0.321218590E+01 0.460748077E+01 0.105860636E+01 0.235683905E+04 0.845974025E-06 0.142149498E-05 0.299335673E-07 6 M 0.247571319E+04 -0.859465537E+01 0.358698280E+01 0.534015308E+00 0.263208762E+04 -0.226744881E-09 0.466316703E-09 0.974265991E-11 7 E 0.307715328E+04 -0.281167090E+01 0.607245682E+01 -0.203826085E-03 0.317784945E+04 -0.171688863E-07 0.756538907E-08 0.933970770E-09 8 E 0.374846106E+04 -0.652955217E+01 -0.167397747E+01 -0.862414523E-03 0.401519797E+04 -0.412399211E-06 -0.151430213E-06 -0.119285253E-08 9 E 0.466824349E+04 -0.273593843E+01 -0.609246014E+01 -0.800641732E+00 0.468122416E+04 -0.236019295E-06 -0.564207993E-07 -0.186133411E-07 10 M 0.475478129E+04 -0.101540805E+01 -0.119785617E+02 0.160893622E+00 0.485116866E+04 -0.287513302E-09 0.137795380E-09 -0.509419042E-10 11 E 0.539736375E+04 0.278121005E+01 -0.410626715E+01 0.431234742E+01 0.564573643E+04 -0.140378209E-04 0.778623194E-05 -0.381349582E-05 12 E 0.612787163E+04 0.560404452E+01 0.223480490E+01 0.260004249E+01 0.639379790E+04 0.115497437E-05 0.365370327E-06 0.593978489E-08 13 E 0.704485876E+04 0.177974815E+01 0.625768332E+01 0.101451263E+01 0.706313028E+04 0.130671872E-05 0.150926892E-06 -0.466542199E-07 14 M 0.716666891E+04 -0.176240043E+01 0.790833426E+01 0.175204794E-01 0.725985242E+04 0.209564583E-09 0.700753119E-10 -0.183547678E-10 15 E 0.778789235E+04 0.353811112E+01 0.745892829E+01 -0.307986422E-03 0.788465756E+04 -0.105665131E-06 0.269187015E-06 -0.126454067E-07 16 E 0.843299375E+04 -0.571052799E+01 0.602035595E+01 -0.139271366E-02 0.868303729E+04 -0.121949197E-07 0.177422793E-07 -0.230362316E-09 17 E 0.935908094E+04 -0.818041667E+01 0.133153056E+01 0.209605765E+00 0.937458004E+04 -0.227926355E-07 -0.541113050E-07 -0.383580630E-08 217 18 M 0.946240830E+04 -0.957865349E+01 -0.597281676E+01 0.795716818E+00 0.955360215E+04 0.200654504E-10 -0.472172984E-10 0.654119464E-13 19 E 0.100703673E+05 -0.226296188E+01 -0.526386952E+01 -0.296849435E+00 0.106255721E+05 0.217014232E-06 -0.759193909E-08 -0.701891434E-06 20 E 0.108008999E+05 0.318793043E+01 -0.441421336E+01 0.186159881E+01 0.110844258E+05 -0.117610853E-07 0.984460996E-08 0.246709785E-09 21 E 0.117154995E+05 0.554299234E+01 0.134948883E+01 -0.420931989E+00 0.117299012E+05 0.240627342E-07 -0.747522351E-08 0.172937540E-08 22 M 0.118115110E+05 0.753272298E+01 0.234320625E+01 -0.651591544E+00 0.118997845E+05 -0.152904817E-10 -0.479398265E-10 0.341924675E-11 23 E 0.124000008E+05 0.549286016E+01 0.386774336E+01 0.313150970E+00 0.125095811E+05 0.249326614E-08 0.141391322E-08 0.191355907E-10 24 E 0.131305358E+05 0.555656066E+00 0.541534364E+01 0.395095781E+01 0.138713462E+05 0.128818340E-09 -0.597664718E-10 0.248428992E-10 25 E 0.140451165E+05 -0.361014487E+01 0.568924633E+01 0.909383458E+00 0.140647708E+05 0.194292288E-09 0.164097555E-08 -0.846929702E-10 26 M 0.141761451E+05 -0.899639482E+01 0.194633976E+01 0.338475840E+00 0.142621227E+05 0.217485938E-10 0.254819316E-10 0.173977177E-11 27 E 0.147493287E+05 -0.468307736E+01 0.205377449E+01 0.112514671E+00 0.148588942E+05 -0.676941236E-07 0.517010403E-08 0.185978790E-09 28 E 0.154797652E+05 -0.287924099E+01 -0.420791716E+01 -0.207209935E-02 0.157811538E+05 0.107766151E-08 0.164281742E-07 0.432121794E-10 29 E 0.163930641E+05 ================ PARENT CYCLER 5.219Ggfh-f3 ======================= Parent cycler number 113 Approximate search space (synodic periods after J2000) 19 Number of steps to walk eccentricity/inclination 243 / 243 Number of cycles 7 Total delta v over 45.02 years (km/s) 0.004426 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 1 193.1 ****** 8.483 8.692 8.692 7 101.0 5.417 5.417 11.636 11.636 13 141.5 5.385 5.385 7.196 7.196 19 118.7 5.994 5.994 11.217 11.217 25 102.8 5.333 5.333 7.514 7.514 31 140.5 5.714 5.714 9.211 9.211 37 99.2 5.319 5.319 11.302 11.302 AVERAGE 128.1 5.527 5.949 9.538 9.538 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 15305.168830 LEG E/M time start vinfx vinfy vinfz time dv dvx dvy dxz (days) (km/s) (km/s) (km/s) (days) (km/s) (km/s) (km/s) 1 E -0.735243551E+02 -0.542817797E+01 0.468094635E+01 0.453782203E+01 -0.445661875E+02 -0.665920502E-04 -0.639287851E-04 -0.418520681E-05 2 M 0.119530095E+03 -0.760842792E+01 0.420185452E+01 0.532019797E-03 0.431370312E+03 0.150687267E-07 -0.340567950E-08 -0.108940698E-08 3 E 0.697011978E+03 -0.894556809E+00 0.528176417E+01 -0.818129500E-03 0.802882610E+03 -0.933273526E-06 0.872482179E-06 0.153058130E-07 4 E 0.140281619E+04 -0.526096799E+01 -0.113340091E+01 -0.412502131E+00 0.165117096E+04 0.330563694E-05 0.105613491E-04 -0.698914012E-04 5 E 0.176804380E+04 0.514913796E-01 -0.500708581E+00 -0.536926097E+01 0.179486063E+04 0.152674612E-05 0.562006878E-06 -0.730385801E-07 6 E 0.194682265E+04 -0.517651478E+01 0.406262716E-01 0.158678998E+01 0.206370265E+04 0.702363467E-06 -0.140027320E-04 -0.237569225E-03 7 E 0.231207266E+04 -0.142368143E+01 -0.517188411E+01 -0.751191630E+00 0.232722524E+04 0.106289607E-05 -0.660044090E-06 0.817641081E-07 8 M 0.241308991E+04 -0.107406863E+01 -0.115817836E+02 0.310514400E+00 0.250757166E+04 -0.647894479E-10 0.141425382E-09 -0.383423414E-10 9 E 0.304296824E+04 0.256721050E+01 -0.477360457E+01 -0.533590738E-01 0.361276755E+04 0.614843660E-05 0.729121219E-05 -0.375919914E-04 10 E 0.377348017E+04 0.496288065E+01 0.981632708E+00 -0.195779505E+01 0.402915858E+04 0.267046996E-05 -0.229372619E-05 0.107336363E-03 11 E 0.413873504E+04 -0.681537593E-01 0.483795396E+00 -0.538170419E+01 0.416670795E+04 0.873835816E-08 0.505356173E-07 0.901607517E-08 12 E 0.432522107E+04 0.533236963E+01 0.212706814E+00 0.696574883E+00 0.443846041E+04 0.402086683E-06 -0.103077306E-04 0.424834630E-03 13 E 0.469050927E+04 0.874586676E+00 0.520160113E+01 0.108713740E+01 0.471173481E+04 -0.141891715E-06 -0.389803450E-08 0.303109536E-08 14 M 0.483201286E+04 -0.201089667E+01 0.687346573E+01 -0.700462326E+00 0.491647206E+04 0.394447471E-10 0.623490553E-11 -0.388772235E-11 15 E 0.539507421E+04 0.430967837E+01 0.379755533E+01 0.344044434E-03 0.549920949E+04 0.129086849E-08 0.125139564E-07 -0.925271241E-10 16 E 0.608930942E+04 -0.370353959E+01 0.402447517E+01 -0.185233632E+01 0.633768185E+04 0.120666420E-07 -0.197954104E-06 -0.671149055E-05 17 E 0.645456300E+04 -0.464112678E+00 -0.326942908E+00 -0.576646834E+01 0.648172884E+04 0.128584717E-07 0.113740153E-08 -0.220713010E-08 18 E 0.663566861E+04 -0.257893065E+01 0.457724132E+01 0.284725872E+01 0.674889859E+04 0.268397951E-04 0.165383807E-04 0.320038169E-03 19 E 0.700092663E+04 -0.598648272E+01 0.199555190E+00 0.237095792E+00 0.701873181E+04 0.104439644E-07 0.244356711E-09 -0.173588632E-09 20 M 0.711962787E+04 -0.962004249E+01 -0.571373734E+01 0.787429717E+00 0.721046811E+04 -0.830659384E-11 0.444032413E-10 0.292777879E-12 21 E 0.772522948E+04 -0.311103299E+01 -0.449416432E+01 -0.499081716E+00 0.830234119E+04 0.102250833E-05 0.188420040E-05 -0.528731696E-04 22 E 0.845575063E+04 0.244264062E+01 -0.329889119E+01 -0.364685852E+01 0.871507281E+04 0.143933030E-04 0.579895117E-04 -0.951330904E-03 23 E 0.882099314E+04 0.443353759E+00 0.240208732E+00 -0.547866429E+01 0.884855576E+04 0.501959629E-07 0.301965420E-07 0.107635575E-07 24 E 0.900474391E+04 0.209715543E+01 -0.458045789E+01 0.173863534E+01 0.911797012E+04 0.137163994E-04 0.643642221E-05 -0.251004472E-03 25 E 0.936998973E+04 0.510904777E+01 0.147440633E+01 -0.400986817E+00 0.938541700E+04 0.891814698E-09 0.210141745E-07 -0.628756576E-10 26 M 0.947283818E+04 0.707102167E+01 0.242653651E+01 -0.753545497E+00 0.964207471E+04 -0.296385353E-10 -0.302899680E-11 0.521882446E-11 27 E 0.100772544E+05 0.531488980E+01 -0.166518243E+01 0.691603420E-03 0.101824538E+05 -0.191718964E-08 -0.845817595E-09 -0.431369360E-11 28 E 0.107785841E+05 0.459017147E+00 0.273328099E+01 -0.479165543E+01 0.109794782E+05 0.168164458E-03 0.676519724E-03 -0.842418054E-04 29 E 0.111438462E+05 -0.479036347E+00 0.171411709E+00 -0.551238700E+01 0.111715594E+05 0.135948717E-08 0.142232088E-07 0.723417687E-09 30 E 0.113286011E+05 0.238546024E+01 0.510370889E+01 0.831677360E+00 0.114418346E+05 0.866763616E-05 -0.392519746E-05 0.331294115E-03 31 E 0.116938706E+05 -0.351700290E+01 0.440124882E+01 0.954414545E+00 0.117149412E+05 -0.607842941E-08 0.158858647E-07 -0.200522544E-09 32 M 0.118343411E+05 -0.896577630E+01 0.210516976E+01 0.162927522E+00 0.119187951E+05 -0.569755621E-11 -0.900432127E-11 0.498246826E-11 33 E 0.123973677E+05 -0.194360210E+01 0.497218916E+01 -0.961900282E-03 0.127360689E+05 0.539527225E-07 -0.753002983E-08 -0.587865124E-11 34 E 0.131029952E+05 -0.488357919E+01 -0.218442399E+01 -0.514882932E+00 0.133513591E+05 -0.894001098E-07 0.572229304E-07 0.114586476E-04 35 E 0.134682362E+05 0.158148754E+00 -0.459541456E+00 -0.533199382E+01 0.134950498E+05 -0.160341731E-09 -0.548784117E-09 0.103988668E-09 36 E 0.136469936E+05 -0.508652090E+01 -0.104959132E+01 0.123629481E+01 0.137638751E+05 0.221860021E-05 -0.970386885E-05 -0.229053477E-03 37 E 0.140122483E+05 -0.935060580E-03 -0.524143880E+01 -0.903208529E+00 0.140271265E+05 -0.343900260E-09 -0.447656627E-09 -0.284188557E-10 38 M 0.141114359E+05 0.120538729E+01 -0.112353963E+02 0.196367453E+00 0.142054033E+05 -0.231539690E-09 0.199116858E-10 -0.669619926E-12 39 E 0.147378849E+05 -0.100398101E+00 -0.517095413E+01 0.445438468E-03 0.148452857E+05 -0.476158307E-06 0.829887576E-06 0.817231812E-08 40 E 0.154538901E+05 0.515990263E+01 0.237712198E+00 -0.179516797E+00 0.157095592E+05 0.219067915E-06 0.125013155E-05 -0.110815320E-03 41 E 0.158191317E+05 0.232499814E-01 0.450580917E+00 -0.516629828E+01 0.158470878E+05 0.467281402E-07 0.241806855E-06 0.424289959E-07 42 E 0.160055056E+05 0.504457123E+01 -0.488478954E+00 0.888150495E+00 0.161150791E+05 -0.978559908E-07 0.358493540E-06 -0.261926564E-05 43 E 0.163707505E+05 ================ PARENT CYCLER 5.219Ggfh+f3 ======================= Parent cycler number 114 Approximate search space (synodic periods after J2000) 16 Number of steps to walk eccentricity/inclination 243 / 243 Number of cycles 7 Total delta v over 44.90 years (km/s) 0.005993 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 1 116.0 ****** 5.635 8.342 8.342 7 144.8 5.499 5.499 8.898 8.898 13 100.1 5.466 5.466 11.633 11.633 19 141.3 5.394 5.394 7.196 7.196 25 118.7 5.995 5.995 11.215 11.215 31 102.9 5.331 5.331 7.514 7.514 37 140.4 5.714 5.714 9.216 9.216 AVERAGE 123.5 5.567 5.576 9.145 9.145 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 12965.276998 LEG E/M time start vinfx vinfy vinfz time dv dvx dvy dxz (days) (km/s) (km/s) (km/s) (days) (km/s) (km/s) (km/s) 1 E -0.259850353E+02 0.546869398E+01 0.105058850E+01 -0.864040475E+00 -0.859166001E+01 0.328547306E-05 0.165914557E-04 -0.103576538E-05 2 M 0.899708001E+02 0.825538952E+01 -0.100331172E+01 -0.658393589E+00 0.183211531E+03 0.178082526E-08 0.265345212E-07 0.958893977E-09 3 E 0.711575670E+03 0.468702057E+01 -0.267647816E+01 0.552950153E-03 0.817618284E+03 0.502781687E-06 0.220402247E-06 -0.307889587E-07 4 E 0.141852643E+04 0.232929377E+01 0.466710191E+01 0.134911677E+01 0.167055191E+04 -0.961314639E-06 0.256807438E-05 -0.361381078E-04 5 E 0.178378075E+04 -0.423072575E+00 0.245328718E+00 0.534796282E+01 0.181158284E+04 -0.453473449E-06 0.489966632E-06 -0.118757862E-06 6 E 0.196912803E+04 0.320365579E+01 0.443153007E+01 -0.573777448E+00 0.208236364E+04 0.165714402E-05 -0.161818619E-05 -0.126516813E-03 7 E 0.233440417E+04 -0.301328212E+01 0.447736457E+01 0.105525290E+01 0.235612703E+04 -0.454701221E-06 -0.383351262E-06 0.354245309E-08 218 8 M 0.247922326E+04 -0.832875820E+01 0.313130476E+01 0.304563498E-01 9 E 0.304052962E+04 -0.104005318E+01 0.530880243E+01 -0.130138693E-02 10 E 0.374386923E+04 -0.508062798E+01 -0.122833432E+01 0.157386941E+01 11 E 0.410909731E+04 0.660936453E-01 -0.508074905E+00 0.542508099E+01 12 E 0.428786783E+04 -0.524050759E+01 -0.609840851E-01 -0.156121528E+01 13 E 0.465311826E+04 -0.150305445E+01 -0.520143351E+01 -0.753742082E+00 14 M 0.475324665E+04 -0.104453789E+01 -0.115816790E+02 0.307445378E+00 15 E 0.538311697E+04 0.258236425E+01 -0.476305169E+01 0.270736108E+00 16 E 0.611362645E+04 0.464601774E+01 0.973907876E+00 0.263236722E+01 17 E 0.647888250E+04 -0.701507888E-01 0.484993562E+00 0.538735611E+01 18 E 0.666536745E+04 0.536645805E+01 0.238822674E+00 -0.436434618E+00 19 E 0.703065516E+04 0.893870765E+00 0.520735190E+01 0.108591907E+01 20 M 0.717196988E+04 -0.201440443E+01 0.687264985E+01 -0.700454674E+00 21 E 0.773498302E+04 0.430900886E+01 0.379854950E+01 0.344567141E-03 22 E 0.842921252E+04 -0.384243824E+01 0.420922460E+01 0.929704886E+00 23 E 0.879446634E+04 -0.464001210E+00 -0.325481577E+00 0.576867741E+01 24 E 0.897557058E+04 -0.260690800E+01 0.461394326E+01 -0.276638299E+01 25 E 0.934082850E+04 -0.598698535E+01 0.200675623E+00 0.237007210E+00 26 M 0.945952851E+04 -0.961852021E+01 -0.571413970E+01 0.787268443E+00 27 E 0.100650688E+05 -0.313843911E+01 -0.445206562E+01 -0.643315595E+00 28 E 0.107955899E+05 0.272133815E+01 -0.379527598E+01 0.288054482E+01 29 E 0.111608345E+05 0.443385745E+00 0.241199337E+00 0.547837709E+01 30 E 0.113445792E+05 0.197699236E+01 -0.438424608E+01 -0.229656002E+01 31 E 0.117098246E+05 0.510687750E+01 0.147700716E+01 -0.401243189E+00 32 M 0.118127059E+05 0.707214762E+01 0.242501047E+01 -0.753465370E+00 33 E 0.124171290E+05 0.531442419E+01 -0.166782725E+01 0.692881014E-03 34 E 0.131184456E+05 0.143130556E+01 0.531169911E+01 0.618320287E+00 35 E 0.134837183E+05 -0.479183239E+00 0.173431067E+00 0.551465595E+01 36 E 0.136684759E+05 0.234541721E+01 0.498618115E+01 -0.143692785E+01 37 E 0.140337459E+05 -0.351955006E+01 0.439926692E+01 0.954693140E+00 38 M 0.141741910E+05 -0.896994147E+01 0.207913441E+01 0.379885577E+00 39 E 0.147525445E+05 -0.355590071E+01 0.376609602E+01 0.695433861E-03 40 E 0.154661780E+05 -0.355584408E+01 -0.378191174E+01 0.171149441E+00 41 E 0.158314102E+05 0.300775302E+00 -0.347539236E+00 0.516673475E+01 42 E 0.160106040E+05 -0.415143323E+01 -0.291430614E+01 -0.594315045E+00 43 E 0.163758476E+05 ================ PARENT CYCLER 5.225Ggg3 ======================= Parent cycler number 117 Approximate search space (synodic periods after J2000) 19 Number of steps to walk eccentricity/inclination 81 / 81 Number of cycles 7 Total delta v over 44.74 years (km/s) 0.000000 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 1 110.4 ****** 11.558 9.003 9.003 5 98.4 5.567 5.567 11.585 11.585 9 144.2 5.302 5.302 7.222 7.222 13 119.8 5.900 5.900 11.138 11.138 17 105.7 5.245 5.245 7.560 7.560 21 142.3 5.560 5.560 9.213 9.213 25 97.3 5.400 5.400 11.268 11.268 AVERAGE 116.9 5.496 6.362 9.570 9.570 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 15305.331809 LEG E/M time start vinfx vinfy vinfz (days) (km/s) (km/s) (km/s) 1 E 0.370130761E+02 -0.414134615E+01 0.107484435E+02 0.954140000E+00 2 M 0.147428592E+03 -0.860047073E+01 0.266046518E+01 0.193795946E-01 3 E 0.700823039E+03 -0.807404834E+00 0.545804854E+01 -0.139547925E-02 4 E 0.140042093E+04 -0.554776984E+01 -0.506839539E+00 -0.333658624E-04 5 E 0.231553401E+04 -0.168371200E+01 -0.525118637E+01 -0.760817852E+00 6 M 0.241398092E+04 -0.944037513E+00 -0.115420394E+02 0.319538820E+00 7 E 0.304156926E+04 0.212963449E+01 -0.486644011E+01 -0.178486983E+00 8 E 0.377209670E+04 0.530555407E+01 0.428025651E+00 -0.829562409E-03 9 E 0.468622258E+04 0.568551208E+00 0.515367307E+01 0.110616018E+01 10 M 0.483042312E+04 -0.192858107E+01 0.692046469E+01 -0.740213295E+00 11 E 0.539261178E+04 0.469846950E+01 0.351449639E+01 0.207853140E-03 12 E 0.608355118E+04 -0.313243363E+01 0.498165622E+01 0.266289541E-02 13 E 0.699998438E+04 -0.589244150E+01 0.179079212E+00 0.233688180E+00 14 M 0.711982647E+04 -0.952597245E+01 -0.572012663E+01 0.774415553E+00 15 E 0.772057733E+04 -0.384395482E+01 -0.353592355E+01 0.383121659E+00 16 E 0.845105598E+04 0.223303995E+01 -0.472305305E+01 0.837574478E-01 17 E 0.936486325E+04 0.494437129E+01 0.169749822E+01 -0.424363326E+00 18 M 0.947059154E+04 0.716466950E+01 0.229303156E+01 -0.747382910E+00 19 E 0.100757847E+05 0.527573780E+01 -0.185062771E+01 0.767935655E-03 20 E 0.107763011E+05 0.208630405E+01 0.515055901E+01 -0.347758714E-01 21 E 0.116914065E+05 -0.354095058E+01 0.417653440E+01 0.967937581E+00 22 M 0.118337473E+05 -0.896022984E+01 0.213854781E+01 0.146352371E+00 23 E 0.123961998E+05 -0.176276873E+01 0.508278120E+01 -0.796315047E-03 24 E 0.131003174E+05 -0.519372012E+01 -0.155897981E+01 -0.445091950E-04 25 E 0.140148145E+05 -0.166638251E+00 -0.532008967E+01 -0.909978196E+00 26 M 0.141120687E+05 0.129043703E+01 -0.111911730E+02 0.230655329E+00 27 E 0.147345066E+05 0.161120902E+01 -0.485885296E+01 -0.645528117E-01 28 E 0.154649702E+05 0.506665110E+01 0.803376736E+00 0.102201698E-02 29 E 0.163783218E+05 ================ PARENT CYCLER 5.333gGf3 ======================= Parent cycler number 122 Approximate search space (synodic periods after J2000) 13 Number of steps to walk eccentricity/inclination 27 / 27 Number of cycles 7 Total delta v over 45.30 years (km/s) 0.145076 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 2 186.8 12.801 12.801 12.593 12.592 6 168.1 8.889 8.889 7.355 7.355 10 162.6 5.966 5.953 10.443 10.443 14 113.6 5.876 5.875 9.928 9.928 18 167.6 5.785 5.785 8.465 8.465 22 127.9 6.400 6.287 13.003 13.003 26 147.0 8.481 8.479 8.854 8.854 AVERAGE 153.4 7.742 7.724 10.092 10.091 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------- 0.269813274E+04 0.314603056E+04 0.399222433E+04 0.413591289E+04 0.440109546E+04 0.466813752E+04 0.510597403E+04 0.549999849E+04 0.635104288E+04 0.650685524E+04 0.677495376E+04 0.705185236E+04 0.725642185E+04 0.783911745E+04 0.867758512E+04 0.882163197E+04 0.908880053E+04 0.935863350E+04 0.955035956E+04 0.101819522E+05 0.110512611E+05 0.111883962E+05 0.114578053E+05 0.117252568E+05 0.119517232E+05 0.125223265E+05 0.133704837E+05 0.135114319E+05 0.137780569E+05 0.140548126E+05 0.142609441E+05 0.148595896E+05 0.155355721E+05 0.158582892E+05 0.160763479E+05 0.148172836E-09 0.122468425E-07 0.393704827E-04 0.310205048E-06 -0.324551769E-05 -0.410269657E-06 0.603554067E-09 -0.103042483E-04 0.224300241E-04 0.147048651E-05 -0.592814211E-05 -0.127044741E-04 0.212002826E-08 -0.891222026E-07 -0.223954371E-04 -0.402460080E-05 0.219663791E-04 -0.330145787E-05 -0.171898710E-08 -0.103038931E-05 -0.270604617E-04 -0.320911045E-05 0.127343839E-04 -0.260612646E-06 -0.226651960E-08 0.147546564E-05 -0.107431841E-04 0.377059974E-06 0.768826908E-04 -0.204476577E-05 -0.967237363E-07 0.278569377E-04 0.230116695E-04 0.157229368E-04 0.125758383E-04 -0.126447389E-09 -0.920215924E-08 0.867667012E-04 0.217142567E-06 0.823328924E-05 0.765894936E-07 -0.275360610E-08 0.132197274E-04 0.566038847E-04 0.112050479E-04 0.963060566E-05 -0.117267732E-05 0.652201541E-09 -0.865911577E-06 -0.103690274E-04 -0.348378744E-06 0.165014410E-04 -0.508892923E-07 0.234200974E-08 -0.576695655E-05 0.258026443E-04 -0.191624051E-05 0.903018213E-06 -0.441634297E-05 0.592031644E-09 0.648385190E-06 0.112683725E-04 0.532771955E-05 -0.533325612E-05 0.534470854E-05 -0.103846047E-07 0.584386459E-06 -0.819041447E-05 0.233568221E-04 -0.140939892E-04 -0.206292539E-10 -0.858934190E-10 0.670966338E-03 0.327078242E-07 -0.383144153E-04 0.218053711E-07 0.551864368E-09 0.379329855E-04 0.843764400E-04 -0.455285974E-05 -0.119935257E-03 0.312988721E-06 -0.433877131E-09 0.690624248E-08 0.261661892E-03 -0.691658634E-06 -0.232140545E-03 0.550886465E-07 0.359407193E-10 0.353941142E-04 0.648356897E-03 0.688215097E-06 0.263217307E-03 0.790479884E-08 0.108706104E-09 -0.213166175E-08 -0.161647102E-03 -0.283246793E-06 -0.290315852E-03 -0.683185581E-07 -0.211890849E-07 -0.112090593E-05 -0.107037393E-03 0.126812415E-05 0.173921581E-04 time dv (days) 0.535754035E+02 0.230437759E+03 0.805762723E+03 0.205930235E+04 0.233030104E+04 0.250811917E+04 0.315114838E+04 0.404633446E+04 0.470785266E+04 0.491475142E+04 0.549625269E+04 0.639513847E+04 0.701796069E+04 0.720993910E+04 0.783014913E+04 0.872519816E+04 0.938072250E+04 0.956137052E+04 0.101808622E+05 0.114992344E+05 0.117127576E+05 0.119181152E+05 0.125018174E+05 0.133655216E+05 0.140294026E+05 0.142054344E+05 0.148659901E+05 0.157115752E+05 dvx (km/s) 0.551899089E-10 0.103766288E-10 0.531496349E-09 -0.335134026E-09 0.192283098E-08 0.245229697E-10 -0.715341905E-08 -0.118018048E-07 -0.212455357E-08 0.477166199E-11 0.238232582E-09 -0.437584221E-10 0.143427628E-09 -0.358122657E-11 -0.525874848E-09 0.835149971E-09 -0.165761519E-08 -0.561158032E-11 -0.321559755E-09 -0.107006918E-10 0.590323699E-10 0.244865953E-11 0.149267375E-10 -0.132336738E-10 -0.602525663E-10 -0.180020290E-10 -0.191131021E-11 0.190271262E-10 dvy (km/s) 0.180245150E-09 0.158063370E-11 -0.400875822E-09 -0.920834868E-10 0.618153434E-09 -0.878194132E-11 0.101650223E-07 -0.122149173E-08 0.121366545E-07 0.255943616E-11 0.115280113E-08 0.168397011E-10 0.893620191E-10 0.237177724E-10 -0.798749102E-09 -0.142614825E-08 -0.249700444E-09 0.234532312E-10 -0.122214730E-09 0.106345565E-10 -0.728000863E-10 0.880260860E-11 -0.111801727E-10 -0.531231808E-11 -0.568763591E-12 0.853145387E-12 0.329601808E-11 -0.122680984E-11 dxz (km/s) 0.503413647E-11 0.126333926E-10 0.339615853E-11 -0.761319524E-11 0.135729686E-09 -0.279943497E-11 0.538319555E-10 -0.287912544E-10 -0.518602859E-09 -0.517767074E-11 -0.152096443E-10 -0.583640522E-11 -0.507547103E-11 0.652349829E-12 0.100516874E-10 -0.835103496E-12 -0.110768673E-09 -0.194032190E-11 -0.657822218E-11 -0.737399568E-11 0.612423224E-11 -0.202663364E-10 0.281887917E-11 -0.545401389E-11 -0.579190235E-11 -0.599046326E-11 -0.699185481E-11 0.274991402E-11 219 EPOCH TIME (days after J2000) 10890.237955 LEG E/M time start vinfx vinfy vinfz (days) (km/s) (km/s) (km/s) 1 E -0.131011506E+03 -0.585741365E+01 -0.114158287E+02 0.103359150E-03 2 E 0.332327578E+03 -0.126496978E+02 0.173139938E+01 -0.918189485E+00 3 M 0.519121475E+03 -0.118826778E+02 -0.406101280E+01 0.927750925E+00 4 E 0.119677122E+04 -0.613545124E+01 0.323204078E+01 -0.554261887E+01 5 E 0.229254685E+04 -0.397431994E+01 0.794279933E+01 -0.101807658E-02 6 E 0.277175343E+04 0.639832607E+01 0.616318177E+01 0.290051977E+00 7 M 0.293983156E+04 0.521248535E+01 0.512509588E+01 -0.811009742E+00 8 E 0.355070374E+04 0.357696299E+01 -0.794856829E+00 -0.472391231E+01 9 E 0.464646951E+04 0.866926011E+00 -0.591489413E+01 0.605833273E-03 10 E 0.513679720E+04 -0.586923721E+01 0.985775727E-01 0.992696798E+00 11 M 0.529942508E+04 -0.103614000E+02 -0.108714708E+01 0.719457359E+00 12 E 0.590549337E+04 -0.304992294E+01 0.168889465E+01 -0.472647276E+01 13 E 0.700126377E+04 0.101630464E+01 0.577830857E+01 -0.526917257E-03 14 E 0.749225536E+04 0.539381493E+01 -0.202431998E+01 -0.115312930E+01 15 M 0.760582185E+04 0.749630132E+01 -0.647637060E+01 -0.659825921E+00 16 E 0.825669932E+04 0.151701060E+01 -0.514414518E+01 0.218572827E+01 17 E 0.935244888E+04 -0.309915962E+01 -0.489511992E+01 0.114947874E-02 18 E 0.984336353E+04 -0.361676377E+01 0.422775905E+01 0.158632704E+01 19 M 0.100109137E+05 -0.659270984E+01 0.527896911E+01 -0.565856374E+00 20 E 0.105640908E+05 0.243808487E+01 0.308360426E+01 0.505697568E+01 21 E 0.116598359E+05 0.630937606E+01 -0.110667064E+01 0.119619834E-02 22 E 0.121488953E+05 -0.228323015E+01 -0.584711532E+01 -0.348837128E+00 23 M 0.122767947E+05 -0.698033875E+01 -0.109570021E+02 0.545801016E+00 24 E 0.129303701E+05 -0.649944445E+01 -0.224293378E+01 0.491435084E+01 25 E 0.140261255E+05 -0.803164730E+01 0.260685379E+01 0.628438816E-03 26 E 0.145066174E+05 0.218439659E+01 0.816560293E+01 0.672343203E+00 27 M 0.146535764E+05 0.436043554E+01 0.769204673E+01 -0.455664431E+00 28 E 0.153188270E+05 0.801138098E+01 0.349378427E+01 0.498037777E+01 29 E 0.164145897E+05 ================ PARENT CYCLER 5.333ggF3 ======================= Parent cycler number 123 Approximate search space (synodic periods after J2000) 16 Number of steps to walk eccentricity/inclination 243 / 243 Number of cycles 7 Total delta v over 44.87 years (km/s) 0.417988 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 3 190.4 8.875 8.875 3.012 3.012 7 158.2 5.265 5.265 7.122 7.122 11 94.8 6.872 6.872 5.785 5.785 15 214.6 5.083 5.140 3.599 3.599 19 107.4 5.700 5.499 10.854 10.854 23 193.9 9.567 9.567 3.181 3.153 27 130.6 5.689 5.694 10.111 10.111 AVERAGE 155.7 6.722 6.702 6.238 6.234 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 12518.073970 LEG E/M time start vinfx vinfy vinfz (days) (km/s) (km/s) (km/s) 1 E -0.100943991E+02 -0.880641690E+01 0.128243752E+01 0.107477443E-02 2 E 0.468604970E+03 -0.326758634E-01 0.891446726E+01 -0.177920665E-02 3 E 0.128750206E+04 0.696705433E+01 0.539215846E+01 0.107593450E+01 4 M 0.147787828E+04 0.216899749E+01 0.817568029E+00 -0.192307787E+01 5 E 0.233010979E+04 0.406283607E+01 -0.339045270E+01 0.104416801E-02 6 E 0.282329579E+04 -0.453209386E+01 -0.271261382E+01 -0.646598070E-01 7 E 0.355382475E+04 -0.400619072E+01 0.339710323E+01 0.355073541E+00 8 M 0.371206071E+04 -0.697929019E+01 -0.120578349E+01 0.744162530E+00 9 E 0.466241030E+04 -0.102448308E+01 0.676128708E+01 -0.149483633E-02 10 E 0.514954362E+04 0.676294202E+01 0.113404259E+01 -0.113878032E-02 11 E 0.593798154E+04 0.426345797E+01 -0.532954283E+01 -0.804835542E+00 12 M 0.603281285E+04 0.513634827E+01 -0.254418612E+01 -0.778685257E+00 13 E 0.702260636E+04 -0.374893595E+00 -0.523882548E+01 -0.727659185E-04 14 E 0.751570532E+04 -0.462344004E+01 0.245890428E+01 -0.277816785E-02 15 E 0.825089635E+04 0.242556028E+00 0.502098670E+01 0.107384547E+01 16 M 0.846549794E+04 -0.300277778E+01 0.197907490E+01 0.127619569E+00 17 E 0.934762264E+04 0.417838834E+01 0.315219519E+01 -0.203030518E-03 18 E 0.984123670E+04 0.142182722E+01 -0.503313345E+01 0.177090112E-02 19 E 0.105636799E+05 -0.398975509E+01 -0.374842767E+01 -0.517322281E+00 20 M 0.106710647E+05 -0.558144229E+01 -0.930211273E+01 0.362276993E+00 21 E 0.116851449E+05 -0.959489585E+01 0.356804239E+00 -0.334936229E-03 22 E 0.121610511E+05 -0.133335386E+01 0.952206201E+01 -0.919474336E-03 23 E 0.129887431E+05 0.638217317E+01 0.704884494E+01 0.105629756E+01 24 M 0.131826239E+05 0.209834819E+01 0.116671599E+01 -0.204443812E+01 25 E 0.140277005E+05 0.489489071E+01 -0.296135926E+01 0.524432870E-03 26 E 0.145192592E+05 -0.383232782E+01 -0.392303373E+01 0.160483526E+01 27 E 0.152499034E+05 -0.535829551E+01 0.187748465E+01 0.431742795E+00 28 M 0.153804889E+05 -0.100134058E+02 -0.105949637E+01 0.916204544E+00 29 E 0.163795091E+05 ================ PARENT CYCLER 5.333gGff3 ======================= Parent cycler number 126 Approximate search space (synodic periods after J2000) 10 Number of steps to walk eccentricity/inclination 9 / 9 Number of cycles 7 Total delta v over 45.05 years (km/s) 0.201402 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 2 201.0 12.165 12.165 7.589 7.589 7 129.1 5.519 5.632 12.383 12.383 12 161.4 7.162 7.162 6.912 6.912 17 153.8 5.286 5.356 10.467 10.467 22 112.9 5.807 5.807 9.864 9.864 27 170.2 5.684 5.684 8.242 8.242 32 127.8 5.259 5.259 11.806 11.806 AVERAGE 150.9 6.698 6.724 9.609 9.609 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 8550.345812 LEG E/M time start vinfx vinfy vinfz (days) (km/s) (km/s) (km/s) 1 E -0.945623671E+02 -0.847910586E+01 0.866607796E+01 0.744454486E-03 time dv (days) 0.105291427E+03 0.360346663E+03 0.620768936E+03 0.136113756E+04 0.254173427E+04 0.279696515E+04 0.316585426E+04 0.371506861E+04 0.495047268E+04 0.516119138E+04 0.539033532E+04 0.661774413E+04 0.725657940E+04 0.750929033E+04 0.770345348E+04 0.842106176E+04 0.960772450E+04 0.986849605E+04 0.103040975E+05 0.112324953E+05 0.119972869E+05 0.121680802E+05 0.123748310E+05 0.130947334E+05 0.142711763E+05 0.145286612E+05 0.147533640E+05 0.154831914E+05 dvx (km/s) -0.145942297E-05 0.669355530E-05 0.500065547E-10 0.258348980E-05 -0.127804502E-04 0.410308767E-05 0.150344961E-06 0.201873531E-04 -0.146025178E-02 0.982640652E-05 0.272300430E-06 -0.238874637E-03 0.809944594E-05 -0.230810666E-04 -0.327502797E-07 -0.236833507E-04 0.623045988E-05 -0.145176755E-04 0.235621561E-08 -0.280920785E-03 -0.239626694E-05 -0.152188922E-03 0.187608390E-07 0.789781838E-05 0.324240137E-04 -0.199665177E-04 0.877372738E-06 -0.212718596E-04 dvy (km/s) -0.182550855E-04 -0.234615570E-04 0.114856364E-06 -0.416779607E-04 0.176649782E-04 0.182256410E-04 0.337270692E-07 0.200824810E-03 0.255441808E-03 0.328063582E-03 0.333338499E-06 -0.111549595E-03 0.290664540E-04 0.138752513E-04 -0.771909082E-08 -0.244345531E-04 0.145505881E-04 0.178865363E-06 0.296520560E-06 0.698764361E-03 0.222196103E-04 0.124225454E-03 -0.200516901E-06 -0.106001835E-03 -0.208217843E-04 -0.270349774E-04 -0.426576556E-06 0.708745852E-04 dxz (km/s) -0.528403630E-05 -0.312312987E-05 -0.236182291E-08 -0.161046902E-03 -0.385295291E-05 -0.477301785E-06 -0.619855805E-07 -0.311630328E-05 0.245793119E-04 0.167872165E-04 -0.883993976E-08 -0.386547202E-04 0.307248968E-05 -0.217843354E-05 0.364887649E-08 -0.403139820E-03 -0.207399184E-05 0.737227068E-06 0.482067065E-06 -0.163249641E-02 0.559633527E-05 0.587990890E-05 -0.117002315E-07 -0.487279407E-03 -0.579968921E-05 0.600203762E-06 -0.354974952E-07 -0.268845240E-03 time dv (days) 0.277125222E+03 0.689707185E+03 0.131605850E+04 0.198069488E+04 0.258163465E+04 0.339310838E+04 0.357756014E+04 0.409220055E+04 0.491084829E+04 0.529146244E+04 0.595220623E+04 0.651781167E+04 0.726915584E+04 0.780978173E+04 0.828308659E+04 0.861545914E+04 0.960430195E+04 0.101807850E+05 0.105797876E+05 0.113504985E+05 0.119897249E+05 0.126080048E+05 0.130178252E+05 0.133178361E+05 0.142833110E+05 0.150745488E+05 0.152694912E+05 0.157401362E+05 dvx (km/s) 0.778241867E-08 -0.105002859E-06 -0.229677273E-06 -0.210889273E-08 -0.158941227E-06 0.231367803E-05 -0.992702512E-07 0.115955685E-08 0.382033637E-07 0.267929383E-07 -0.162080033E-05 0.145907470E-07 0.224301141E-06 0.304714900E-01 0.618702245E-06 -0.230463662E-06 0.489708268E-04 0.740142747E-01 0.519302752E-04 -0.183464534E-07 -0.221529451E-08 -0.110525036E-06 0.519341498E-05 0.791853835E-08 -0.314946885E-05 -0.681782414E-03 0.731391418E-05 0.113956084E-09 dvy (km/s) 0.113459607E-07 -0.260701638E-07 0.136653614E-06 -0.778078513E-09 0.137961935E-06 -0.113942019E-05 0.124332077E-06 -0.301147095E-09 -0.764212196E-06 0.163657719E-06 -0.186266054E-05 -0.416957613E-08 0.633115018E-06 0.449342285E-02 -0.414389556E-05 0.200517161E-06 0.375422155E-04 -0.519470875E-01 -0.190991427E-04 -0.946274638E-08 -0.402954071E-06 0.175855587E-05 0.154884851E-04 -0.139998843E-07 0.182688195E-05 0.174361214E-02 -0.289774416E-05 -0.946561500E-09 dxz (km/s) -0.419939164E-08 -0.806681579E-09 0.131382817E-07 0.130892095E-10 -0.913781593E-08 -0.162191569E-04 0.637476847E-08 -0.479733075E-10 -0.604758803E-07 -0.776825841E-09 -0.145871544E-08 -0.139623009E-08 0.103877495E-06 -0.405256750E-04 -0.659986809E-07 0.173110687E-06 -0.591049065E-05 0.209532160E-04 -0.707475771E-06 0.168549587E-09 0.131614311E-06 0.406637666E-06 -0.296679808E-06 0.818160681E-08 -0.423212657E-06 -0.243499370E-02 -0.487274710E-06 -0.202219741E-09 time dv dvx (days) (km/s) 0.152659238E+03 -0.379862849E-04 dvy (km/s) 0.287283077E-04 dxz (km/s) 0.479314349E-05 220 2 E 0.371893491E+03 0.469203358E+01 0.110998054E+02 -0.166450206E+01 3 M 0.572922177E+03 -0.109914410E+01 0.749900883E+01 -0.391605728E+00 4 E 0.120977607E+04 0.439791245E+01 -0.571476040E+00 0.333193311E+01 5 E 0.157503752E+04 0.183761617E+01 -0.517926899E+01 0.627967718E+00 6 E 0.230551498E+04 0.534440879E+01 -0.142821997E+01 0.162675923E-03 7 E 0.279792363E+04 -0.315247702E+01 -0.466169138E+01 -0.214414204E+00 8 M 0.292701340E+04 -0.754538959E+01 -0.979586936E+01 0.660097862E+00 9 E 0.356657378E+04 -0.641530559E+01 0.274874570E+01 -0.153735360E+01 10 E 0.393182136E+04 -0.189283952E+01 0.619336779E+01 -0.299586903E+01 11 E 0.466235348E+04 -0.606879113E+01 0.378064088E+01 0.214621316E-03 12 E 0.514806920E+04 0.308523580E+01 0.642793497E+01 0.680350698E+00 13 M 0.530942609E+04 0.338421174E+01 0.597840845E+01 -0.763820491E+00 14 E 0.591494871E+04 -0.395555272E+01 0.303487742E+01 -0.183166356E+01 15 E 0.628019297E+04 -0.449420944E+01 -0.281083442E+01 -0.601533936E-03 16 E 0.699636560E+04 0.172319960E+01 -0.501485368E+01 -0.438169384E-03 17 E 0.748945027E+04 -0.528481532E+01 0.338938104E+00 0.804502770E+00 18 M 0.764321469E+04 -0.103466098E+02 -0.140579730E+01 0.724976010E+00 19 E 0.824668103E+04 -0.213353794E-01 0.579262101E+01 0.323554514E+00 20 E 0.861195332E+04 0.481858662E+01 0.325011363E+01 0.816965459E-01 21 E 0.934243261E+04 0.872647076E+00 0.573080379E+01 -0.212208327E-03 22 E 0.983368819E+04 0.533349243E+01 -0.198935810E+01 -0.114583173E+01 23 M 0.994661748E+04 0.751398447E+01 -0.635704859E+01 -0.645501377E+00 24 E 0.105958043E+05 -0.231085178E+01 -0.518307448E+01 -0.458284992E+00 25 E 0.109610482E+05 -0.564903512E+01 -0.415753003E+00 -0.406057248E+00 26 E 0.116915653E+05 -0.310057045E+01 -0.477381044E+01 0.130122059E-02 27 E 0.121828795E+05 -0.350251072E+01 0.417449326E+01 0.161677072E+01 28 M 0.123530746E+05 -0.659769997E+01 0.493864705E+01 -0.611962080E-01 29 E 0.129259439E+05 -0.170337258E+00 -0.611767323E+00 0.519539649E+01 30 E 0.132912068E+05 0.835679304E+00 -0.516702385E+01 0.192684382E-02 31 E 0.140051711E+05 0.525440647E+01 -0.316564001E+00 0.632004575E-03 32 E 0.144985939E+05 -0.218898962E+01 -0.476203955E+01 -0.437545262E+00 33 M 0.146264106E+05 -0.514007008E+01 -0.106113223E+02 0.607371119E+00 34 E 0.152487907E+05 -0.451759803E+01 0.234104960E+01 -0.148693494E+01 35 E 0.156140481E+05 0.558575820E+00 0.529132993E+01 -0.463274831E-03 36 E 0.163595383E+05 ================ PARENT CYCLER 5.333gGff3 ======================= Parent cycler number 127 Approximate search space (synodic periods after J2000) 13 Number of steps to walk eccentricity/inclination 81 / 81 Number of cycles 7 Total delta v over 45.23 years (km/s) 0.089069 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 2 178.5 11.156 11.156 12.777 12.777 7 158.5 8.793 8.793 7.925 7.925 12 179.0 7.745 7.745 10.364 10.364 17 116.1 6.023 6.023 10.004 10.004 22 168.1 5.828 5.828 8.460 8.460 27 128.1 6.339 6.310 13.014 13.014 32 146.9 8.494 8.484 8.859 8.859 AVERAGE 153.6 7.768 7.763 10.200 10.200 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 10890.237955 LEG E/M time start vinfx vinfy vinfz (days) (km/s) (km/s) (km/s) 1 E -0.107248984E+03 -0.465928833E+00 -0.111736290E+02 -0.348626553E-03 2 E 0.362736393E+03 -0.105731072E+02 -0.290800274E+01 -0.204912760E+01 3 M 0.541255905E+03 -0.111286825E+02 -0.621572646E+01 0.877032941E+00 4 E 0.120869771E+04 -0.542300988E+01 -0.138265219E+01 0.674672321E+01 5 E 0.193921041E+04 -0.523644185E+01 0.507333502E+01 0.486868854E+01 6 E 0.230447318E+04 -0.545604801E+01 0.683899103E+01 0.223134783E-03 7 E 0.278400182E+04 0.544640246E+01 0.689946452E+01 0.241942764E+00 8 M 0.294254506E+04 0.564038090E+01 0.550079360E+01 -0.855231468E+00 9 E 0.353045704E+04 0.445192267E+01 0.187172851E+01 -0.606949088E+01 10 E 0.426097036E+04 -0.788773133E+00 -0.381541552E+01 -0.670775149E+01 11 E 0.462622232E+04 -0.175121776E+01 -0.756550976E+01 -0.357959697E-03 12 E 0.510951446E+04 -0.740928138E+01 0.832344924E+00 0.209769752E+01 13 M 0.528849159E+04 -0.103387887E+02 -0.203404799E+00 0.699170925E+00 14 E 0.590155551E+04 -0.485788832E+01 0.341416813E+01 -0.111934717E+01 15 E 0.663207307E+04 0.562444126E+00 0.578340651E+01 0.161997888E+01 16 E 0.699732117E+04 0.145541647E+01 0.583830016E+01 -0.137267690E-02 17 E 0.748776949E+04 0.551209333E+01 -0.212369474E+01 -0.117614005E+01 18 M 0.760389334E+04 0.741688419E+01 -0.668116011E+01 -0.653885614E+00 19 E 0.825600226E+04 0.382577550E+01 -0.436823437E+01 0.573299184E+00 20 E 0.898649961E+04 -0.236660337E+01 -0.526500708E+01 -0.737148288E+00 21 E 0.935174867E+04 -0.318513749E+01 -0.488971853E+01 0.131111669E-02 22 E 0.984250010E+04 -0.365590424E+01 0.424445227E+01 0.160745840E+01 23 M 0.100106389E+05 -0.657916923E+01 0.528858830E+01 -0.562760034E+00 24 E 0.105641135E+05 0.343937603E+01 0.517254939E+01 0.152330876E+01 25 E 0.112939740E+05 0.566016417E+01 -0.247770812E+00 -0.286449274E+01 26 E 0.116592120E+05 0.623971873E+01 -0.115692339E+01 0.114104770E-02 27 E 0.121485037E+05 -0.229017384E+01 -0.586985268E+01 -0.347715569E+00 28 M 0.122766143E+05 -0.700683024E+01 -0.109530559E+02 0.546684393E+00 29 E 0.129303948E+05 -0.695518906E+01 -0.420940541E+01 0.239933918E+01 30 E 0.136609004E+05 -0.796148833E+01 0.130773008E+01 -0.258837117E+01 31 E 0.140261532E+05 -0.804397794E+01 0.260868915E+01 0.632356131E-03 32 E 0.145065967E+05 0.218682393E+01 0.816950829E+01 0.671361204E+00 33 M 0.146535179E+05 0.436788037E+01 0.769348465E+01 -0.455816638E+00 34 E 0.153188478E+05 0.837217796E+01 0.559117137E+01 -0.173655482E+00 35 E 0.160493642E+05 0.990376434E+01 0.259440259E+00 -0.174221168E+01 36 E 0.164146130E+05 ================ PARENT CYCLER 5.333ggFf3 ======================= Parent cycler number 128 Approximate search space (synodic periods after J2000) 12 Number of steps to walk eccentricity/inclination 9 / 9 Number of cycles 7 Total delta v over 44.99 years (km/s) 0.002351 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 3 138.8 5.557 5.557 11.700 11.700 8 77.0 8.010 8.010 8.020 8.020 13 137.0 5.996 5.996 9.230 9.230 0.450294679E+03 0.668450261E+03 0.137779633E+04 0.189644760E+04 0.253202296E+04 0.281728710E+04 0.302294746E+04 0.369075796E+04 0.450163641E+04 0.491492566E+04 0.517227274E+04 0.546686197E+04 0.616696725E+04 0.655950029E+04 0.725276963E+04 0.751251493E+04 0.773373464E+04 0.835991544E+04 0.893336421E+04 0.959788551E+04 0.985062759E+04 0.100439955E+05 0.107126824E+05 0.112824757E+05 0.119470487E+05 0.122220244E+05 0.126223232E+05 0.131779753E+05 0.134625583E+05 0.142568168E+05 0.145177664E+05 0.147197676E+05 0.153620205E+05 0.159047893E+05 0.119437250E-04 -0.183008941E-06 0.788566996E-04 0.975676476E-05 0.169803580E-05 -0.166340049E-04 -0.123948004E-07 -0.662799675E-05 0.568457835E-04 -0.123226846E-05 -0.193972088E-05 0.619884526E-08 -0.224571427E-04 -0.169765358E-05 -0.161136877E-05 -0.105914128E-05 0.735516809E-08 0.112018598E-05 0.348130169E-05 0.100258569E-06 -0.500594173E-06 -0.121965979E-08 0.181848114E-05 -0.429309338E-05 0.754235315E-07 -0.178477013E-06 -0.192123050E-10 -0.798198438E-03 -0.101678631E-06 0.491216710E-06 0.171004326E-06 -0.858105534E-11 -0.213119294E-05 -0.367661401E-06 -0.461883983E-04 -0.118977108E-06 -0.119172157E-03 -0.163563037E-04 0.176900611E-05 0.291950953E-05 0.416627482E-07 -0.207267281E-04 0.150611690E-03 0.631652489E-06 0.613838691E-06 0.567163282E-08 -0.250656796E-04 -0.362135115E-06 0.393118436E-05 -0.466623615E-05 0.777299439E-08 -0.217538030E-06 0.141719016E-05 0.527323635E-06 0.492841026E-07 -0.333576540E-09 -0.124722961E-05 0.136101372E-05 0.139714805E-06 0.202003764E-07 0.350169887E-10 -0.315100230E-03 -0.881937886E-07 -0.606977370E-07 0.406846193E-06 0.642008437E-11 -0.637152087E-05 0.661380532E-06 0.186976041E-05 0.518870410E-07 -0.122950423E-04 -0.511575077E-03 0.696953192E-06 -0.855140837E-06 -0.610545298E-09 0.341854410E-03 -0.136573378E-02 -0.209712082E-06 0.451940458E-07 0.293810866E-08 -0.457764105E-03 0.979098936E-07 0.210375225E-06 -0.249264634E-06 -0.272639783E-09 0.142784532E-03 -0.160408424E-04 0.547550107E-07 -0.372988663E-07 0.103365368E-09 0.188728903E-03 -0.222832300E-03 -0.150336108E-07 0.851780602E-08 0.251369947E-10 -0.164183993E-03 0.777226196E-08 -0.354937211E-07 0.370770571E-07 -0.955603436E-12 0.940017315E-04 -0.428889228E-07 time dv (days) 0.104244435E+03 0.389514320E+03 0.641372176E+03 0.147168228E+04 0.206339976E+04 0.255382808E+04 0.280936874E+04 0.324238017E+04 0.401259583E+04 0.443629130E+04 0.488236715E+04 0.513636103E+04 0.538045118E+04 0.606957455E+04 0.674529998E+04 0.724254533E+04 0.750518807E+04 0.770170968E+04 0.842401665E+04 0.910337931E+04 0.960693941E+04 0.987780925E+04 0.102541677E+05 0.107903703E+05 0.114035454E+05 0.119283224E+05 0.121677203E+05 0.123746814E+05 0.131276313E+05 0.137741288E+05 0.142711794E+05 0.145286348E+05 0.147533174E+05 0.154284252E+05 0.161370239E+05 dvx (km/s) -0.369162657E-05 0.127013677E-04 -0.212749675E-07 0.724277494E-04 0.274383071E-04 0.423171653E-05 -0.517900047E-05 0.385618904E-08 0.165297475E-04 0.405050580E-03 0.295825584E-06 0.212199438E-05 0.151112368E-07 -0.107041436E-05 -0.217729788E-04 -0.608356518E-06 0.232384701E-07 -0.238401310E-08 0.336909897E-04 -0.119884255E-04 -0.699884233E-05 0.826851568E-05 -0.623133952E-05 0.418905701E-02 0.709851665E-04 -0.384937631E-04 0.158519990E-03 -0.122927975E-05 0.125259517E-04 0.195969613E-04 0.263450500E-04 0.234862890E-04 0.319903471E-05 0.208019730E-04 0.238594434E-04 dvy (km/s) -0.659967564E-06 -0.573342659E-05 0.490031764E-07 0.333697497E-03 0.223542725E-04 -0.531730746E-05 -0.263798460E-05 0.981453493E-08 0.309558771E-04 0.407316487E-03 0.275964577E-05 0.119095629E-04 0.183265351E-07 0.393030423E-06 0.142884681E-05 -0.135969407E-05 -0.140067924E-05 -0.882818643E-08 0.797480145E-05 -0.199942957E-05 -0.135063679E-04 -0.304334748E-05 0.790355308E-06 -0.136256753E-01 -0.862459617E-04 -0.253376967E-04 -0.153570016E-03 0.827862755E-06 -0.500368933E-05 -0.130524224E-03 -0.178294159E-04 0.308769167E-04 -0.911490556E-06 -0.753589043E-04 -0.123692626E-03 dxz (km/s) 0.151314210E-07 0.535061532E-06 -0.586808582E-10 0.290739361E-04 0.276686157E-03 0.113255805E-05 0.298634454E-06 0.260080160E-09 -0.158162157E-05 -0.113431875E-03 -0.195815083E-06 0.718611182E-06 -0.417844978E-09 0.355753973E-04 -0.393589163E-03 -0.110252290E-06 0.576150483E-07 -0.290660335E-09 -0.163408820E-03 0.143420768E-03 0.152489546E-05 -0.432364604E-06 0.129899970E-05 -0.116846902E-01 0.746241765E-03 0.236778937E-04 -0.801965293E-05 0.411454920E-07 0.903216133E-04 0.222963579E-03 -0.500053847E-05 -0.385673281E-06 -0.129798489E-06 0.902820460E-05 0.167090512E-03 221 18 89.6 6.351 6.351 12.744 12.744 23 120.2 8.060 8.060 7.139 7.139 28 125.8 5.393 5.393 11.219 11.219 33 120.5 5.583 5.583 7.685 7.685 AVERAGE 115.6 6.421 6.421 9.677 9.677 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 9383.726209 LEG E/M time start vinfx vinfy vinfz (days) (km/s) (km/s) (km/s) 1 E -0.550696517E+02 0.981908015E+00 -0.550182275E+01 0.118184488E-02 2 E 0.436767285E+03 -0.556402390E+01 0.408827368E+00 0.449281304E-03 3 E 0.119796096E+04 -0.543802490E+01 -0.101325280E+01 0.529421414E+00 4 M 0.133678400E+04 -0.111397713E+02 -0.346667556E+01 0.885748342E+00 5 E 0.196575210E+04 -0.383999007E+01 0.685063372E+01 -0.142021050E+01 6 E 0.233099246E+04 -0.279803294E+01 0.745966313E+01 -0.335937491E-03 7 E 0.281372398E+04 0.708138057E+01 0.372728440E+01 -0.142361896E-02 8 E 0.362039655E+04 0.768254714E+01 -0.217044853E+01 -0.655003303E+00 9 M 0.369737952E+04 0.788478513E+01 0.111168588E+01 -0.959029235E+00 10 E 0.433600312E+04 0.269525023E+01 -0.537569164E+01 0.514146836E+00 11 E 0.470127020E+04 0.180084107E+01 -0.574561226E+01 0.110957237E-02 12 E 0.519147171E+04 -0.588226058E+01 -0.118330014E+01 0.354229049E+00 13 E 0.592195519E+04 -0.305638940E+01 0.505413313E+01 0.103202959E+01 14 M 0.605898369E+04 -0.856219259E+01 0.340933762E+01 0.501378846E+00 15 E 0.665600619E+04 0.287464164E+01 0.555517811E+01 0.905155583E+00 16 E 0.702125060E+04 0.371112978E+01 0.512522896E+01 0.318650331E-03 17 E 0.751064120E+04 0.489571550E+01 -0.384422076E+01 0.116808317E+01 18 E 0.824115749E+04 -0.187949743E+01 -0.602017295E+01 -0.746536672E+00 19 M 0.833077626E+04 -0.194641876E+01 -0.125944393E+02 0.103919650E+00 20 E 0.898573692E+04 -0.739573632E+01 -0.243641142E+01 -0.226066101E+01 21 E 0.935097907E+04 -0.801363365E+01 -0.133876021E+01 0.216175212E-03 22 E 0.983269156E+04 -0.212440116E+01 0.783934640E+01 -0.203052878E-02 23 E 0.106408104E+05 0.343040348E+01 0.722032437E+01 0.103283254E+01 24 M 0.107610388E+05 -0.242684438E+01 0.668999711E+01 -0.569247612E+00 25 E 0.113251061E+05 0.642935595E+00 -0.620164391E+00 0.532990325E+01 26 E 0.116903583E+05 0.494828680E+01 -0.216504074E+01 -0.177720706E-03 27 E 0.121832301E+05 -0.340991470E+01 -0.415115262E+01 0.396910542E+00 28 E 0.129137256E+05 -0.537274631E+01 -0.391460078E+00 0.254886064E+00 29 M 0.130395119E+05 -0.969528868E+01 -0.558900750E+01 0.789178609E+00 30 E 0.136463505E+05 -0.215811473E+01 0.486094451E+01 -0.152722991E+01 31 E 0.140116103E+05 -0.146096980E+01 0.535221527E+01 0.454372014E-03 32 E 0.145037756E+05 0.555992465E+01 -0.151858439E+00 0.140287655E-02 33 E 0.152614997E+05 0.496867774E+01 0.248284123E+01 -0.559447237E+00 34 M 0.153820172E+05 0.746063328E+01 0.163429062E+01 -0.854677097E+00 35 E 0.160128617E+05 0.112499167E+01 -0.417113948E+01 -0.296013173E+01 36 E 0.163781074E+05 ================ PARENT CYCLER 5.333gfGf3 ======================= Parent cycler number 129 Approximate search space (synodic periods after J2000) 15 Number of steps to walk eccentricity/inclination 81 / 81 Number of cycles 7 Total delta v over 44.91 years (km/s) 0.004897 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 3 131.2 6.765 6.765 8.801 8.801 8 162.7 5.419 5.419 8.968 8.968 13 98.3 5.677 5.677 12.214 12.214 18 174.0 6.874 6.874 7.067 7.067 23 127.9 5.323 5.323 11.466 11.466 28 131.5 6.408 6.408 7.929 7.929 33 155.0 5.273 5.273 9.385 9.385 AVERAGE 140.1 5.963 5.963 9.404 9.404 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 11719.652253 LEG E/M time start vinfx vinfy vinfz (days) (km/s) (km/s) (km/s) 1 E -0.291476088E+02 0.294579873E+00 0.674428933E+01 0.407052128E-03 2 E 0.458398685E+03 0.640148187E+01 -0.454147107E+00 0.213505317E+01 3 E 0.118891699E+04 0.665406695E+01 0.383930552E+00 -0.115693139E+01 4 M 0.132009492E+04 0.842243653E+01 -0.242347684E+01 -0.806383809E+00 5 E 0.196731272E+04 -0.448484259E+00 -0.458362603E+01 -0.293202532E+01 6 E 0.233256719E+04 -0.129117507E+01 -0.528634184E+01 -0.105448646E-03 7 E 0.282486857E+04 -0.460683799E+01 0.287959451E+01 -0.119297599E+00 8 E 0.355534327E+04 -0.407617809E+01 0.331017250E+01 0.133996569E+01 9 M 0.371803809E+04 -0.821769113E+01 0.356980172E+01 0.377845112E+00 10 E 0.430954414E+04 0.340742355E+01 0.451901768E+01 -0.202747617E+00 11 E 0.467481589E+04 0.407987364E+01 0.394246534E+01 -0.302904672E-03 12 E 0.516672259E+04 0.316475929E+01 -0.471844768E+01 0.882189860E-02 13 E 0.589720699E+04 -0.133221327E+01 -0.546987664E+01 -0.733384436E+00 14 M 0.599551519E+04 -0.163789245E+01 -0.121022235E+02 0.187579633E+00 15 E 0.664202239E+04 -0.650798319E+01 -0.138893370E+01 0.175574443E+01 16 E 0.700728426E+04 -0.687809369E+01 -0.363816730E+00 -0.107722503E-02 17 E 0.749382837E+04 -0.745313619E-01 0.663313811E+01 -0.185133923E+01 18 E 0.822430509E+04 -0.449790114E+00 0.660684084E+01 0.184505639E+01 19 M 0.839834661E+04 -0.988404541E+00 0.697414757E+01 -0.576120081E+00 20 E 0.898414842E+04 0.174835061E+00 -0.552063429E+00 0.529156287E+01 21 E 0.934939620E+04 0.467347984E+01 -0.254352395E+01 -0.100299475E-02 22 E 0.984259915E+04 -0.383214973E+01 -0.365001304E+01 -0.391012526E+00 23 E 0.105731385E+05 -0.525422800E+01 -0.807117978E+00 0.283126971E+00 24 M 0.107010687E+05 -0.100614199E+02 -0.543832822E+01 0.819543641E+00 25 E 0.113183813E+05 -0.325871692E+01 0.539654488E+01 0.908271802E+00 26 E 0.116836197E+05 -0.244471340E+01 0.589108852E+01 -0.102219337E-02 27 E 0.121725356E+05 0.598299064E+01 0.203373780E+01 -0.961144455E+00 28 E 0.129030548E+05 0.585001465E+01 0.249953405E+01 -0.766054650E+00 29 M 0.130346034E+05 0.784530440E+01 0.783574930E+00 -0.839486128E+00 30 E 0.136700170E+05 0.748369191E+00 -0.467224560E+01 -0.231533601E+01 31 E 0.140352867E+05 0.344362747E-01 -0.527235584E+01 0.556943678E-03 32 E 0.145283547E+05 -0.489329938E+01 0.195100142E+01 0.212770505E+00 33 E 0.152588396E+05 -0.419261553E+01 0.299012593E+01 0.113434167E+01 34 M 0.154138372E+05 -0.900995635E+01 0.257481543E+01 0.516927971E+00 35 E 0.160097545E+05 0.215589215E+01 0.427483517E+01 -0.348495247E+01 36 E 0.163749993E+05 ================ PARENT CYCLER 5.333gFgf3 ======================= Parent cycler number 130 time dv (days) 0.200685555E+03 0.550946336E+03 0.121878442E+04 0.143112922E+04 0.208993382E+04 0.257235822E+04 0.293472487E+04 0.363194399E+04 0.379317306E+04 0.439079318E+04 0.495617499E+04 0.530104423E+04 0.594250946E+04 0.617838819E+04 0.677288440E+04 0.728062762E+04 0.804391810E+04 0.825460031E+04 0.842902036E+04 0.911357168E+04 0.960628669E+04 0.101478579E+05 0.106588447E+05 0.108456489E+05 0.115369524E+05 0.119170794E+05 0.122928044E+05 0.129325935E+05 0.131305377E+05 0.137595811E+05 0.142626146E+05 0.146174343E+05 0.152795774E+05 0.156091213E+05 0.160676486E+05 dvx (km/s) -0.890703787E-05 -0.141641174E-04 -0.250526745E-04 0.146607159E-06 -0.318194533E-05 0.747843520E-07 -0.129110279E-05 -0.139129457E-05 0.904157033E-08 -0.245301843E-06 0.449769123E-06 0.114721173E-05 0.312934727E-06 0.977661625E-10 0.130093365E-05 -0.253244667E-07 -0.118504114E-04 -0.385384669E-07 0.629277392E-10 -0.104443503E-06 0.353102987E-09 -0.181589350E-08 0.434728036E-07 -0.128943997E-09 -0.134043035E-02 -0.956572996E-07 -0.740406376E-06 0.121153652E-06 -0.532187463E-09 0.130158547E-04 0.384993868E-07 0.174688501E-05 -0.390196671E-06 -0.108405679E-09 -0.214788075E-05 dvy (km/s) 0.347860228E-04 -0.149642694E-04 0.215344150E-04 0.251399584E-06 0.321018471E-05 -0.591798768E-06 -0.149322673E-05 -0.829165464E-06 0.132232773E-07 -0.132012721E-06 -0.946909160E-06 0.243767697E-06 -0.123257698E-05 -0.319254940E-09 -0.355496685E-06 -0.531267653E-07 -0.194981039E-04 -0.904969803E-07 -0.140198572E-10 0.269812080E-06 0.122833095E-09 -0.307972754E-08 0.155609552E-07 -0.222049274E-10 0.272213885E-03 0.210452595E-06 -0.129103850E-05 0.791480177E-06 0.751607859E-09 0.105265818E-04 -0.876163408E-06 0.580796866E-06 -0.477333131E-06 -0.404446302E-10 -0.817489938E-06 dxz (km/s) 0.130189968E-05 0.569371360E-06 -0.552750752E-07 -0.306933381E-08 0.205540130E-03 -0.101589256E-06 0.113910177E-07 0.128951523E-06 0.154639655E-08 -0.702397361E-08 0.128353108E-06 -0.105133751E-07 -0.784487346E-07 -0.264577373E-10 0.252080067E-04 -0.598997342E-08 0.638260319E-04 -0.239571130E-08 0.212991838E-11 -0.336168101E-05 0.755649492E-10 -0.324111494E-09 -0.234608761E-08 0.761661665E-11 0.152681977E-03 -0.842327176E-07 0.143362853E-07 0.504498990E-07 0.115281387E-10 -0.154435874E-03 -0.277086466E-08 -0.201885799E-07 -0.288523830E-07 0.388275704E-10 0.227958877E-06 time dv (days) 0.224376464E+03 0.567976432E+03 0.120859368E+04 0.144306631E+04 0.206958397E+04 0.258856391E+04 0.294174452E+04 0.357974749E+04 0.387774472E+04 0.442277838E+04 0.491585017E+04 0.527629525E+04 0.591195322E+04 0.609249127E+04 0.676255881E+04 0.723109455E+04 0.804168591E+04 0.825041132E+04 0.850379094E+04 0.913755249E+04 0.960586173E+04 0.995218005E+04 0.105923281E+05 0.107936656E+05 0.113841242E+05 0.119427451E+05 0.127277302E+05 0.129227871E+05 0.131299154E+05 0.137832506E+05 0.144001570E+05 0.150762183E+05 0.152820892E+05 0.155032248E+05 0.161156755E+05 dvx (km/s) 0.309509878E-05 -0.112569287E-03 0.462183053E-04 -0.100590292E-06 -0.678756371E-04 -0.397562643E-05 0.197365505E-04 -0.759711645E-05 0.289295979E-08 -0.628652224E-05 -0.143788318E-05 0.326063052E-05 -0.270410272E-05 0.235742588E-09 -0.335055817E-04 0.179361507E-06 0.365203328E-05 -0.152778932E-05 0.120369555E-09 -0.526303009E-03 -0.280777008E-05 -0.477096647E-06 -0.266475779E-05 0.774705613E-09 0.707901518E-08 0.846570215E-07 0.236965239E-05 0.160724794E-06 -0.428713824E-09 -0.703157089E-04 -0.213177452E-06 0.312533636E-06 0.119122918E-06 0.362347049E-09 0.272511059E-04 dvy (km/s) 0.427077799E-04 0.198540843E-04 0.290042741E-04 -0.474632556E-07 -0.166129951E-04 -0.123638532E-04 -0.648308759E-05 -0.188797173E-04 -0.562220165E-08 0.627880505E-06 -0.627150372E-06 -0.405355677E-05 0.210986513E-05 -0.343066955E-09 0.540743749E-04 -0.168689074E-06 -0.146617584E-04 -0.603539563E-06 0.480782147E-09 0.878183694E-03 0.120440314E-05 -0.677648427E-06 -0.414410187E-06 -0.120303255E-08 0.532914290E-08 -0.152622511E-06 0.297733178E-05 -0.367075488E-06 -0.371372864E-09 -0.130610551E-04 -0.592743947E-06 -0.272500195E-05 0.676115603E-07 0.716255233E-09 0.865315714E-05 dxz (km/s) 0.419085118E-05 0.895734350E-04 -0.219558872E-05 0.985917807E-08 -0.131750778E-03 0.142856939E-05 0.184333035E-03 -0.659512106E-06 -0.195699575E-09 -0.160138734E-03 0.309439204E-06 0.132525548E-08 -0.210723281E-06 0.902587725E-10 0.272788899E-03 -0.378775315E-07 -0.468857701E-03 0.911214682E-08 -0.263390244E-09 -0.234146311E-04 0.102478383E-07 0.119767751E-08 0.178755679E-07 -0.146333719E-10 0.797893228E-08 -0.327716067E-07 0.214887206E-03 0.103195483E-07 0.269716034E-10 -0.874671756E-03 -0.707392911E-08 -0.607995659E-05 -0.814987591E-08 -0.267818261E-10 0.160918306E-03 222 Approximate search space (synodic periods after J2000) 21 Number of steps to walk eccentricity/inclination 3 / 3 Number of cycles 7 Total delta v over 45.09 years (km/s) 0.498304 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 2 156.8 8.041 8.044 12.146 12.146 7 165.3 5.983 5.983 7.088 7.088 12 145.2 5.266 5.270 11.190 11.190 17 119.7 5.594 5.594 7.733 7.733 22 163.8 5.481 5.502 9.262 9.262 27 128.8 6.032 6.030 11.713 11.713 32 165.4 5.689 5.689 7.274 7.274 AVERAGE 149.3 6.012 6.016 9.487 9.487 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 17129.190330 LEG E/M time start vinfx vinfy vinfz (days) (km/s) (km/s) (km/s) 1 E -0.792246096E+02 0.678990554E+01 -0.436503404E+01 0.865019182E-03 2 E 0.403376809E+03 -0.249291153E+01 -0.756937593E+01 -0.109077869E+01 3 M 0.560207430E+03 -0.437398715E+01 -0.113201662E+02 0.499142238E+00 4 E 0.120592632E+04 0.262752316E+01 -0.533109960E+01 0.171975644E-02 5 E 0.197615351E+04 -0.360101702E+01 -0.151769201E+01 0.450798594E+01 6 E 0.234141330E+04 -0.572251422E+01 -0.174058839E+01 0.573764373E-03 7 E 0.283151829E+04 -0.612670579E+00 0.577467641E+01 0.144008437E+01 8 M 0.299679466E+04 -0.142349528E+01 0.691892913E+01 -0.584737293E+00 9 E 0.357527376E+04 0.281152002E+01 0.447929975E+01 0.923132326E+00 10 E 0.428514069E+04 -0.112120354E+00 -0.519203459E+00 0.529423562E+01 11 E 0.465039878E+04 0.353247298E+01 -0.392847713E+01 -0.250946335E-03 12 E 0.514363674E+04 -0.495438720E+01 -0.176085269E+01 0.353550645E+00 13 M 0.528882865E+04 -0.989878822E+01 -0.515785755E+01 0.789443534E+00 14 E 0.589821163E+04 -0.328456752E+01 -0.450807930E+01 -0.618270059E-03 15 E 0.665960378E+04 -0.443142145E+01 0.341596599E+01 0.192866017E+00 16 E 0.702486846E+04 -0.383532333E+01 0.405375111E+01 0.455099650E-03 17 E 0.751667314E+04 0.497853103E+01 0.248799876E+01 -0.559236585E+00 18 M 0.763639711E+04 0.753048980E+01 0.159537569E+01 -0.741360493E+00 19 E 0.824980951E+04 0.523229434E+01 -0.165319094E+01 0.185172918E-01 20 E 0.898099428E+04 -0.472510278E-01 -0.548253600E+01 0.266257945E+00 21 E 0.934623667E+04 -0.889393464E+00 -0.542347876E+01 0.102376105E-02 22 E 0.983838275E+04 -0.460889195E+01 0.270764616E+01 0.130158928E+01 23 M 0.100022036E+05 -0.885012445E+01 0.273049496E+01 -0.107124382E+00 24 E 0.105592433E+05 0.696186288E+00 0.601166273E+01 -0.449211536E-01 25 E 0.112898602E+05 0.551323732E+01 0.242801855E+01 -0.537779322E+00 26 E 0.116551204E+05 0.582447734E+01 0.156858427E+01 0.801959374E-03 27 E 0.121456098E+05 0.121329420E+01 -0.583466436E+01 -0.922308031E+00 28 M 0.122743599E+05 -0.289561081E+00 -0.117079317E+02 0.210470787E+00 29 E 0.129150829E+05 0.370927515E+01 -0.427139433E+01 0.497317702E-03 30 E 0.136777684E+05 -0.438924679E+01 -0.358218667E+01 -0.554328545E+00 31 E 0.140430126E+05 -0.490927881E+01 -0.288432168E+01 0.957617834E-03 32 E 0.145342196E+05 -0.125803868E+01 0.534290910E+01 0.149482873E+01 33 M 0.146996411E+05 -0.284399094E+01 0.668314956E+01 -0.405384372E+00 34 E 0.152781734E+05 -0.136459226E+01 0.507342129E+01 -0.106502677E-03 35 E 0.160250341E+05 0.513835632E+01 0.106374644E+01 -0.159536085E+00 36 E 0.163903137E+05 ================ PARENT CYCLER 5.333gGfff3 ======================= Parent cycler number 131 Approximate search space (synodic periods after J2000) 13 Number of steps to walk eccentricity/inclination 243 / 243 Number of cycles 7 Total delta v over 45.23 years (km/s) 0.122209 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 2 178.5 11.147 11.147 12.780 12.780 8 158.3 8.800 8.800 7.942 7.942 14 179.4 7.798 7.798 10.359 10.359 20 116.3 6.033 6.033 10.009 10.009 26 168.2 5.830 5.830 8.460 8.460 32 127.9 6.391 6.284 13.002 13.002 38 147.0 8.478 8.478 8.852 8.852 AVERAGE 153.6 7.782 7.767 10.201 10.201 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 10890.237955 LEG E/M time start vinfx vinfy vinfz (days) (km/s) (km/s) (km/s) 1 E -0.107058773E+03 -0.427806741E+00 -0.111673112E+02 -0.347678464E-03 2 E 0.362960848E+03 -0.105494795E+02 -0.293652625E+01 -0.208334227E+01 3 M 0.541414931E+03 -0.111233138E+02 -0.623118835E+01 0.876610455E+00 4 E 0.120880978E+04 -0.614513094E+01 0.566064532E+01 -0.266266790E+01 5 E 0.157408378E+04 -0.178953652E+01 0.275403726E+01 -0.813179249E+01 6 E 0.193933301E+04 -0.502048636E+01 0.491088933E+01 -0.526015114E+01 7 E 0.230459625E+04 -0.547386314E+01 0.683372511E+01 0.193546604E-03 8 E 0.278409618E+04 0.544430167E+01 0.691021904E+01 0.237879790E+00 9 M 0.294243618E+04 0.566241161E+01 0.550305850E+01 -0.855808585E+00 10 E 0.352997919E+04 -0.138619138E+01 0.579035463E+01 0.503871080E+01 11 E 0.389524868E+04 -0.941485420E+00 -0.146757126E+01 0.760493435E+01 12 E 0.426050517E+04 -0.643258418E+00 -0.704183159E+01 0.330698339E+01 13 E 0.462575129E+04 -0.183292966E+01 -0.760208233E+01 -0.321155643E-03 14 E 0.510882845E+04 -0.744214704E+01 0.873169783E+00 0.215778401E+01 15 M 0.528821559E+04 -0.103334491E+02 -0.182381277E+00 0.697379412E+00 16 E 0.590131557E+04 0.571720970E+00 0.599903105E+01 0.308416234E+00 17 E 0.626658763E+04 0.589189020E+00 0.572966480E+01 0.175148220E+01 18 E 0.663182884E+04 0.589990442E+00 0.559337207E+01 -0.220205438E+01 19 E 0.699707582E+04 0.148352415E+01 0.584126612E+01 -0.141119294E-02 20 E 0.748748936E+04 0.551957353E+01 -0.213097354E+01 -0.117768280E+01 21 M 0.760377163E+04 0.741167909E+01 -0.669404613E+01 -0.653475692E+00 22 E 0.825595644E+04 -0.238139730E+01 -0.531959869E+01 0.329751650E+00 23 E 0.862120086E+04 -0.204714050E+01 -0.436361299E+01 -0.326704274E+01 24 E 0.898646392E+04 -0.230659326E+01 -0.507182951E+01 0.168768014E+01 25 E 0.935171289E+04 -0.318966655E+01 -0.488957404E+01 0.131787810E-02 26 E 0.984245511E+04 -0.365805910E+01 0.424544999E+01 0.160855425E+01 27 M 0.100106218E+05 -0.657844438E+01 0.528946122E+01 -0.563014819E+00 28 E 0.105641017E+05 -0.734119399E+00 -0.749087696E+00 0.630939806E+01 time dv (days) 0.171728128E+03 0.426901402E+03 0.657065263E+03 0.132146040E+04 0.210764704E+04 0.259626789E+04 0.285630974E+04 0.335545170E+04 0.412896996E+04 0.453716877E+04 0.491181490E+04 0.516541552E+04 0.538023610E+04 0.601242045E+04 0.671439348E+04 0.730027908E+04 0.753463173E+04 0.777134784E+04 0.868120852E+04 0.903578064E+04 0.960707409E+04 0.986295588E+04 0.102194491E+05 0.110414504E+05 0.113446492E+05 0.119052700E+05 0.121649223E+05 0.123704683E+05 0.130294857E+05 0.137325550E+05 0.143082644E+05 0.145590328E+05 0.147864210E+05 0.153902025E+05 0.160871317E+05 dvx (km/s) -0.551454804E-03 -0.532816907E-03 -0.119621866E-05 0.153287309E-03 -0.822627265E-04 0.196103929E-04 0.402971726E-04 -0.823808586E-05 -0.804772639E-02 -0.244434323E-03 -0.402188376E-03 0.651751807E-03 0.629272769E-06 0.238840592E-04 0.136434862E-04 -0.137491673E-03 -0.822382344E-04 0.165148354E-05 0.363205002E-02 -0.989106459E-04 -0.688481344E-05 -0.287703292E-03 0.700925816E-06 -0.830371552E-03 0.390661661E-03 0.586318804E-03 0.958125360E-03 0.897553102E-06 -0.167988163E-03 -0.284866864E-04 0.958889655E-05 -0.964007479E-04 -0.143984643E-06 -0.383417263E-05 0.213469868E-05 dvy (km/s) 0.253084202E-03 0.721814053E-03 0.313872649E-05 -0.101537017E-03 0.117564911E-03 0.374054411E-05 0.236623962E-04 0.403958910E-05 -0.609950131E-02 0.202260488E-03 0.337606929E-03 -0.120193638E-02 -0.385327016E-06 -0.170143708E-03 0.882957856E-04 0.389075958E-04 -0.163608311E-03 0.370088460E-06 0.990374006E-03 -0.444805611E-05 0.129644704E-03 -0.681923112E-03 0.247875525E-05 0.499512852E-03 -0.693741324E-03 0.602939892E-04 0.437699219E-03 0.335715583E-06 0.670790365E-04 0.315816227E-04 0.194652540E-04 -0.288412399E-04 -0.269780047E-07 0.487314797E-05 -0.686321174E-05 dxz (km/s) 0.914232698E-04 -0.256701945E-04 -0.450751667E-07 -0.106686459E-05 0.931183866E-04 0.262108261E-05 -0.700329841E-06 0.422315110E-05 0.197111143E+00 0.107773541E-03 0.570368579E-04 -0.718203896E-04 -0.642170147E-07 -0.213746657E-06 -0.818826397E-05 0.287046105E-04 -0.628093113E-05 0.945621877E-06 -0.129197359E-04 -0.823184647E-05 -0.480523676E-04 0.258876506E-04 0.238382852E-05 0.251335147E-04 0.126617656E-03 -0.217785203E-04 0.242234089E-04 0.178796667E-07 -0.429772022E-05 -0.115672884E-04 0.131158131E-04 -0.507247981E-05 -0.631039040E-08 0.572851004E-07 0.632331737E-05 time dv (days) 0.109150253E+03 0.389728961E+03 0.641524159E+03 0.133665568E+04 0.170922600E+04 0.205986988E+04 0.255393621E+04 0.280943058E+04 0.306582021E+04 0.377836244E+04 0.415458078E+04 0.438468885E+04 0.486728987E+04 0.513573652E+04 0.538018058E+04 0.601820263E+04 0.637981240E+04 0.674505540E+04 0.724228259E+04 0.750493170E+04 0.773420859E+04 0.837283465E+04 0.873443241E+04 0.910334359E+04 0.960689884E+04 0.987777011E+04 0.102596878E+05 0.108124801E+05 dvx (km/s) 0.176733366E-07 -0.633908781E-07 0.101482966E-09 0.194138425E-04 -0.219035144E-03 0.704756424E-06 -0.195994842E-07 0.239511242E-07 -0.320227129E-10 0.298054223E-04 0.135564640E-03 0.363155551E-04 -0.235094356E-05 0.714953691E-08 -0.444718567E-10 -0.239283117E-07 -0.766717390E-07 -0.264056482E-04 -0.249387624E-08 -0.268806932E-09 -0.118051513E-11 0.526212915E-07 -0.816568972E-04 0.159977800E-05 -0.121104351E-07 0.149156797E-07 -0.371068642E-10 0.235488042E-03 dvy (km/s) 0.790078764E-08 0.211283786E-07 -0.198219072E-09 0.233113084E-04 -0.760039178E-03 0.914702224E-06 0.238877666E-07 0.133236922E-07 -0.992244459E-10 0.278838481E-04 -0.425706250E-03 -0.163008340E-05 0.103929432E-05 0.258890010E-07 -0.481597264E-10 -0.784510171E-09 0.149085933E-07 0.222987562E-05 -0.512784027E-08 -0.104971571E-07 0.482126347E-11 0.447131513E-08 0.340376782E-04 -0.695610504E-06 -0.231371905E-07 -0.584260746E-08 0.227009422E-11 0.811326334E-04 dxz (km/s) 0.336448240E-09 -0.212272699E-08 0.515170746E-12 -0.300811402E-03 -0.201040103E-04 -0.561459018E-05 -0.530764941E-08 -0.142547352E-08 -0.109700898E-10 -0.452308738E-03 0.423810460E-04 -0.290681806E-03 0.821894818E-09 0.154904335E-08 0.179446256E-11 -0.171222494E-05 -0.156664231E-05 0.349162394E-03 -0.402232293E-09 0.396745053E-09 0.357543973E-12 0.568740078E-06 -0.768682896E-03 -0.293232883E-04 0.259728708E-08 -0.819457718E-09 0.116428098E-10 0.208545167E-04 223 29 E 0.109293641E+05 -0.741656784E+00 -0.755085847E+00 0.631244912E+01 30 E 0.112946166E+05 0.350050908E+01 -0.388073271E+00 0.534160538E+01 31 E 0.116598434E+05 0.630102413E+01 -0.110330782E+01 0.120210459E-02 32 E 0.121489360E+05 -0.228268270E+01 -0.584477809E+01 -0.348927222E+00 33 M 0.122768119E+05 -0.697793955E+01 -0.109574867E+02 0.545706772E+00 34 E 0.129303698E+05 -0.834846924E+01 0.129623651E+01 0.110346043E+00 35 E 0.132956341E+05 -0.831311753E+01 0.131591555E+01 0.618156602E+00 36 E 0.136608751E+05 -0.811256730E+01 0.130893423E+01 -0.200046454E+01 37 E 0.140261281E+05 -0.802847956E+01 0.260488275E+01 0.632347612E-03 38 E 0.145066339E+05 0.218285664E+01 0.816415758E+01 0.672614419E+00 39 M 0.146535988E+05 0.435802417E+01 0.769193825E+01 -0.455564269E+00 40 E 0.153188251E+05 0.933939722E+01 0.160250975E+00 -0.372751219E+01 41 E 0.156840859E+05 0.456370181E+01 -0.806522906E+00 -0.890468386E+01 42 E 0.160493347E+05 0.455397273E+01 -0.823122484E+00 -0.891892566E+01 43 E 0.164145929E+05 ================ PARENT CYCLER 5.333gfGff3 ======================= Parent cycler number 132 Approximate search space (synodic periods after J2000) 14 Number of steps to walk eccentricity/inclination 3 / 3 Number of cycles 7 Total delta v over 45.05 years (km/s) 0.454025 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 3 149.9 9.098 9.098 12.962 12.962 9 188.4 8.169 8.168 7.489 7.489 15 131.2 5.598 5.362 12.000 12.001 21 146.7 6.882 6.882 7.241 7.242 27 142.9 5.288 5.290 9.878 9.879 33 103.7 5.510 5.511 11.108 11.109 39 175.2 6.246 6.246 7.703 7.705 AVERAGE 148.3 6.684 6.651 9.769 9.770 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 11304.945104 LEG E/M time start vinfx vinfy vinfz (days) (km/s) (km/s) (km/s) 1 E -0.833363365E+02 0.864724432E+01 -0.287675994E+01 -0.675048562E-03 2 E 0.395271183E+03 0.466476650E+01 -0.766306743E+01 0.156003176E+01 3 E 0.760541801E+03 -0.413794316E+00 -0.895487956E+01 -0.155086181E+01 4 M 0.910474744E+03 -0.226507750E+01 -0.127619655E+02 0.137267777E+00 5 E 0.158361400E+04 -0.761120994E+01 -0.223889870E+01 -0.191964055E+01 6 E 0.194888079E+04 -0.761599905E+01 -0.224679876E+01 0.180927050E+01 7 E 0.231415363E+04 -0.808786436E+01 -0.104453070E+01 0.124520171E-04 8 E 0.279569148E+04 -0.668894167E+01 0.457325199E+01 -0.871587004E+00 9 E 0.316095103E+04 -0.180343508E+01 0.702944256E+01 0.374938799E+01 10 M 0.334934537E+04 -0.291828620E+01 0.687171544E+01 -0.596418089E+00 11 E 0.392430222E+04 -0.155252156E+01 -0.267712504E+00 0.551359867E+01 12 E 0.428955915E+04 0.368563314E+01 -0.124718969E+01 0.420881029E+01 13 E 0.465481036E+04 0.521083289E+01 -0.240750355E+01 0.311264025E-03 14 E 0.514639758E+04 0.286144349E+01 -0.481973996E+01 0.359337723E+00 15 E 0.551165764E+04 -0.441050473E+01 -0.304847741E+01 0.627334543E-01 16 M 0.564287170E+04 -0.923476369E+01 -0.762391569E+01 0.782990363E+00 17 E 0.627243384E+04 -0.516069031E+01 0.450061114E+01 -0.360524985E+00 18 E 0.663767917E+04 -0.253437907E+01 0.267262771E+01 0.578207948E+01 19 E 0.700293943E+04 -0.440715559E+01 0.524383220E+01 0.664729567E-03 20 E 0.748981952E+04 -0.214464477E+00 0.683664639E+01 -0.581081288E+00 21 E 0.785508301E+04 0.486224489E+01 0.486643451E+01 -0.193970705E+00 22 M 0.800177375E+04 0.596968581E+01 0.401512625E+01 -0.829496710E+00 23 E 0.863163069E+04 0.297352317E+01 -0.434283768E+01 -0.344402218E+00 24 E 0.899687557E+04 0.295059925E+01 -0.429041211E+01 0.959033915E+00 25 E 0.936213032E+04 0.227735856E+01 -0.477114918E+01 0.409265077E-03 26 E 0.985520964E+04 0.387150585E+00 -0.312792507E+00 0.526383116E+01 27 E 0.102204654E+05 -0.447520827E+01 0.269213474E+01 0.843617150E+00 28 M 0.103633202E+05 -0.985473021E+01 0.336087157E+00 0.604353827E+00 29 E 0.109552126E+05 0.142062678E+01 0.515825185E+01 0.127684309E+01 30 E 0.113204897E+05 0.145852975E+01 0.526442574E+01 0.723745287E+00 31 E 0.116857317E+05 0.224141119E+01 0.504226058E+01 0.473757242E-04 32 E 0.121781657E+05 0.447406592E+01 0.282371067E+01 -0.156871498E+01 33 E 0.125434434E+05 0.327327518E+01 -0.430077054E+01 -0.107786900E+01 34 M 0.126471907E+05 0.459749470E+01 -0.101068281E+02 -0.362864183E+00 35 E 0.132989006E+05 -0.442815001E+01 -0.439732563E+01 -0.450832903E-01 36 E 0.136641470E+05 -0.441025682E+01 -0.440663807E+01 -0.394242665E+00 37 E 0.140294048E+05 -0.505129252E+01 -0.364438214E+01 0.515029513E-03 38 E 0.145184528E+05 -0.619685978E+01 0.280413946E+00 -0.550000834E+00 39 E 0.148837189E+05 -0.277815222E+01 0.525865987E+01 0.190761270E+01 40 M 0.150588690E+05 -0.466972280E+01 0.612511439E+01 -0.196927813E+00 41 E 0.156393601E+05 0.271364029E+01 0.302756287E+00 0.448366518E+01 42 E 0.160046209E+05 0.513787709E+01 0.100835689E+01 -0.250164349E+00 43 E 0.163698935E+05 ================ PARENT CYCLER 5.658Gfh-f3 ======================= Parent cycler number 145 Approximate search space (synodic periods after J2000) 10 Number of steps to walk eccentricity/inclination 81 / 81 Number of cycles 7 Total delta v over 44.79 years (km/s) 0.008902 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 1 132.1 ****** 5.576 7.558 7.558 6 108.0 7.708 7.708 11.482 11.482 11 81.1 6.884 6.884 8.228 8.228 16 131.6 6.718 6.718 9.119 9.119 21 89.3 6.406 6.406 11.822 11.822 26 128.9 6.086 6.086 7.701 7.701 31 107.7 7.653 7.653 11.073 11.073 AVERAGE 111.3 6.909 6.719 9.569 9.569 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 8294.078101 LEG E/M time start vinfx vinfy vinfz (days) (km/s) (km/s) (km/s) 1 E 0.458601114E+01 0.143797694E+01 0.530575296E+01 0.933512191E+00 2 M 0.136734376E+03 -0.209180941E+00 0.744630533E+01 -0.127742264E+01 3 E 0.686544409E+03 -0.100932128E+01 0.508858744E+01 -0.542014450E+01 4 E 0.178231748E+04 -0.730422470E+00 -0.638710042E+00 -0.743466449E+01 0.111777358E+05 0.113968801E+05 0.119924264E+05 0.121681174E+05 0.125055572E+05 0.130618650E+05 0.134271209E+05 0.137887137E+05 0.142711861E+05 0.145286787E+05 0.147533828E+05 0.154503190E+05 0.159580225E+05 0.163232783E+05 0.142485660E-03 -0.244847576E-05 -0.165740191E-08 0.385569237E-06 -0.453479674E-06 -0.221958007E-07 -0.784094081E-07 -0.139838993E-05 -0.119935906E-07 -0.276932216E-07 0.655268565E-09 0.101203627E-05 0.591198036E-03 -0.982933076E-03 0.467351835E-04 0.446623385E-03 -0.434498844E-07 -0.312648605E-06 0.160901310E-05 0.562405431E-06 -0.801682171E-06 -0.120687337E-04 0.754501609E-08 -0.379089366E-07 -0.161574493E-08 -0.171027434E-04 0.209944294E-03 -0.347481284E-03 0.121720874E-04 -0.250730231E-02 -0.115566209E-07 -0.148184325E-07 0.155948328E-07 0.826275826E-04 -0.327293741E-04 0.160392311E-03 0.210055443E-08 0.777018178E-09 -0.286090076E-10 -0.120628785E-03 -0.996791551E-04 0.165593056E-03 time dv (days) 0.165539574E+03 0.450061776E+03 0.783031742E+03 0.101144563E+04 0.163840402E+04 0.204385173E+04 0.256455331E+04 0.285048041E+04 0.319862990E+04 0.343558890E+04 0.412519353E+04 0.435530437E+04 0.491043572E+04 0.520483919E+04 0.553133975E+04 0.573730602E+04 0.632722064E+04 0.675456245E+04 0.733888669E+04 0.754460904E+04 0.787708662E+04 0.809625229E+04 0.868641742E+04 0.905166378E+04 0.961360077E+04 0.990999801E+04 0.102418937E+05 0.104521041E+05 0.110100042E+05 0.113752760E+05 0.119417974E+05 0.122366101E+05 0.125590055E+05 0.127449472E+05 0.133536875E+05 0.137189357E+05 0.142934907E+05 0.145732427E+05 0.149099914E+05 0.151459427E+05 0.156941493E+05 0.160922863E+05 dvx (km/s) -0.401187041E-03 0.986256618E-03 -0.104089391E-02 -0.659497105E-03 0.163671054E-03 0.116632529E-03 0.167737583E-03 0.198433946E-04 -0.183394542E-04 0.456598668E-03 0.263629504E-03 -0.112655906E-03 0.313302009E-03 0.403243287E-04 -0.931917183E-03 -0.195829718E-03 0.119450545E-03 0.301908657E-03 -0.435369301E-04 -0.254744025E-03 0.145740555E-03 0.144639738E-03 -0.518489116E-03 0.262320220E-03 -0.734201290E-04 -0.484628525E-03 0.425752038E-03 0.599784684E-03 -0.513845540E-03 0.273104218E-03 0.198451424E-03 -0.616104333E-03 -0.382517143E-02 -0.167965510E-02 -0.227626880E-03 0.445123199E-03 -0.250514594E-03 0.128171078E-03 0.121208094E-02 0.129586100E-02 0.107248542E-03 -0.229125495E-04 dvy (km/s) 0.152294521E-03 0.671099161E-03 0.149888597E-03 0.332131934E-03 -0.516758210E-03 -0.121963989E-03 0.244516041E-04 0.622365382E-04 -0.128140968E-03 -0.159375889E-03 -0.581793861E-03 -0.556625863E-03 -0.154518254E-03 0.315727940E-03 0.132420085E-02 0.893380876E-03 0.217209286E-03 -0.140569283E-03 -0.657060040E-04 -0.969579765E-04 -0.188002783E-02 -0.947500036E-03 -0.295837258E-03 0.562772416E-03 0.140409581E-03 -0.944570653E-04 0.617649963E-03 0.699311198E-03 -0.244452235E-04 -0.141918284E-04 0.463825400E-03 0.494233791E-03 -0.165857783E-02 -0.284641867E-03 0.141492612E-03 -0.245494477E-03 -0.297356063E-03 -0.418286238E-03 0.366343030E-03 -0.313552607E-03 0.151938274E-02 0.297975818E-03 dxz (km/s) -0.123609231E-04 0.172194945E-03 0.135491800E-04 -0.550821548E-04 0.493145293E-05 0.294303306E-03 -0.145304123E-04 -0.600137312E-04 -0.130741472E-03 0.171625748E-03 -0.122323487E-03 -0.693337831E-03 -0.133323363E-04 0.268468679E-03 -0.104274784E-03 -0.285828221E-04 -0.496015600E-04 0.121539088E-03 0.180666903E-04 0.285754291E-04 0.109519310E-03 -0.392069602E-04 0.224964062E-03 -0.358771691E-03 -0.636063581E-04 0.884007285E-04 -0.172847385E-03 0.938520125E-04 0.385883239E-03 -0.353531563E-06 -0.117908586E-04 -0.918025684E-03 -0.166389346E-04 -0.110483335E-03 -0.167543451E-03 0.164806840E-03 -0.789991353E-04 -0.235567801E-03 -0.141716455E-03 0.184693601E-03 0.151605542E-03 0.953190270E-04 time dv dvx dvy dxz (days) (km/s) (km/s) (km/s) 0.244082658E+02 -0.121132007E-05 0.179981692E-05 0.181444534E-06 0.252194482E+03 0.809329096E-09 -0.204541772E-08 -0.207934141E-09 0.109198045E+04 0.787175846E-04 0.626355701E-04 0.468909677E-03 0.180943871E+04 -0.405770645E-06 0.125501664E-06 0.587861565E-07 224 5 E 0.196312566E+04 -0.308504952E+01 0.513790508E+01 0.486760423E+01 0.208000557E+04 -0.640483388E-04 -0.453281338E-04 -0.462951100E-03 6 E 0.232837537E+04 -0.742436798E+01 0.203553703E+01 0.383035189E+00 0.234457573E+04 -0.270486570E-06 0.194383892E-06 -0.135405214E-07 7 M 0.243637776E+04 -0.105921494E+02 -0.435003746E+01 0.849472854E+00 0.252915319E+04 -0.267542095E-09 -0.430884715E-09 0.143235241E-10 8 E 0.305488063E+04 -0.508990917E+01 0.245422896E+01 -0.430256671E+01 0.375617442E+04 -0.152419796E-04 -0.108461890E-04 0.130882494E-03 9 E 0.415065217E+04 0.660809671E+00 0.511201142E+00 -0.705925632E+01 0.417828080E+04 -0.108364093E-05 -0.328503350E-06 -0.255522509E-06 10 E 0.433484302E+04 0.286661847E+01 -0.508446299E+01 0.366423936E+01 0.445173244E+04 -0.107733461E-03 -0.705636034E-04 0.931857056E-03 11 E 0.470012246E+04 0.670945348E+01 -0.137460246E+01 -0.698303228E+00 0.471228795E+04 -0.279165175E-06 -0.932600771E-06 0.867798966E-07 12 M 0.478122573E+04 0.819038441E+01 0.557201354E+00 -0.555440882E+00 0.487010548E+04 -0.202962957E-08 -0.112282993E-07 0.490519640E-09 13 E 0.537375735E+04 0.289320309E+01 -0.348403889E+00 -0.584031673E+01 0.612983523E+04 -0.794524713E-03 -0.334734269E-03 -0.325092967E-04 14 E 0.646952240E+04 -0.689230318E+00 0.229130442E+00 -0.648229440E+01 0.649721120E+04 -0.436544777E-05 0.192374675E-05 0.107299686E-05 15 E 0.665411445E+04 0.255863259E+01 0.579286771E+01 0.223082553E+01 0.677099205E+04 -0.198310854E-04 0.660139665E-05 -0.229271434E-03 16 E 0.701935695E+04 -0.295072516E+01 0.595216676E+01 0.998504614E+00 0.703909585E+04 -0.335059858E-05 -0.336366745E-05 -0.166536030E-07 17 M 0.715094961E+04 -0.850106926E+01 0.328356781E+01 -0.331994192E+00 0.733669846E+04 -0.885831106E-09 0.103104108E-08 0.142688463E-09 18 E 0.769726974E+04 -0.586210688E+01 0.810886334E+00 -0.250467513E+01 0.809174506E+04 0.639931559E-05 -0.226597230E-04 -0.226212909E-03 19 E 0.879303450E+04 0.234802486E+00 -0.641456372E+00 -0.639874440E+01 0.881985088E+04 -0.436035432E-07 -0.198842000E-07 0.985939285E-09 20 E 0.897181036E+04 -0.554201550E+01 -0.117394078E+01 0.300085236E+01 0.908870198E+04 -0.405737530E-04 0.142508788E-03 0.131057146E-02 21 E 0.933709667E+04 -0.252475473E+01 -0.583424042E+01 -0.792316091E+00 0.935048806E+04 0.503193995E-07 -0.152177924E-07 0.412276732E-08 22 M 0.942637260E+04 -0.900320647E+00 -0.117858080E+02 0.207734516E+00 0.952201343E+04 -0.122515646E-10 0.130526284E-10 -0.558894964E-12 23 E 0.100639781E+05 -0.233202588E+01 -0.367126203E+01 -0.422502401E+01 0.108638895E+05 -0.224148656E-03 0.169475637E-03 0.615021222E-04 24 E 0.111597471E+05 -0.164119664E+00 0.583146232E+00 -0.602863388E+01 0.111877221E+05 -0.125163012E-06 -0.190455908E-06 -0.306293413E-07 25 E 0.113462473E+05 0.525329930E+01 0.964524915E+00 0.289281826E+01 0.114594767E+05 0.287796427E-05 -0.137761060E-04 0.141256814E-03 26 E 0.117115035E+05 0.151022380E+01 0.580520282E+01 0.103078519E+01 0.117308456E+05 0.658257616E-07 -0.128169665E-06 0.432120371E-08 27 M 0.118404505E+05 -0.177752571E+01 0.737706253E+01 -0.131134945E+01 0.119385219E+05 0.821565780E-10 0.162262962E-10 0.241201644E-10 28 E 0.123852919E+05 -0.197662273E+01 0.489274611E+01 -0.528658207E+01 0.127797675E+05 0.265103165E-04 0.282654509E-04 0.178356213E-03 29 E 0.134810574E+05 -0.591297141E+00 -0.748103391E+00 -0.741006044E+01 0.135080942E+05 -0.161829241E-06 -0.534344499E-07 0.373759381E-07 30 E 0.136613031E+05 -0.364999065E+01 0.426666126E+01 0.522438029E+01 0.137781878E+05 0.582741460E-04 0.619177767E-04 0.450954099E-03 31 E 0.140265677E+05 -0.755848326E+01 0.118184390E+01 0.200605910E+00 0.140427229E+05 -0.145157896E-06 -0.483797255E-07 -0.827513411E-09 32 M 0.141342691E+05 -0.931462886E+01 -0.594065864E+01 0.744103416E+00 0.142810572E+05 -0.143653480E-08 -0.233581121E-08 -0.323236634E-09 33 E 0.147214216E+05 -0.199578892E+01 -0.440273887E+01 -0.121895898E+01 0.151816489E+05 0.275077347E-04 0.143434904E-05 -0.354910119E-04 34 E 0.158172009E+05 0.389990589E+00 0.113032182E+00 -0.497537323E+01 0.158446123E+05 0.573659473E-05 0.268235843E-05 0.103732446E-05 35 E 0.159999435E+05 0.106145335E+00 -0.189869389E+01 0.443815882E+01 0.162665796E+05 0.507325095E-03 -0.127046212E-02 0.135736548E-03 36 E 0.163651985E+05 ================ PARENT CYCLER 5.658Gfh+f3 ======================= Parent cycler number 146 Approximate search space (synodic periods after J2000) 13 Number of steps to walk eccentricity/inclination 9 / 9 Number of cycles 7 Total delta v over 45.17 years (km/s) 0.007219 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 1 164.7 ****** 6.408 10.873 10.873 6 95.0 5.483 5.483 8.382 8.382 11 133.1 6.524 6.524 9.116 9.116 16 89.3 6.402 6.402 11.823 11.823 21 128.9 6.090 6.090 7.709 7.709 26 106.7 7.678 7.678 11.274 11.274 31 100.9 5.427 5.427 7.472 7.472 AVERAGE 116.9 6.268 6.288 9.521 9.521 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 10633.970176 LEG E/M time start vinfx vinfy vinfz time dv dvx dvy dxz (days) (km/s) (km/s) (km/s) (days) (km/s) (km/s) (km/s) 1 E -0.814350096E+02 -0.619590199E+01 -0.140773750E+01 0.829886849E+00 -0.567344575E+02 -0.471761027E-07 0.864622333E-07 -0.414027137E-09 2 M 0.832353376E+02 -0.103689986E+02 -0.318217590E+01 0.763709842E+00 0.174698649E+03 0.235193665E-10 0.574484295E-10 -0.611123768E-11 3 E 0.692990744E+03 -0.212766569E+01 -0.357838611E+01 0.381379375E+01 0.154769319E+04 -0.132394397E-04 0.257759349E-03 0.218713355E-04 4 E 0.178876311E+04 0.497609240E+00 0.129410201E+00 0.562858277E+01 0.181617094E+04 0.100907765E-06 -0.128750518E-06 -0.123277764E-07 5 E 0.197148198E+04 0.812246572E+00 -0.516856227E+01 -0.158928496E+01 0.208471642E+04 -0.351100640E-04 -0.609354130E-05 -0.630145548E-03 6 E 0.233675438E+04 0.542936699E+01 0.151829838E+00 -0.748258768E+00 0.235101000E+04 0.120415334E-06 -0.142204004E-06 0.528300423E-08 7 M 0.243179185E+04 0.836198320E+01 -0.216779654E+00 -0.538361153E+00 0.252182357E+04 0.120080213E-09 0.591752222E-09 0.183698033E-10 8 E 0.303200333E+04 0.445128667E+01 0.380130491E+01 0.240255393E+01 0.342647232E+04 -0.684833423E-04 0.302060884E-04 0.767944946E-03 9 E 0.412775051E+04 -0.636611293E+00 0.235965063E+00 0.629409125E+01 0.415545489E+04 0.177230141E-07 -0.184539266E-07 0.496587244E-08 10 E 0.431244633E+04 0.258207549E+01 0.538816073E+01 -0.259714172E+01 0.442567182E+04 -0.334066298E-04 0.149622239E-04 0.405211249E-03 11 E 0.467768985E+04 -0.294676197E+01 0.573383301E+01 0.100423268E+01 0.469765323E+04 0.175614026E-07 0.344285664E-08 -0.645006433E-09 12 M 0.481077902E+04 -0.849099459E+01 0.329931120E+01 -0.331859357E+00 0.495288022E+04 0.172199791E-10 0.720213431E-11 -0.726919964E-11 13 E 0.535732210E+04 -0.598011515E+01 0.772950308E+00 0.221112287E+01 0.575179746E+04 -0.337156207E-05 0.119476371E-04 -0.134480407E-03 14 E 0.645308698E+04 0.234134905E+00 -0.639379164E+00 0.639521620E+01 0.654068693E+04 -0.326969623E-08 0.227146322E-08 -0.357575663E-07 15 E 0.663186238E+04 -0.489430103E+01 -0.949808442E+00 -0.402308656E+01 0.674510235E+04 -0.651863745E-04 0.245657604E-03 -0.179076120E-02 16 E 0.699715261E+04 -0.251971615E+01 -0.583188796E+01 -0.792046865E+00 0.701054883E+04 0.813639299E-08 -0.438599861E-07 0.394334627E-09 17 M 0.708646072E+04 -0.903002989E+00 -0.117870748E+02 0.207596742E+00 0.733514209E+04 0.143476178E-06 -0.113596970E-06 -0.768819865E-08 18 E 0.772410526E+04 0.825836190E+00 -0.280670908E+01 0.531468306E+01 0.861167661E+04 -0.294291123E-03 -0.138717357E-04 0.428211491E-04 19 E 0.881987236E+04 -0.164469414E+00 0.585638405E+00 0.603502590E+01 0.884784761E+04 0.109986015E-07 0.172502029E-07 -0.269111320E-08 20 E 0.900637402E+04 0.486532354E+01 0.844527879E+00 -0.354239613E+01 0.911595090E+04 0.272925018E-05 -0.161366986E-04 -0.864038348E-04 21 E 0.937163029E+04 0.150912839E+01 0.580981531E+01 0.103046500E+01 0.939095802E+04 0.180879619E-07 -0.348835871E-07 0.117193056E-08 22 M 0.950048180E+04 -0.177152571E+01 0.738533739E+01 -0.132202032E+01 0.985451583E+04 0.987844808E-09 0.481559567E-08 -0.109624511E-06 23 E 0.100451495E+05 -0.298390248E+01 0.571429484E+01 0.382306132E+01 0.109765502E+05 0.105875534E-04 -0.140802898E-04 0.160871204E-03 24 E 0.111409150E+05 -0.598826407E+00 -0.748774445E+00 0.743243667E+01 0.111679540E+05 0.302974696E-06 0.881288405E-07 0.679143905E-07 25 E 0.113211750E+05 -0.410193991E+01 0.467973671E+01 -0.452411627E+01 0.114380592E+05 0.338317487E-04 0.341467024E-04 -0.299154931E-03 26 E 0.116864381E+05 -0.759636158E+01 0.109594770E+01 0.213415461E+00 0.117024406E+05 0.247075186E-06 0.700970770E-07 0.436859316E-09 27 M 0.117931212E+05 -0.956629168E+01 -0.591337720E+01 0.792125582E+00 0.118841172E+05 0.227009422E-10 -0.643827158E-10 -0.167468921E-11 28 E 0.123997609E+05 -0.185690522E+01 -0.254404622E+01 0.465069398E+01 0.133311572E+05 -0.684540345E-04 0.176391275E-03 0.424815042E-04 29 E 0.134955212E+05 0.448678390E+00 0.287848404E+00 0.558917685E+01 0.135691074E+05 0.117816402E-08 -0.108288950E-08 -0.312512033E-10 30 E 0.136794866E+05 0.254967537E+01 -0.474421538E+01 -0.766474476E+00 0.137927145E+05 0.269978955E-05 0.147795680E-05 0.113331139E-03 31 E 0.140447377E+05 0.525344358E+01 0.130665302E+01 -0.384997133E+00 0.140598667E+05 0.188099709E-08 -0.171514298E-08 0.138975689E-09 32 M 0.141455973E+05 0.696524098E+01 0.255831841E+01 -0.873967971E+00 0.142964343E+05 -0.200786775E-10 0.307777158E-11 0.143301764E-12 33 E 0.147740847E+05 0.468265530E+01 -0.158917557E+01 0.209094419E+01 0.154644113E+05 0.217850337E-04 0.121522278E-04 -0.385101694E-03 34 E 0.158698413E+05 -0.404962591E+00 -0.266324529E+00 0.534960293E+01 0.158988593E+05 -0.202006095E-07 -0.380807991E-07 -0.689635775E-08 35 E 0.160512038E+05 -0.247758231E+01 0.491119359E+01 -0.675178498E+00 0.161644395E+05 0.828679459E-08 0.988971093E-08 -0.720915710E-07 36 E 0.164164805E+05 ================ PARENT CYCLER 5.658Gfh-3 ======================= Parent cycler number 147 Approximate search space (synodic periods after J2000) 17 Number of steps to walk eccentricity/inclination 81 / 81 Number of cycles 7 Total delta v over 44.58 years (km/s) 0.012485 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 1 92.0 ****** 11.902 7.454 7.454 5 115.4 7.894 7.894 10.383 10.383 9 93.0 5.402 5.402 9.506 9.506 13 140.9 5.788 5.788 8.643 8.643 17 93.7 6.851 6.854 11.812 11.812 21 131.0 5.494 5.494 7.457 7.457 225 25 112.6 7.579 7.579 10.962 10.962 AVERAGE 111.2 6.501 7.273 9.460 9.460 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 13753.826276 LEG E/M time start vinfx vinfy vinfz (days) (km/s) (km/s) (km/s) 1 E 0.567449975E+02 0.917752847E+01 0.753848070E+01 0.781623212E+00 2 M 0.148735594E+03 0.139063851E+01 0.720455151E+01 -0.131454828E+01 3 E 0.692209369E+03 0.144253657E+01 0.693563806E+01 -0.311053663E+01 4 E 0.142274075E+04 -0.283159744E+01 0.396416059E+01 -0.602820457E+01 5 E 0.233486984E+04 -0.662158732E+01 0.426087923E+01 0.556410159E+00 6 M 0.245023652E+04 -0.101278082E+02 -0.223399395E+01 0.502369324E+00 7 E 0.301680342E+04 -0.538195381E+01 -0.213545035E+00 0.266066316E+00 8 E 0.374736790E+04 -0.189365116E+01 -0.505115280E+01 -0.152547740E-02 9 E 0.466178239E+04 0.466903413E+01 -0.248914813E+01 -0.108661312E+01 10 M 0.475480232E+04 0.792572684E+01 -0.524092838E+01 -0.294285916E+00 11 E 0.536790398E+04 0.440687983E+01 -0.258885108E+01 0.272641227E+01 12 E 0.609841973E+04 0.381603138E+01 0.433289096E+01 -0.744517468E-02 13 E 0.701439832E+04 -0.196208424E+01 0.532543803E+01 0.113603285E+01 14 M 0.715526945E+04 -0.699611131E+01 0.502267282E+01 -0.723383833E+00 15 E 0.769860188E+04 -0.506058498E+01 0.446224741E+01 -0.107196172E+01 16 E 0.842911146E+04 -0.409600670E+01 -0.677587830E+00 -0.542279406E+01 17 E 0.933844756E+04 -0.518039383E+01 -0.445926641E+01 -0.504715802E+00 18 M 0.943216525E+04 -0.424778512E+01 -0.110088761E+02 0.527448620E+00 19 E 0.100554332E+05 0.132075575E+01 -0.534636218E+01 0.218512397E-01 20 E 0.107859883E+05 0.540572212E+01 -0.110161371E+01 0.275267962E-02 21 E 0.117008676E+05 0.176512114E+01 0.513859674E+01 0.813374365E+00 22 M 0.118318586E+05 0.101002233E+01 0.729039574E+01 -0.119734462E+01 23 E 0.123864604E+05 0.568396929E+00 0.741036534E+01 -0.226792926E+00 24 E 0.131170074E+05 -0.311302446E+01 0.361673281E+01 -0.572123949E+01 25 E 0.140287730E+05 -0.697409421E+01 0.293058729E+01 0.457713753E+00 26 M 0.141413790E+05 -0.103884959E+02 -0.340505007E+01 0.803023552E+00 27 E 0.147484250E+05 -0.587641888E+01 0.183522934E+01 -0.785606560E-03 28 E 0.154305779E+05 -0.168757558E+01 -0.411625010E+01 0.426603001E+01 29 E 0.163412448E+05 ================ PARENT CYCLER 5.658gFh-3 ======================= Parent cycler number 148 Approximate search space (synodic periods after J2000) 15 Number of steps to walk eccentricity/inclination 3 / 3 Number of cycles 7 Total delta v over 44.86 years (km/s) 0.030461 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 2 144.6 7.892 7.892 9.285 9.285 6 167.9 5.605 5.605 8.845 8.845 10 100.1 5.506 5.506 11.855 11.855 14 165.5 6.118 6.118 7.154 7.154 18 121.8 5.676 5.676 11.277 11.277 22 119.7 6.073 6.073 8.266 8.266 26 180.1 6.721 6.721 9.041 9.041 AVERAGE 142.8 6.227 6.227 9.389 9.389 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 12238.309690 LEG E/M time start vinfx vinfy vinfz (days) (km/s) (km/s) (km/s) 1 E -0.108334520E+02 0.462031884E+01 -0.642914227E+01 -0.483504525E-03 2 E 0.639963015E+03 0.748738902E+01 -0.175946042E+01 -0.176881553E+01 3 M 0.784514692E+03 0.830197096E+01 -0.413218055E+01 -0.468890542E+00 4 E 0.141828901E+04 0.264278210E+01 0.489656853E+01 -0.162644435E-01 5 E 0.233343958E+04 -0.303069144E+01 0.469461477E+01 0.130505013E-03 6 E 0.303070874E+04 -0.431061280E+01 0.327017847E+01 0.146471518E+01 7 M 0.319861885E+04 -0.808685566E+01 0.358262598E+01 -0.476186094E-01 8 E 0.376263617E+04 -0.477818240E+01 -0.282802426E+01 -0.839472385E-03 9 E 0.467764446E+04 0.807121185E+00 -0.547345378E+01 0.135368441E-02 10 E 0.537860439E+04 -0.137620483E+01 -0.527923531E+01 -0.743927882E+00 11 M 0.547871509E+04 -0.135268040E+01 -0.117591884E+02 0.666653707E+00 12 E 0.607270957E+04 0.528415806E+01 -0.308044426E+01 -0.244511334E-02 13 E 0.698999402E+04 0.497595512E+01 0.289150283E+01 0.204447147E+01 14 E 0.772050550E+04 -0.639264049E+00 0.590722243E+01 0.145698074E+01 15 M 0.788598107E+04 -0.131024927E+01 0.699896458E+01 -0.693152090E+00 16 E 0.845912129E+04 -0.419448336E+01 0.141480841E+01 -0.356165474E+01 17 E 0.936963259E+04 -0.457118264E+01 -0.343076626E+01 0.550072188E-04 18 E 0.100635653E+05 -0.566974606E+01 -0.112452849E+00 0.249010514E+00 19 M 0.101853534E+05 -0.978292922E+01 -0.557256749E+01 0.644399659E+00 20 E 0.107526015E+05 -0.946571210E+00 -0.599445586E+01 -0.126671059E-02 21 E 0.116697511E+05 0.464418807E+01 -0.392305749E+01 0.218970977E+00 22 E 0.124002983E+05 0.540061124E+01 0.269976058E+01 -0.650807698E+00 23 M 0.125199770E+05 0.818693081E+01 0.978562576E+00 -0.581988595E+00 24 E 0.131199618E+05 0.125138055E+01 0.539972353E+01 -0.365087399E+01 25 E 0.140349464E+05 -0.322160443E+01 0.586886564E+01 -0.774599998E-03 26 E 0.147063948E+05 -0.557206161E+01 0.313381372E+01 0.207377246E+01 27 M 0.148864489E+05 -0.852742156E+01 0.299554037E+01 0.201940992E+00 28 E 0.154657176E+05 -0.291882235E+01 -0.306689696E+01 0.302171906E+01 29 E 0.163755320E+05 ================ PARENT CYCLER 5.658Gfh+3 ======================= Parent cycler number 149 Approximate search space (synodic periods after J2000) 14 Number of steps to walk eccentricity/inclination 243 / 243 Number of cycles 7 Total delta v over 44.84 years (km/s) 0.068829 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 1 90.9 ****** 8.085 11.785 11.785 5 125.7 5.493 5.489 7.383 7.383 9 117.8 7.459 7.477 10.377 10.377 13 95.3 5.353 5.354 9.533 9.533 17 141.2 5.765 5.765 8.643 8.643 21 93.7 6.854 6.854 11.826 11.826 25 129.7 5.537 5.538 7.537 7.537 AVERAGE 113.5 6.077 6.366 9.584 9.583 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 11413.934201 time dv (days) 0.705435870E+02 0.241126135E+03 0.121819196E+04 0.169637948E+04 0.235217484E+04 0.253522155E+04 0.358664372E+04 0.399425982E+04 0.467573538E+04 0.484676757E+04 0.557244839E+04 0.640069267E+04 0.703552899E+04 0.743236899E+04 0.780817831E+04 0.913839361E+04 0.935250521E+04 0.956928421E+04 0.101796276E+05 0.110604521E+05 0.117205162E+05 0.119372329E+05 0.128686214E+05 0.133905371E+05 0.140456639E+05 0.142324359E+05 0.148848556E+05 0.157219913E+05 dvx (km/s) -0.310961168E-04 0.129426272E-06 0.232173598E-03 -0.340545892E-05 0.462370372E-05 -0.141202506E-08 -0.254851679E-04 -0.113560576E-05 -0.335065756E-05 -0.202161761E-07 -0.224955764E-03 0.558626932E-05 -0.268865131E-08 0.105674060E-07 -0.465607957E-04 -0.162590006E-02 -0.125972179E-04 0.474330317E-08 0.206050305E-05 0.946405320E-05 0.133510290E-05 -0.452744747E-08 0.978935026E-04 -0.497810704E-06 0.600562879E-06 -0.378646094E-09 0.270437790E-05 0.179744895E-05 dvy (km/s) -0.198393244E-04 -0.985768449E-07 0.835961470E-05 0.439860940E-05 -0.899111260E-06 -0.193866708E-08 0.361480576E-04 -0.144552201E-05 -0.401358417E-05 -0.875550704E-08 0.413707442E-04 0.140050748E-04 -0.598857802E-04 0.563549716E-07 0.507729205E-04 0.212167689E-02 0.290900461E-04 -0.185159813E-07 -0.394610988E-05 -0.176070030E-05 -0.676291695E-05 0.904307656E-08 -0.138643554E-04 0.491965973E-06 -0.184004383E-06 -0.160064128E-08 0.654424719E-06 0.712594147E-05 dxz (km/s) 0.263588991E-05 0.921249846E-08 0.599818881E-03 0.196011197E-06 0.179078164E-07 -0.190704345E-09 -0.354848015E-03 0.127883877E-07 -0.578762641E-07 0.384347029E-08 0.119462016E-04 0.123398492E-07 0.126848438E-06 0.993161421E-07 0.874982724E-04 -0.446957080E-03 0.216711561E-05 -0.762132879E-09 0.293511967E-04 0.524754457E-08 0.525549203E-06 0.579491564E-09 0.168089722E-03 0.274693086E-07 -0.417042620E-07 -0.552694780E-10 -0.392760391E-06 0.219462404E-06 time dv (days) 0.867860180E+02 0.676100934E+03 0.110140185E+04 0.169283418E+04 0.243802996E+04 0.306596986E+04 0.337910439E+04 0.404628874E+04 0.478278845E+04 0.539362099E+04 0.556781426E+04 0.633872206E+04 0.715070654E+04 0.774532684E+04 0.797195211E+04 0.872316957E+04 0.947372250E+04 0.100818335E+05 0.102704406E+05 0.110185749E+05 0.118450824E+05 0.124182501E+05 0.126099747E+05 0.138153501E+05 0.141356637E+05 0.147334029E+05 0.149733392E+05 0.157204657E+05 dvx (km/s) -0.256719625E-06 -0.776277616E-06 0.803937416E-09 -0.177660516E-07 -0.117829543E-06 -0.158450963E-06 0.474719193E-10 0.371405601E-08 0.852229082E-08 0.172718771E-07 0.263086229E-10 0.166666948E-06 0.422819210E-05 -0.452754321E-06 -0.931780260E-10 0.156622809E-02 0.744984408E-08 -0.536864551E-08 0.148639090E-11 0.568072477E-09 -0.167991932E-10 0.149357980E-08 -0.941601352E-11 -0.415749205E-07 0.752017898E-09 0.278836284E-07 0.830196437E-10 -0.227089776E-08 dvy (km/s) -0.912128784E-07 0.122176765E-06 0.691423908E-09 -0.420455542E-07 0.298883647E-07 0.149045458E-07 -0.112856585E-09 0.281419595E-08 -0.867685729E-08 -0.130543479E-07 -0.116679205E-10 -0.104921648E-06 0.644877351E-05 -0.235979465E-06 -0.527561299E-10 0.359406540E-02 0.176271378E-07 0.493027759E-08 -0.207135764E-10 0.474022788E-08 -0.143423329E-08 -0.153608496E-08 -0.233292275E-10 -0.515609327E-08 -0.404185894E-09 -0.140633160E-06 0.167841474E-09 -0.185332939E-08 dxz (km/s) 0.245979905E-07 -0.711895470E-09 0.156564889E-09 0.153196262E-10 0.248809133E-08 0.849007921E-08 -0.341467928E-10 0.156594786E-10 -0.109274722E-09 0.134097270E-08 -0.485836124E-12 -0.235233713E-09 -0.436893362E-04 0.112215341E-07 0.277318752E-10 0.296280648E-01 -0.870581255E-09 0.436308378E-09 0.325501936E-11 -0.196071627E-10 -0.316791927E-06 0.108508809E-09 -0.103716686E-10 0.268159633E-08 0.832936600E-10 -0.266389604E-08 0.199387600E-12 0.415842241E-10 226 LEG E/M time start vinfx vinfy vinfz (days) (km/s) (km/s) (km/s) 1 E -0.123124890E+02 -0.726359024E+01 -0.353299234E+01 -0.358400240E+00 2 M 0.785536570E+02 -0.572727405E+01 -0.102809210E+02 0.624440735E+00 3 E 0.697166904E+03 -0.633804853E+00 -0.507119206E+01 0.202137286E+01 4 E 0.142767812E+04 0.512053544E+01 -0.202384960E+01 -0.256648174E+00 5 E 0.234249544E+04 0.255686131E+01 0.481977962E+01 0.602359807E+00 6 M 0.246819681E+04 0.253381206E+01 0.685055090E+01 -0.107666920E+01 7 E 0.302903021E+04 0.180986924E+01 0.709334442E+01 -0.997821087E-01 8 E 0.375956508E+04 -0.253843333E+01 0.421612384E+01 0.543475618E+01 9 E 0.467188462E+04 -0.632730617E+01 0.394474052E+01 0.560451140E+00 10 M 0.478964084E+04 -0.101297628E+02 -0.219521724E+01 0.501737299E+00 11 E 0.535665321E+04 -0.538061264E+01 -0.201109537E+00 -0.108351970E+00 12 E 0.608726127E+04 -0.151899258E+01 -0.426094228E+01 0.292276662E+01 13 E 0.699821000E+04 0.469520255E+01 -0.233169434E+01 -0.108622500E+01 14 M 0.709351738E+04 0.788439585E+01 -0.535133108E+01 -0.291976476E+00 15 E 0.770754067E+04 0.525375516E+01 -0.167510105E+01 0.169170352E+01 16 E 0.843804223E+04 0.382285178E+01 0.429787431E+01 -0.664115980E-02 17 E 0.935393281E+04 -0.197772688E+01 0.529405525E+01 0.113814002E+01 18 M 0.949508833E+04 -0.699317637E+01 0.502670271E+01 -0.724720887E+00 19 E 0.100384435E+05 -0.469266654E+01 0.448669647E+01 0.213414792E+01 20 E 0.107689527E+05 -0.408156505E+01 -0.673817037E+00 0.543686638E+01 21 E 0.116782877E+05 -0.517510737E+01 -0.446504567E+01 -0.504020310E+00 22 M 0.117719707E+05 -0.426532181E+01 -0.110179103E+02 0.524299487E+00 23 E 0.123958852E+05 0.146222795E+01 -0.534482630E+01 0.330778976E+00 24 E 0.131264404E+05 0.545856578E+01 -0.105158757E+01 0.257614717E-02 25 E 0.140414921E+05 0.180024383E+01 0.517428062E+01 0.806651190E+00 26 M 0.141711910E+05 0.108189503E+01 0.735134940E+01 -0.126112620E+01 27 E 0.147240205E+05 0.753927549E+00 0.744796542E+01 0.164819267E+01 28 E 0.154545700E+05 -0.302970471E+01 0.374379628E+01 0.597724984E+01 29 E 0.163664791E+05 ================ PARENT CYCLER 5.658gFh+3 ======================= Parent cycler number 150 Approximate search space (synodic periods after J2000) 7 Number of steps to walk eccentricity/inclination 81 / 81 Number of cycles 7 Total delta v over 44.91 years (km/s) 0.414088 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 2 109.8 5.421 5.421 10.231 10.231 6 159.3 5.278 5.280 8.050 8.050 10 108.9 5.604 5.604 12.086 12.086 14 148.1 6.614 6.998 7.639 7.639 18 182.8 7.830 7.830 10.117 10.117 22 101.0 5.324 5.324 9.568 9.568 26 170.0 5.672 5.672 8.242 8.242 AVERAGE 140.0 5.963 6.018 9.419 9.419 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 5998.597490 LEG E/M time start vinfx vinfy vinfz (days) (km/s) (km/s) (km/s) 1 E -0.163434076E+02 0.312110647E+01 -0.448206429E+01 0.669082653E-03 2 E 0.688414031E+03 0.417001008E+01 -0.327540771E+01 -0.112897359E+01 3 M 0.798262835E+03 0.610484784E+01 -0.820700212E+01 -0.231565547E+00 4 E 0.142922257E+04 0.387558034E+01 0.352892185E+01 -0.475294374E-02 5 E 0.234310108E+04 -0.202966084E+01 0.485443490E+01 0.428051572E-03 6 E 0.305309026E+04 -0.228121993E+01 0.456146079E+01 0.136469653E+01 7 M 0.321240276E+04 -0.583700085E+01 0.553067123E+01 -0.368322062E+00 8 E 0.377235284E+04 -0.401740961E+01 -0.883846842E+00 0.389352146E+01 9 E 0.468162424E+04 -0.206442513E+01 -0.527231378E+01 0.630266122E-03 10 E 0.537800154E+04 -0.431308170E+01 -0.356765237E+01 -0.277801753E+00 11 M 0.548688840E+04 -0.666331611E+01 -0.100482418E+02 0.842202841E+00 12 E 0.606253261E+04 0.221290407E+01 -0.621768827E+01 -0.200320036E-01 13 E 0.698180125E+04 0.536733554E+01 -0.105404529E+00 0.388582960E+01 14 E 0.771232245E+04 0.249282302E+01 0.651860050E+01 0.518992748E+00 15 M 0.786039371E+04 0.402229526E+01 0.641718868E+01 -0.996021397E+00 16 E 0.843295521E+04 -0.178563618E+01 0.477885560E+01 0.571135328E+01 17 E 0.934600773E+04 -0.577080031E+01 0.525451176E+01 -0.134077875E-03 18 E 0.999832940E+04 -0.737098233E+01 0.100009723E+01 0.244438821E+01 19 M 0.101811181E+05 -0.101037680E+02 -0.314272973E+00 0.411560084E+00 20 E 0.107641537E+05 -0.203206540E+01 -0.391422301E+01 0.306511065E+01 21 E 0.116746737E+05 0.364902649E+01 -0.387832884E+01 -0.397162886E+00 22 E 0.124052123E+05 0.479797892E+01 -0.203227247E+01 -0.109176708E+01 23 M 0.125062191E+05 0.777015156E+01 -0.557441024E+01 -0.307087964E+00 24 E 0.131238077E+05 0.370461776E+01 0.425210184E+01 -0.134723898E-02 25 E 0.140392383E+05 -0.184302985E+01 0.534632942E+01 -0.105731500E-02 26 E 0.147348662E+05 -0.349129686E+01 0.417019625E+01 0.161069206E+01 27 M 0.149049030E+05 -0.660110127E+01 0.493525937E+01 -0.619339646E-01 28 E 0.154776558E+05 -0.361013492E+01 -0.195390511E+01 0.324834118E+01 29 E 0.163870208E+05 ================ PARENT CYCLER 5.658Gffh-3 ======================= Parent cycler number 151 Approximate search space (synodic periods after J2000) 14 Number of steps to walk eccentricity/inclination 243 / 243 Number of cycles 7 Total delta v over 44.84 years (km/s) 0.021349 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 1 89.4 ****** 8.398 11.782 11.782 6 125.4 5.496 5.496 7.389 7.389 11 117.9 7.454 7.454 10.379 10.379 16 93.3 5.391 5.391 9.504 9.504 21 140.6 5.807 5.807 8.643 8.643 26 93.6 6.865 6.864 11.829 11.829 31 129.7 5.536 5.536 7.531 7.531 AVERAGE 112.9 6.092 6.421 9.580 9.580 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 11413.934201 LEG E/M time start vinfx vinfy vinfz (days) (km/s) (km/s) (km/s) 1 E -0.103379281E+02 -0.758021304E+01 -0.359660039E+01 -0.366940353E+00 2 M 0.790912082E+02 -0.567621294E+01 -0.103061558E+02 0.621466340E+00 time dv (days) 0.131743293E+01 0.171345644E+03 0.112086341E+04 0.169297514E+04 0.236135064E+04 0.259718850E+04 0.313861044E+04 0.444380473E+04 0.468954805E+04 0.487469269E+04 0.556122347E+04 0.633321743E+04 0.701250611E+04 0.718562087E+04 0.783172594E+04 0.874028612E+04 0.937510614E+04 0.983740206E+04 0.104913592E+05 0.114236739E+05 0.116923401E+05 0.121712760E+05 0.125273852E+05 0.134009559E+05 0.140609469E+05 0.143812662E+05 0.148336030E+05 0.157372618E+05 dvx (km/s) -0.199126820E-02 -0.266287037E-05 -0.122119745E-04 0.109069903E-03 0.302549352E-04 0.125133870E-06 -0.732073254E-04 0.235234286E-02 0.873166072E-04 -0.831386774E-07 0.668455150E-03 -0.643723518E-06 -0.407842410E-05 -0.850007101E-08 -0.180943714E-04 -0.125934568E-06 0.163648463E-06 0.342227037E-09 0.958992920E-04 0.255978559E-06 -0.562266595E-06 0.911057590E-07 -0.123975961E-04 -0.709015176E-04 -0.185778461E-04 0.278258332E-05 0.428685866E-05 0.771629201E-04 dvy (km/s) 0.815939405E-03 0.454995382E-05 -0.318238233E-03 0.386399213E-04 0.154779629E-04 -0.602836201E-07 -0.172354150E-03 0.222655350E-02 -0.789612099E-04 -0.909001543E-07 0.200017049E-03 -0.945194870E-06 0.197971097E-07 -0.263163607E-08 -0.740632403E-05 -0.517242710E-06 0.257596935E-05 0.108296556E-10 0.685685867E-04 0.681434749E-07 0.135544792E-05 -0.151232579E-06 0.182018985E-03 0.145142371E-04 0.431916517E-04 0.136036079E-06 0.934848878E-04 -0.909884501E-04 dxz (km/s) -0.171850157E-03 0.355338523E-07 0.770126254E-04 -0.552408272E-02 -0.196380607E-05 0.630928041E-07 -0.300257620E-03 0.983692498E-03 -0.456443060E-05 -0.268834316E-07 0.140491500E-03 0.261631111E-07 -0.279171889E-06 0.683457549E-09 -0.333398615E-03 -0.727481911E-09 -0.177338008E-07 0.496091022E-09 0.171303368E-03 0.108848568E-07 0.102214897E-06 -0.125323540E-07 0.216902342E-03 -0.435915258E-07 -0.363298087E-05 0.309845962E-05 0.723887315E-03 0.290354887E-05 time dv (days) 0.893702083E+02 0.704891352E+03 0.981241160E+03 0.209635388E+04 0.247799902E+04 0.307698714E+04 0.329639527E+04 0.405422697E+04 0.488357366E+04 0.539433457E+04 0.574592829E+04 0.632912052E+04 0.731784100E+04 0.773453314E+04 0.810086954E+04 0.915426670E+04 0.979610968E+04 0.100257477E+05 0.102685735E+05 0.110190993E+05 0.118280869E+05 0.124203634E+05 0.125988574E+05 0.133892826E+05 0.141435825E+05 0.147688736E+05 0.149908160E+05 0.157686526E+05 dvx (km/s) 0.242009807E-05 0.456974051E-04 -0.178350277E-05 0.216648512E-04 -0.117816967E-04 0.334972035E-04 0.110018932E-06 0.480243231E-05 0.162723452E-06 0.378176352E-05 -0.206790058E-07 0.280541539E-04 0.555560340E-04 -0.154845542E-04 -0.119721545E-07 0.228518060E-02 -0.826224609E-06 0.114911559E-05 0.450134387E-08 -0.113402082E-06 -0.376577008E-05 0.111603949E-06 -0.851153061E-09 0.107086842E-07 0.374973005E-07 0.992718947E-07 0.462103884E-10 0.326364055E-07 dvy (km/s) -0.523218888E-06 -0.690033873E-04 0.188698353E-06 -0.288673119E-04 0.115071152E-04 -0.328657774E-04 0.155519112E-07 0.161146394E-05 -0.729938320E-06 0.375358072E-06 -0.250142591E-07 -0.151352073E-04 0.414456690E-04 0.630645158E-05 -0.167276233E-07 0.305292651E-02 0.272706767E-05 0.173074715E-05 0.293432742E-08 -0.175524363E-06 0.439090009E-06 0.700126014E-06 -0.200277533E-09 0.128846745E-07 -0.167667043E-07 -0.130789172E-08 -0.820077736E-12 0.362305582E-07 dxz (km/s) -0.415020074E-07 -0.301087086E-06 -0.227027779E-06 -0.485486094E-04 0.812835937E-07 -0.289810071E-05 -0.567273274E-08 0.112159224E-06 -0.240937699E-06 0.203257100E-06 -0.112215925E-07 0.130960754E-02 0.406407249E-04 0.407312121E-06 0.146212255E-08 0.118883288E-02 -0.180251839E-03 0.206224199E-06 -0.719122199E-10 0.141503029E-08 -0.144521241E-03 -0.206300865E-07 0.939804332E-10 -0.119167708E-11 -0.753748859E-09 -0.455736832E-08 -0.327576414E-12 0.199194154E-08 time dv dvx dvy dxz (days) (km/s) (km/s) (km/s) 0.307644236E+01 -0.293834990E-05 -0.486248633E-06 -0.175200004E-06 0.171854237E+03 -0.905927971E-09 0.207814147E-08 -0.130173782E-08 227 3 E 0.697511399E+03 -0.208179055E+01 0.137809933E+01 -0.489403753E+01 4 E 0.106276775E+04 0.446091929E+01 -0.123078819E+01 -0.296960023E+01 5 E 0.142802494E+04 0.514177935E+01 -0.199576941E+01 0.384393532E-02 6 E 0.234288037E+04 0.257853229E+01 0.481559097E+01 0.602316384E+00 7 M 0.246826767E+04 0.253389671E+01 0.685664245E+01 -0.108066948E+01 8 E 0.302886009E+04 -0.334220077E+01 0.491539380E+01 -0.431322377E+01 9 E 0.339412288E+04 -0.781030369E+00 -0.489922058E+00 -0.727100933E+01 10 E 0.375937906E+04 -0.251992186E+01 0.421298168E+01 -0.543862082E+01 11 E 0.467171201E+04 -0.631159509E+01 0.392552420E+01 0.560883701E+00 12 M 0.478959534E+04 -0.101330528E+02 -0.219065106E+01 0.498909756E+00 13 E 0.535638173E+04 -0.146076276E+01 -0.511401155E+01 -0.931246744E+00 14 E 0.572162458E+04 0.671932791E-01 -0.118796407E+01 -0.524825285E+01 15 E 0.608687955E+04 -0.192366988E+01 -0.502715247E+01 -0.151699143E-02 16 E 0.700125084E+04 0.466864752E+01 -0.246801241E+01 -0.108618249E+01 17 M 0.709458588E+04 0.791864458E+01 -0.524677392E+01 -0.297509366E+00 18 E 0.770802516E+04 0.341980341E+01 0.463639934E+01 -0.422404889E+00 19 E 0.807328510E+04 0.348767307E+00 0.123377081E+01 -0.565145475E+01 20 E 0.843853195E+04 0.381075460E+01 0.436215678E+01 -0.874633802E-02 21 E 0.935458531E+04 -0.194949118E+01 0.535160114E+01 0.113433344E+01 22 M 0.949521928E+04 -0.699853981E+01 0.501972750E+01 -0.722768185E+00 23 E 0.100385240E+05 -0.672727960E+01 -0.117136439E+01 0.168283766E+00 24 E 0.104037891E+05 -0.674412073E+01 -0.116716248E+01 -0.285078046E+00 25 E 0.107690361E+05 -0.411889710E+01 -0.682193788E+00 -0.542114157E+01 26 E 0.116783713E+05 -0.518570894E+01 -0.446891641E+01 -0.504224192E+00 27 M 0.117719785E+05 -0.426590176E+01 -0.110200810E+02 0.523646389E+00 28 E 0.123959888E+05 -0.321648871E+01 0.118190618E+01 -0.437603779E+01 29 E 0.127612439E+05 0.105749260E+01 0.297552960E+00 -0.544449572E+01 30 E 0.131264973E+05 0.545755087E+01 -0.104588903E+01 0.252999618E-02 31 E 0.140415412E+05 0.180440242E+01 0.517065755E+01 0.807084061E+00 32 M 0.141712646E+05 0.107469402E+01 0.734722810E+01 -0.125670232E+01 33 E 0.147241944E+05 -0.293452865E+01 0.277437568E+01 0.649621061E+01 34 E 0.150894545E+05 -0.486702763E+01 0.588347494E+01 -0.207932220E+00 35 E 0.154547190E+05 -0.303411593E+01 0.373393741E+01 -0.595916465E+01 36 E 0.163666167E+05 ================ PARENT CYCLER 5.658Gffh+3 ======================= Parent cycler number 152 Approximate search space (synodic periods after J2000) 14 Number of steps to walk eccentricity/inclination 3 / 3 Number of cycles 7 Total delta v over 44.84 years (km/s) 0.023134 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 1 89.4 ****** 8.416 11.782 11.782 6 125.4 5.495 5.495 7.387 7.387 11 117.8 7.470 7.462 10.379 10.379 16 93.3 5.393 5.393 9.505 9.505 21 140.7 5.800 5.800 8.642 8.642 26 93.6 6.862 6.862 11.820 11.820 31 130.3 5.518 5.518 7.503 7.503 AVERAGE 112.9 6.090 6.421 9.574 9.574 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 11413.934201 LEG E/M time start vinfx vinfy vinfz (days) (km/s) (km/s) (km/s) 1 E -0.102273512E+02 -0.759809692E+01 -0.360042825E+01 -0.367433574E+00 2 M 0.791236549E+02 -0.567291393E+01 -0.103074483E+02 0.621343095E+00 3 E 0.697518849E+03 -0.255532750E+01 0.156503272E+01 -0.460370889E+01 4 E 0.106277513E+04 0.413044764E+01 -0.109951258E+01 -0.345936157E+01 5 E 0.142803230E+04 0.514112462E+01 -0.199483324E+01 0.383026231E-02 6 E 0.234288395E+04 0.257881818E+01 0.481431665E+01 0.602512988E+00 7 M 0.246828999E+04 0.253121431E+01 0.685547925E+01 -0.107966791E+01 8 E 0.302892776E+04 -0.387428996E+01 0.603076217E+01 -0.157036710E+01 9 E 0.339418977E+04 -0.384210621E+01 0.599026337E+01 -0.171135410E+01 10 E 0.375945567E+04 -0.254273680E+01 0.424811965E+01 0.541826242E+01 11 E 0.467177329E+04 -0.631740097E+01 0.393218223E+01 0.560767001E+00 12 M 0.478960735E+04 -0.101326566E+02 -0.219171234E+01 0.499235535E+00 13 E 0.535641264E+04 -0.120463514E+01 -0.445752936E+01 -0.279601042E+01 14 E 0.572165384E+04 -0.145939652E+01 -0.515022468E+01 -0.554067205E+00 15 E 0.608693776E+04 -0.191923971E+01 -0.503135029E+01 -0.151675620E-02 16 E 0.700131778E+04 0.466919762E+01 -0.247147520E+01 -0.108629293E+01 17 M 0.709459706E+04 0.792011366E+01 -0.524713486E+01 -0.296409855E+00 18 E 0.770794507E+04 0.324373150E+01 0.443119363E+01 0.180747906E+01 19 E 0.807320516E+04 0.293584633E+01 0.409797840E+01 -0.286487062E+01 20 E 0.843845245E+04 0.381249586E+01 0.435133983E+01 -0.814806043E-02 21 E 0.935447757E+04 -0.195375482E+01 0.534191985E+01 0.113494231E+01 22 M 0.949520368E+04 -0.699762249E+01 0.501998463E+01 -0.721929776E+00 23 E 0.100385382E+05 -0.672703749E+01 -0.117252298E+01 -0.447347638E-01 24 E 0.104038032E+05 -0.674246056E+01 -0.116828768E+01 -0.271242161E+00 25 E 0.107690503E+05 -0.407978011E+01 -0.682623757E+00 0.544799361E+01 26 E 0.116783840E+05 -0.518661783E+01 -0.446497687E+01 -0.504567766E+00 27 M 0.117720248E+05 -0.425610844E+01 -0.110146898E+02 0.525474670E+00 28 E 0.123956658E+05 -0.360213325E+01 0.127799802E+01 0.400008152E+01 29 E 0.127609110E+05 0.358989715E+01 -0.236136406E+00 0.419698721E+01 30 E 0.131261614E+05 0.543494864E+01 -0.107692360E+01 0.268865787E-02 31 E 0.140411376E+05 0.178227584E+01 0.515954042E+01 0.809468307E+00 32 M 0.141714117E+05 0.105237499E+01 0.732560891E+01 -0.123372545E+01 33 E 0.147249851E+05 -0.446457125E+01 0.507768260E+01 -0.340147107E+01 34 E 0.150902622E+05 -0.488302516E+01 0.576066312E+01 -0.263673881E+00 35 E 0.154555246E+05 -0.306290154E+01 0.369419854E+01 0.586327981E+01 36 E 0.163673757E+05 ================ PARENT CYCLER 5.658Gfh-f3 ======================= Parent cycler number 153 Approximate search space (synodic periods after J2000) 16 Number of steps to walk eccentricity/inclination 81 / 81 Number of cycles 7 Total delta v over 44.95 years (km/s) 0.141477 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 1 116.9 ****** 5.980 8.686 8.686 6 136.5 6.134 6.134 9.099 9.099 11 89.8 6.369 6.369 11.680 11.680 16 133.7 5.700 5.699 7.709 7.709 0.850919068E+03 0.131114264E+04 0.170248157E+04 0.236168846E+04 0.258038616E+04 0.327723879E+04 0.364980221E+04 0.404220227E+04 0.468939451E+04 0.487461330E+04 0.560839929E+04 0.592616736E+04 0.634290351E+04 0.701525109E+04 0.718660177E+04 0.796005452E+04 0.820112150E+04 0.873166903E+04 0.937568041E+04 0.977230471E+04 0.102430725E+05 0.104585761E+05 0.114783175E+05 0.116924124E+05 0.119841420E+05 0.125603536E+05 0.129146503E+05 0.134010104E+05 0.140609997E+05 0.144643174E+05 0.149506557E+05 0.152940026E+05 0.157374073E+05 0.199862789E-03 -0.496043381E-04 0.320724445E-05 -0.286424341E-05 0.581994620E-08 0.160458551E-04 -0.118823026E-03 -0.292651827E-06 0.398103106E-06 -0.169288184E-09 0.547520544E-05 -0.611529623E-03 -0.663284346E-07 -0.234344098E-06 -0.365770377E-09 -0.490278928E-06 -0.560076613E-04 -0.211748155E-07 -0.835487261E-10 0.425564130E-09 -0.108897704E-04 0.159664279E-05 -0.430184547E-02 -0.134485658E-05 0.704519194E-08 0.542330668E-03 -0.310087379E-03 -0.136708163E-04 -0.535875599E-05 0.647490558E-08 -0.867778464E-04 -0.160991049E-03 -0.243492782E-04 -0.104417619E-02 -0.307197775E-04 -0.624589251E-06 -0.170666434E-05 -0.407110232E-08 0.260768611E-04 0.467152462E-03 0.384171669E-06 -0.382366892E-06 -0.245196630E-09 -0.315265854E-06 -0.179840012E-02 -0.952109392E-07 -0.228966855E-06 -0.100790199E-09 0.197136354E-06 0.103648531E-04 -0.599504867E-07 0.173716935E-08 -0.311460895E-09 -0.154852917E-04 -0.343556853E-05 0.558944655E-02 0.310977310E-05 -0.148482614E-07 -0.126345348E-02 0.377949893E-03 0.280341363E-05 0.110350390E-04 0.415005431E-07 0.628772658E-03 0.783071154E-05 0.276966356E-04 -0.295350144E-04 -0.198415392E-03 0.414105549E-08 0.198341493E-06 0.409044875E-09 0.370585740E-03 0.119150507E-03 0.192998963E-07 -0.225623465E-07 -0.140344276E-10 -0.248084941E-03 -0.338798328E-03 0.668162107E-09 -0.620346626E-08 0.414158374E-10 -0.529185541E-04 -0.551188728E-05 -0.328278858E-10 0.346796986E-11 0.222812310E-09 -0.212891108E-04 -0.182956160E-08 -0.117873351E-02 0.229882359E-06 0.157699962E-09 -0.962973059E-04 0.263869042E-05 -0.885407172E-08 -0.100272574E-05 0.449663877E-07 -0.217709385E-03 0.210729857E-03 0.125753487E-05 time dv (days) 0.317529975E+01 0.277010117E+03 0.847273922E+03 0.132210772E+04 0.170248780E+04 0.236169486E+04 0.271497061E+04 0.327000069E+04 0.363526527E+04 0.448018659E+04 0.468944840E+04 0.487462815E+04 0.561208148E+04 0.597369975E+04 0.633382037E+04 0.701530967E+04 0.718659926E+04 0.795997453E+04 0.832522579E+04 0.873158049E+04 0.937558649E+04 0.964190401E+04 0.102796131E+05 0.106448663E+05 0.111145971E+05 0.116924302E+05 0.118655709E+05 0.125198492E+05 0.129727562E+05 0.134006542E+05 0.140606787E+05 0.142765906E+05 0.149697208E+05 0.153313354E+05 0.157381984E+05 dvx (km/s) -0.300511038E-08 0.491249711E-09 0.115443939E-03 -0.386611112E-05 0.655941492E-08 -0.101891130E-05 0.125005226E-08 0.190303154E-05 0.768766915E-05 0.389890859E-02 0.192679910E-08 0.856121475E-11 0.301101197E-04 -0.909290981E-06 0.271478798E-09 0.388847795E-08 0.274792177E-10 -0.753720789E-06 0.550098554E-05 -0.174736080E-09 -0.118097808E-09 -0.104229235E-10 -0.714343923E-08 0.582699622E-08 -0.363291130E-09 -0.235522523E-08 -0.380145712E-10 -0.157395542E-03 -0.286087780E-03 -0.702614166E-07 -0.366454018E-07 -0.365883634E-09 0.162113698E-04 -0.342768328E-05 -0.443814745E-06 dvy (km/s) -0.270982784E-09 -0.186563221E-08 -0.106383932E-02 0.108527440E-04 -0.128208490E-08 0.361306898E-05 -0.466316041E-08 0.353283969E-05 0.112773509E-04 0.775017861E-02 -0.179880745E-08 0.253000595E-10 0.892669238E-05 -0.276457259E-06 0.472874018E-09 0.392423400E-08 0.107271459E-10 0.404371337E-07 -0.334865755E-06 -0.597049660E-09 0.186613318E-08 -0.123540743E-10 -0.578492623E-07 0.419839268E-07 0.998352054E-09 0.537672063E-08 0.433351562E-11 -0.107793881E-02 0.422048225E-06 0.131643127E-07 0.110222118E-06 0.703696966E-09 0.462833850E-04 -0.108476398E-06 0.523674745E-06 dxz (km/s) -0.183836349E-09 0.111493009E-10 -0.475738983E-04 -0.192433362E-03 0.877542134E-11 -0.297543779E-07 0.182571304E-09 0.134053724E-03 0.388670638E-03 0.326321466E-02 -0.105320467E-09 -0.132270603E-12 -0.645491805E-03 0.979218324E-04 -0.649159236E-11 0.967381720E-10 0.204864429E-11 0.234362963E-04 0.105560034E-03 -0.541929043E-11 0.120159576E-11 -0.122662383E-10 -0.137846190E-04 0.157254371E-05 0.193798753E-09 0.400755125E-09 0.220361016E-11 -0.687821097E-03 -0.750903503E-05 -0.361695593E-10 -0.926285212E-08 0.198292854E-09 0.790151323E-03 0.196043340E-03 -0.188300725E-07 228 21 106.5 7.622 7.622 11.388 11.388 26 125.1 6.323 6.323 8.221 8.221 31 134.5 6.355 6.355 9.198 9.198 AVERAGE 120.4 6.417 6.355 9.426 9.426 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 12973.862251 LEG E/M time start vinfx vinfy vinfz time dv dvx dvy dxz (days) (km/s) (km/s) (km/s) (days) (km/s) (km/s) (km/s) 1 E -0.401904111E+02 0.580865954E+01 0.108287728E+01 -0.917539801E+00 -0.226523319E+02 0.153993774E-02 -0.461863348E-03 -0.371269379E-04 2 M 0.767301173E+02 0.853346029E+01 -0.154000563E+01 -0.502044345E+00 0.168409331E+03 -0.519415357E-05 -0.269102854E-05 0.524650840E-07 3 E 0.687924879E+03 0.184414632E+01 0.571910972E+01 -0.101243627E+01 0.742711628E+03 -0.138132370E-03 0.894942703E-05 -0.707320621E-04 4 E 0.105316987E+04 0.214110503E+01 0.523497017E+01 -0.230533755E+01 0.132779948E+04 -0.679518212E-04 -0.167617258E-03 0.587866422E-06 5 E 0.196860189E+04 0.291419303E+01 0.535015913E+01 0.503019918E+00 0.202339002E+04 -0.131508137E-03 0.627998628E-04 -0.582680346E-05 6 E 0.233385609E+04 -0.295912150E+01 0.527545490E+01 0.101847951E+01 0.235432816E+04 0.118566546E-03 -0.160289653E-03 -0.137847495E-04 7 M 0.247033657E+04 -0.846449342E+01 0.332268751E+01 -0.316943436E+00 0.265646116E+04 0.198257133E-07 -0.292325505E-07 0.889621176E-08 8 E 0.301776186E+04 -0.490538243E+01 -0.260720587E+01 -0.313178439E+01 0.308716046E+04 0.235316851E-04 -0.686719961E-05 0.248034786E-03 9 E 0.338301765E+04 -0.373067628E+01 -0.178850275E+01 -0.487964458E+01 0.366489726E+04 -0.451298633E-04 -0.239501546E-04 0.112485564E-05 10 E 0.429230671E+04 -0.620328458E+01 -0.144021517E+01 -0.255061230E+00 0.434709920E+04 -0.418049877E-05 0.156544086E-04 0.820576971E-07 11 E 0.465758994E+04 -0.256282167E+01 -0.577623812E+01 -0.796121877E+00 0.467105566E+04 -0.370497978E-04 0.838349879E-05 -0.301104006E-05 12 M 0.474736143E+04 -0.767961001E+00 -0.116519368E+02 0.243132942E+00 0.504491936E+04 0.135241559E-06 -0.507414528E-06 -0.978773100E-07 13 E 0.538046342E+04 -0.127598232E+01 0.284420298E+00 -0.554626519E+01 0.543525174E+04 -0.120287819E-03 0.312850853E-03 0.623917824E-04 14 E 0.574571888E+04 0.521403623E+01 0.116215495E+01 -0.201136607E+01 0.601082739E+04 -0.128538094E-02 0.553325381E-02 -0.821067558E-01 15 E 0.665988615E+04 0.271827001E+01 -0.351316756E-01 0.499869902E+01 0.690460396E+04 0.778103692E-03 -0.377844282E-03 0.258341108E-04 16 E 0.702513662E+04 0.963598734E+00 0.551873843E+01 0.104696876E+01 0.704519530E+04 -0.249764307E-03 -0.210975153E-03 0.164407399E-04 17 M 0.715886121E+04 -0.167089423E+01 0.740886914E+01 -0.131854588E+01 0.728437590E+04 0.688582851E-07 -0.582793369E-08 0.467610016E-07 18 E 0.770457726E+04 -0.634719752E+01 0.393255881E+01 0.749181245E+00 0.776667314E+04 -0.123554107E-04 -0.535524461E-04 0.112215950E-03 19 E 0.806984712E+04 -0.382648059E+01 0.240418863E+01 -0.598765004E+01 0.833398057E+04 -0.373901186E-05 0.353604496E-05 -0.686503908E-07 20 E 0.898065212E+04 -0.457841607E+01 0.515921974E+01 0.328969466E+01 0.909753555E+04 0.225396806E-04 0.395086063E-04 0.125620882E-03 21 E 0.934591283E+04 -0.755096281E+01 0.101505100E+01 0.221398150E+00 0.936189057E+04 -0.913354941E-05 -0.146713674E-05 0.722569909E-07 22 M 0.945243115E+04 -0.977944215E+01 -0.579966389E+01 0.634915263E+00 0.963780069E+04 -0.145942257E-07 0.787308852E-08 0.103525483E-07 23 E 0.100141570E+05 0.732426298E-02 -0.303710400E+01 0.563929984E+01 0.102150471E+05 -0.111353248E-03 -0.949204288E-03 0.740060190E-04 24 E 0.103794117E+05 -0.753888870E+00 -0.459962838E+01 0.440383892E+01 0.106619513E+05 -0.369050073E-06 -0.343573888E-04 -0.108324492E-05 25 E 0.112908299E+05 0.230882576E-01 0.292710839E+01 -0.559677884E+01 0.114807622E+05 -0.383426632E-03 -0.182637572E-02 0.234550920E-03 26 E 0.116560843E+05 0.568272963E+01 0.267853103E+01 -0.716141126E+00 0.116748423E+05 -0.232448808E-04 -0.220702247E-04 -0.177684119E-05 27 M 0.117811374E+05 0.816435080E+01 0.742475217E+00 -0.607300996E+00 0.118720448E+05 0.559650081E-07 0.374735712E-07 -0.479426972E-08 28 E 0.123871869E+05 0.726523127E+00 0.609391825E+01 -0.145704422E+01 0.124419733E+05 -0.983963804E-04 -0.114397556E-04 -0.707048998E-04 29 E 0.127524296E+05 0.102739668E+01 0.539689245E+01 -0.309069165E+01 0.128896544E+05 0.480953169E-06 -0.194154254E-06 -0.545457809E-07 30 E 0.136672618E+05 0.198854402E+01 0.597032364E+01 -0.834757344E+00 0.137512689E+05 0.518534426E-05 -0.430984064E-06 0.670537213E-05 31 E 0.140325104E+05 -0.353392754E+01 0.520112158E+01 0.921299772E+00 0.140526845E+05 -0.323069096E-05 -0.173674857E-05 0.295314093E-06 32 M 0.141670044E+05 -0.897632099E+01 0.198593020E+01 0.284695768E+00 0.144917426E+05 -0.910266622E-06 -0.884883601E-06 0.592941842E-06 33 E 0.147367205E+05 -0.262545851E+01 -0.401306666E+01 0.180637834E+01 0.148024699E+05 -0.344624827E-03 0.712160442E-04 0.859414414E-03 34 E 0.151019952E+05 -0.263096909E+01 -0.341356356E+01 -0.279983924E+01 0.153932168E+05 -0.905755829E-04 -0.218427749E-03 0.580391450E-05 35 E 0.160120627E+05 -0.358060074E+01 -0.359784264E+01 0.586193006E+00 0.160668539E+05 0.167744844E-03 -0.153447258E-03 -0.172879283E-04 36 E 0.163773375E+05 ================ PARENT CYCLER 5.658Gfh+f3 ======================= Parent cycler number 154 Approximate search space (synodic periods after J2000) 17 Number of steps to walk eccentricity/inclination 3 / 3 Number of cycles 7 Total delta v over 44.64 years (km/s) 0.243765 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 1 90.6 ****** 12.972 7.476 7.476 6 115.0 7.962 7.962 10.386 10.386 11 93.1 5.397 5.397 9.504 9.504 16 140.7 5.801 5.801 8.641 8.641 21 93.8 6.856 6.856 11.786 11.786 26 132.3 5.409 5.476 7.443 7.443 31 114.7 7.517 7.517 10.610 10.610 AVERAGE 111.5 6.490 7.426 9.407 9.407 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 13753.826276 LEG E/M time start vinfx vinfy vinfz time dv dvx dvy dxz (days) (km/s) (km/s) (km/s) (days) (km/s) (km/s) (km/s) 1 E 0.616339349E+02 0.996884630E+01 0.825985267E+01 0.814270461E+00 0.752269337E+02 -0.194388019E-08 -0.223774248E-07 0.631179919E-09 2 M 0.152253927E+03 0.119460633E+01 0.725169794E+01 -0.137178077E+01 0.238724356E+03 0.614033205E-10 -0.468857952E-10 0.938170584E-11 3 E 0.692694108E+03 -0.465458931E+01 0.626650613E+01 0.463957356E+00 0.930119405E+03 -0.196649959E-06 -0.478659934E-06 0.602862492E-04 4 E 0.105796380E+04 -0.285640386E+01 0.391311679E+01 0.612638036E+01 0.133159276E+04 -0.315832603E-08 0.593544819E-08 -0.917420633E-12 5 E 0.197006035E+04 -0.283162182E+01 0.745822839E+01 -0.291054330E+00 0.209424774E+04 0.111557476E-06 0.516602844E-07 -0.103256002E-04 6 E 0.233531738E+04 -0.667064881E+01 0.431088248E+01 0.555944919E+00 0.235256591E+04 -0.714430230E-08 0.677516481E-09 -0.816696569E-10 7 M 0.245030762E+04 -0.101288664E+02 -0.223924554E+01 0.501280779E+00 0.253526573E+04 0.157512794E-09 0.275899943E-09 -0.214090304E-10 8 E 0.301669499E+04 -0.114326580E+01 -0.433868856E+01 0.300482673E+01 0.327236403E+04 0.306250433E-04 0.109739959E-05 0.479606954E-03 9 E 0.338193648E+04 -0.188535548E+01 -0.504531937E+01 -0.249322174E-02 0.366541589E+04 -0.797965423E-07 -0.839630562E-06 -0.122258472E-09 10 E 0.429638619E+04 -0.213326661E+01 -0.475308600E+01 -0.141546928E+01 0.441326237E+04 0.158034281E-04 -0.726208863E-05 0.368864716E-03 11 E 0.466162428E+04 0.466799592E+01 -0.248097777E+01 -0.108636865E+01 0.467559647E+04 -0.745879523E-06 -0.827895684E-06 -0.156175519E-07 12 M 0.475477222E+04 0.792247025E+01 -0.524087168E+01 -0.296504134E+00 0.484676611E+04 -0.141331635E-08 -0.368816735E-09 0.161435030E-09 13 E 0.536806485E+04 0.329980898E+01 0.449530037E+01 -0.153162748E+01 0.562009431E+04 -0.233756306E-05 -0.227972104E-06 -0.928918413E-04 14 E 0.573332494E+04 0.383318843E+01 0.435409557E+01 -0.419782787E-01 0.600812717E+04 -0.333235215E-07 -0.544588030E-07 -0.210889895E-10 15 E 0.664933239E+04 0.417734472E+01 0.399138584E+01 -0.186402617E+00 0.676256496E+04 -0.152500882E-06 0.240353244E-07 -0.990681744E-05 16 E 0.701459875E+04 -0.195267599E+01 0.534297759E+01 0.113485208E+01 0.703570740E+04 0.477231012E-08 -0.138304106E-06 -0.869518007E-10 17 M 0.715532305E+04 -0.699767776E+01 0.501852353E+01 -0.720172141E+00 0.726399615E+04 0.107072226E-09 -0.231870366E-10 0.152582407E-10 18 E 0.769868854E+04 -0.650100720E+01 -0.116340320E+01 -0.172067116E+01 0.793976338E+04 -0.159667375E-05 -0.266205674E-05 0.256949824E-04 19 E 0.806395345E+04 -0.409568322E+01 -0.665710546E+00 0.544181233E+01 0.834584529E+04 -0.515747503E-06 -0.135043147E-06 -0.211338751E-07 20 E 0.897328198E+04 -0.663514048E+01 0.608793047E+00 0.166378662E+01 0.909382006E+04 0.402753314E-06 0.611206913E-05 0.102268285E-03 21 E 0.933854889E+04 -0.519101102E+01 -0.444960643E+01 -0.505952812E+00 0.935261426E+04 -0.162829217E-06 0.359076268E-06 0.264204626E-07 22 M 0.943231800E+04 -0.421673234E+01 -0.109935969E+02 0.530244920E+00 0.963777572E+04 0.282289579E-02 -0.612081636E-02 -0.869289395E-05 23 E 0.100549171E+05 -0.954902139E+00 0.724546515E+00 -0.528157325E+01 0.102156292E+05 0.784600013E-03 -0.125950136E-02 -0.492679071E-04 24 E 0.104201718E+05 0.506573629E+01 -0.108814441E+01 -0.157492539E+01 0.110689033E+05 0.348277253E-02 0.957626160E-02 -0.650430020E-01 25 E 0.113338782E+05 0.258353076E+01 -0.113068155E+01 0.462771262E+01 0.114909404E+05 0.932733972E-04 -0.232521312E-03 -0.112805181E-04 26 E 0.116991391E+05 0.166066615E+01 0.515439625E+01 0.815734383E+00 0.117189815E+05 -0.237966003E-06 -0.665974330E-06 0.344880348E-07 27 M 0.118314213E+05 0.102930289E+01 0.727605363E+01 -0.118297649E+01 0.119369283E+05 -0.390905925E-09 0.796047474E-09 -0.169990458E-10 28 E 0.123867209E+05 -0.493047519E+01 0.550818385E+01 -0.624939740E-01 0.126278046E+05 -0.941387405E-08 -0.445354028E-07 -0.378410162E-04 29 E 0.127519992E+05 -0.309520202E+01 0.360059597E+01 0.565790468E+01 0.130164068E+05 0.128851970E-07 -0.176896720E-07 0.145042311E-09 30 E 0.136637496E+05 -0.338347867E+01 0.672873711E+01 0.281760299E+00 0.137879369E+05 0.132622742E-05 0.709009046E-06 0.147669706E-03 31 E 0.140290063E+05 -0.685618441E+01 0.305057198E+01 0.434746053E+00 0.140462096E+05 0.341920830E-07 -0.899844846E-08 -0.237476852E-08 32 M 0.141436946E+05 -0.997323084E+01 -0.356109839E+01 0.658911253E+00 0.142306014E+05 0.996989667E-11 -0.313663200E-10 0.177906544E-11 33 E 0.147230730E+05 0.192068525E+00 -0.496985673E+01 0.690267799E+00 0.149751197E+05 -0.417648121E-05 -0.276798358E-05 -0.395818523E-03 34 E 0.150883581E+05 -0.252519993E+00 -0.502267113E+01 -0.122027929E-01 0.154170183E+05 -0.322951573E-08 -0.531939455E-09 0.188860122E-10 35 E 0.160013031E+05 -0.623901152E+00 -0.498153731E+01 -0.342039241E+00 0.161145356E+05 0.272384672E-06 -0.312414037E-07 0.228114839E-04 36 E 0.163665691E+05 ================ PARENT CYCLER 5.658Gfh-f3 ======================= Parent cycler number 155 Approximate search space (synodic periods after J2000) 1 229 Number of steps to walk eccentricity/inclination 243 / 243 Number of cycles 7 Total delta v over 44.72 years (km/s) 0.019589 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 1 68.4 ****** 9.614 8.538 8.538 6 132.7 6.603 6.605 8.901 8.901 11 88.8 6.866 6.865 12.676 12.676 16 169.1 7.842 7.841 7.391 7.391 21 113.5 7.117 7.117 11.022 11.022 26 115.0 5.837 5.837 8.625 8.625 31 136.9 6.216 6.216 8.913 8.913 AVERAGE 117.8 6.747 7.156 9.438 9.438 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 1274.401875 LEG E/M time start vinfx vinfy vinfz (days) (km/s) (km/s) (km/s) 1 E 0.358979740E+02 0.785351490E+01 -0.546175490E+01 -0.959996855E+00 2 M 0.104317685E+03 0.846284597E+01 -0.103710397E+01 -0.453968754E+00 3 E 0.695967871E+03 0.556091035E+01 0.154896854E+00 -0.328236555E+01 4 E 0.142647240E+04 -0.636414625E+00 0.301195492E+00 -0.640745764E+01 5 E 0.161156290E+04 -0.487573262E+01 0.774946689E+00 0.439928694E+01 6 E 0.234207286E+04 -0.222525482E+01 0.612714006E+01 0.106440765E+01 7 M 0.247477841E+04 -0.775173998E+01 0.432979735E+01 -0.630827813E+00 8 E 0.301706875E+04 -0.403923177E+01 0.413008503E+01 -0.363575779E+01 9 E 0.374758221E+04 0.106006444E+00 -0.787426074E+00 -0.678685633E+01 10 E 0.392634201E+04 0.801023292E+00 -0.463539508E+01 0.498168738E+01 11 E 0.465685283E+04 -0.396216592E+01 -0.557429657E+01 -0.599002223E+00 12 M 0.474561194E+04 -0.395032551E+01 -0.120007530E+02 0.103371183E+01 13 E 0.531391610E+04 -0.212532345E+01 -0.584531856E+01 -0.515665986E+01 14 E 0.604443567E+04 0.982874504E+00 0.433055340E+00 -0.801711435E+01 15 E 0.622782820E+04 0.271655689E+01 0.499502990E+01 0.538431248E+01 16 E 0.695833824E+04 0.131190187E+01 0.757083165E+01 0.156344969E+01 17 M 0.712744318E+04 0.143922959E+01 0.718095658E+01 -0.995136125E+00 18 E 0.770257475E+04 0.151537193E+01 0.583902280E+01 -0.333193735E+01 19 E 0.843313508E+04 -0.635985596E+00 -0.485300105E+00 -0.685001380E+01 20 E 0.861415621E+04 -0.421219111E+01 -0.204913048E+01 0.534950103E+01 21 E 0.934466104E+04 -0.684707260E+01 0.190813100E+01 0.359350593E+00 22 M 0.945820478E+04 -0.100848877E+02 -0.441235786E+01 0.566960039E+00 23 E 0.100213980E+05 -0.515988284E+01 -0.348704655E+00 -0.308375488E+01 24 E 0.107519176E+05 0.583063510E+00 -0.176235192E+00 -0.597538116E+01 25 E 0.109325831E+05 0.554690314E+01 0.915383655E+00 0.156762047E+01 26 E 0.116630913E+05 0.566489768E+01 0.108878119E+01 -0.890182428E+00 27 M 0.117780625E+05 0.850142035E+01 -0.136086353E+01 -0.516695280E+00 28 E 0.123888106E+05 0.462072711E+01 -0.184794661E+01 -0.346339733E+01 29 E 0.131193168E+05 -0.584471380E+00 0.221476320E+00 -0.601861667E+01 30 E 0.133041641E+05 -0.338365708E+01 0.456026716E+01 0.253621012E+01 31 E 0.140346965E+05 -0.285006249E+01 0.543168342E+01 0.100572242E+01 32 M 0.141716453E+05 -0.837836757E+01 0.304104027E+01 0.368969767E-01 33 E 0.147319088E+05 -0.202649969E+01 0.466993001E+01 -0.181873764E+01 34 E 0.154624249E+05 0.267958697E+00 -0.398755980E+00 -0.540205078E+01 35 E 0.156414421E+05 0.397859758E+01 -0.343498103E+01 0.104187820E+01 36 E 0.163719124E+05 ================ PARENT CYCLER 5.658Gfh+f3 ======================= Parent cycler number 156 Approximate search space (synodic periods after J2000) 1 Number of steps to walk eccentricity/inclination 81 / 81 Number of cycles 7 Total delta v over 44.74 years (km/s) 0.024816 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 1 69.8 ****** 8.954 8.590 8.589 6 133.1 6.550 6.550 8.900 8.900 11 88.8 6.865 6.864 12.677 12.677 16 169.1 7.842 7.841 7.390 7.390 21 113.5 7.117 7.117 11.023 11.023 26 115.0 5.840 5.840 8.627 8.627 31 137.1 6.207 6.207 8.897 8.897 AVERAGE 118.1 6.737 7.053 9.443 9.443 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 1274.401875 LEG E/M time start vinfx vinfy vinfz (days) (km/s) (km/s) (km/s) 1 E 0.316434779E+02 0.736646228E+01 -0.499707620E+01 -0.972694079E+00 2 M 0.101460671E+03 0.848092442E+01 -0.128489683E+01 -0.448661983E+00 3 E 0.695370293E+03 0.539864793E+01 -0.244897752E+00 0.342077565E+01 4 E 0.142587747E+04 -0.622648242E+00 0.302004770E+00 0.634386441E+01 5 E 0.161100014E+04 -0.265729199E+01 0.533606799E+01 -0.271029692E+01 6 E 0.234155529E+04 -0.223334810E+01 0.606423663E+01 0.106627086E+01 7 M 0.247469389E+04 -0.774816509E+01 0.433397310E+01 -0.630582782E+00 8 E 0.301705138E+04 -0.425434751E+01 0.409687189E+01 0.342041313E+01 9 E 0.374757351E+04 0.105781831E+00 -0.786510308E+00 0.678548026E+01 10 E 0.392633200E+04 0.171132267E+01 -0.451955641E+01 -0.485684802E+01 11 E 0.465684152E+04 -0.396092834E+01 -0.557395734E+01 -0.598944229E+00 12 M 0.474560804E+04 -0.395091309E+01 -0.120007992E+02 0.103380251E+01 13 E 0.531390884E+04 -0.217328503E+01 -0.573820485E+01 0.525696968E+01 14 E 0.604442518E+04 0.983260292E+00 0.433791234E+00 0.801891576E+01 15 E 0.622781748E+04 0.231588549E+01 0.587435387E+01 -0.463015974E+01 16 E 0.695833129E+04 0.131232320E+01 0.757053822E+01 0.156404868E+01 17 M 0.712744886E+04 0.143845494E+01 0.718036114E+01 -0.994666449E+00 18 E 0.770260354E+04 0.215067146E+01 0.502037606E+01 0.419859344E+01 19 E 0.843312728E+04 -0.635564789E+00 -0.484104427E+00 0.685023351E+01 20 E 0.861414865E+04 -0.411829487E+01 -0.217464507E+01 -0.537207358E+01 21 E 0.934465786E+04 -0.684692119E+01 0.190770082E+01 0.359389171E+00 22 M 0.945820086E+04 -0.100856371E+02 -0.441200088E+01 0.566687302E+00 23 E 0.100213666E+05 -0.549684695E+01 -0.146445793E+01 0.198059123E+01 24 E 0.107518550E+05 0.583805018E+00 -0.176415230E+00 0.598041987E+01 25 E 0.109325150E+05 0.546590506E+01 0.655577075E+00 -0.194613946E+01 26 E 0.116630235E+05 0.566751511E+01 0.108903100E+01 -0.890673292E+00 27 M 0.117780273E+05 0.850229207E+01 -0.136433964E+01 -0.516302742E+00 28 E 0.123887732E+05 0.542366658E+01 0.978947549E-01 0.271219571E+01 29 E 0.131192832E+05 -0.583889608E+00 0.222715682E+00 0.601707545E+01 time dv (days) 0.461609307E+02 0.193065213E+03 0.114888068E+04 0.145423598E+04 0.185993629E+04 0.236197870E+04 0.258323648E+04 0.347729223E+04 0.378869696E+04 0.424776677E+04 0.467016670E+04 0.498429969E+04 0.574492264E+04 0.607194455E+04 0.668074442E+04 0.698370398E+04 0.721371291E+04 0.825780060E+04 0.846028825E+04 0.887713795E+04 0.936169260E+04 0.975106525E+04 0.102916903E+05 0.107790175E+05 0.114804643E+05 0.116803370E+05 0.118757822E+05 0.126737080E+05 0.131470439E+05 0.134648812E+05 0.140552388E+05 0.145358166E+05 0.150095049E+05 0.154892775E+05 0.157729267E+05 dvx (km/s) 0.196757279E-03 -0.488977769E-06 0.489791939E-04 0.209420227E-04 0.912343612E-03 -0.225732439E-04 0.176547476E-07 -0.120465080E-03 0.548396994E-07 0.271504469E-03 0.519121144E-07 -0.281430508E-10 0.154006300E-03 0.182143719E-06 -0.815935717E-04 -0.158655083E-06 -0.188482302E-09 0.134904557E-03 0.450088932E-06 0.487167908E-04 -0.734049399E-07 -0.268864470E-08 0.440530130E-03 0.132000991E-05 0.267291577E-04 0.201054279E-05 -0.186621916E-08 -0.439125946E-03 -0.155354759E-06 0.536776298E-04 0.473375634E-06 -0.866562024E-07 -0.407771287E-03 0.171638687E-04 -0.194096616E-04 dvy (km/s) 0.594971485E-04 -0.583816194E-06 0.138197329E-04 -0.110182625E-04 0.231782834E-03 0.931601318E-05 0.734571736E-08 0.174792463E-04 -0.379571658E-07 -0.200293399E-03 0.908519813E-07 -0.608643178E-10 -0.850779119E-03 -0.430360791E-07 0.425332496E-03 0.497236610E-07 0.482435529E-09 0.109751994E-03 -0.188618594E-06 0.131381548E-03 0.406868157E-06 -0.603902104E-08 0.242457121E-03 -0.390104209E-05 -0.524076727E-04 0.247765819E-05 0.290716235E-08 -0.149283803E-03 0.172800246E-06 -0.185117790E-04 -0.339762366E-06 -0.423724883E-07 0.115838927E-03 0.532599424E-04 -0.830732407E-05 dxz (km/s) -0.984736863E-05 -0.424407321E-07 -0.152501354E-04 -0.528129556E-05 0.135403188E-03 -0.148014955E-05 0.840421782E-09 0.312350097E-05 0.106964748E-07 0.900046990E-04 -0.778635765E-08 -0.174338854E-11 0.160936176E-03 0.406532639E-07 -0.817356092E-04 0.752862363E-09 -0.334070074E-10 0.170906705E-02 -0.581825760E-07 -0.699021901E-04 0.666574746E-08 0.267816014E-08 -0.569887848E-04 0.490459696E-06 0.169651867E-03 -0.211780903E-06 0.244037195E-09 -0.383218560E-04 0.435431773E-07 0.692989219E-03 -0.395416049E-07 0.717367064E-07 -0.696618033E-04 -0.620833089E-05 -0.640715110E-03 time dv (days) 0.421160569E+02 0.190547114E+03 0.804946369E+03 0.145364587E+04 0.176441672E+04 0.236152608E+04 0.258316539E+04 0.312662970E+04 0.379226313E+04 0.423314600E+04 0.467015649E+04 0.486495121E+04 0.578874446E+04 0.607193403E+04 0.670265146E+04 0.698369893E+04 0.742077775E+04 0.821397016E+04 0.846028048E+04 0.880408104E+04 0.936168931E+04 0.954267572E+04 0.105692329E+05 0.107789540E+05 0.110567014E+05 0.116802741E+05 0.118696392E+05 0.125933160E+05 0.131470110E+05 dvx (km/s) 0.835908002E-03 -0.704204373E-05 -0.158273659E-03 0.999878622E-04 0.482839811E-03 -0.970240531E-04 0.783473021E-07 0.158414593E-03 0.205125806E-06 0.236913742E-04 0.200976858E-06 0.592271384E-09 0.983577993E-05 0.931191931E-05 0.268272792E-03 -0.851189519E-05 -0.845366673E-06 0.658273019E-04 0.435462713E-05 0.970876485E-04 -0.639038179E-06 -0.460761337E-09 -0.530630372E-04 -0.242164945E-05 -0.133149341E-04 -0.419482654E-05 0.478455837E-10 -0.715766045E-03 0.880151521E-05 dvy (km/s) -0.626965977E-03 -0.775474615E-05 -0.267599090E-04 -0.736273423E-04 -0.130892349E-04 0.578510660E-04 0.309226150E-07 -0.162050903E-03 -0.127664762E-06 -0.193901832E-03 0.349757427E-06 -0.293164068E-08 -0.784481905E-03 -0.209540814E-05 0.202325051E-03 0.267621768E-05 0.189903990E-05 0.539628867E-03 -0.184279837E-05 0.176446028E-03 0.357389951E-05 -0.494748269E-09 0.112281201E-03 0.779979905E-05 -0.135741879E-05 -0.500470987E-05 -0.202038352E-07 -0.193178922E-03 -0.999039941E-05 dxz (km/s) -0.747298559E-04 -0.379477569E-06 -0.729081451E-06 0.269594396E-04 -0.116661969E-02 -0.576192753E-05 0.108746144E-07 0.214087247E-03 -0.365706598E-07 -0.866627013E-04 -0.301797233E-07 -0.549333350E-09 -0.129660737E-03 -0.209327178E-05 0.364635838E-03 0.396353306E-07 -0.151547432E-06 -0.421987015E-03 0.561075617E-06 0.613728320E-04 0.586594146E-07 -0.226219880E-09 0.519525550E-03 0.934628236E-06 -0.958059788E-04 0.426528682E-06 -0.762368692E-09 -0.117911088E-05 0.248844934E-05 230 30 E 0.133041353E+05 -0.411810836E+01 0.267786320E+01 -0.380703397E+01 0.135159837E+05 0.535396300E-03 0.625696794E-04 0.115797102E-04 31 E 0.140346470E+05 -0.284163909E+01 0.542604415E+01 0.100527175E+01 0.140552175E+05 -0.968059762E-05 0.715053706E-05 0.816032113E-06 32 M 0.141717835E+05 -0.836957583E+01 0.301520184E+01 0.850842210E-01 0.145375999E+05 -0.799298547E-06 -0.300690345E-06 0.542754166E-06 33 E 0.147345780E+05 -0.185456554E+01 0.489102026E+01 0.800519094E+00 0.148441495E+05 0.142621070E-03 -0.789846141E-04 -0.201692096E-03 34 E 0.154650546E+05 0.272184295E+00 -0.374870546E+00 0.528519791E+01 0.154919174E+05 0.109508801E-04 -0.578659827E-03 -0.839060167E-04 35 E 0.156441400E+05 0.255455910E+01 -0.443363698E+01 -0.110009960E+01 0.157537156E+05 -0.597678092E-03 0.423161440E-03 0.296888802E-04 36 E 0.163746442E+05 ================ PARENT CYCLER 5.658Gfh-ff3 ======================= Parent cycler number 157 Approximate search space (synodic periods after J2000) 17 Number of steps to walk eccentricity/inclination 27 / 27 Number of cycles 7 Total delta v over 44.73 years (km/s) 0.018009 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 1 97.0 ****** 9.708 7.412 7.412 7 115.7 7.823 7.823 10.394 10.394 13 96.3 5.333 5.333 9.528 9.528 19 140.3 5.835 5.835 8.646 8.646 25 93.6 6.864 6.864 11.849 11.849 31 129.0 5.538 5.538 7.438 7.438 37 113.4 7.627 7.627 10.735 10.735 AVERAGE 112.2 6.503 6.961 9.429 9.429 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 13753.826276 LEG E/M time start vinfx vinfy vinfz time dv dvx dvy dxz (days) (km/s) (km/s) (km/s) (days) (km/s) (km/s) (km/s) 1 E 0.458762234E+02 0.742231840E+01 0.621612328E+01 0.719232916E+00 0.604280498E+02 -0.403622041E-07 -0.111815073E-06 0.526502658E-08 2 M 0.142888399E+03 0.171042462E+01 0.710795761E+01 -0.122228492E+01 0.241675686E+03 -0.281332958E-09 0.208608101E-09 0.312332227E-10 3 E 0.691706658E+03 0.283400119E+01 0.449547108E+01 -0.539473189E+01 0.114462511E+04 -0.971850360E-04 0.423144745E-03 0.101339198E-03 4 E 0.142222028E+04 -0.874861424E+00 -0.444582386E+00 -0.751954625E+01 0.144946762E+04 0.200153265E-07 0.389270399E-08 -0.512305869E-08 5 E 0.160386918E+04 -0.763325197E+00 0.370031390E+01 0.686201631E+01 0.187780835E+04 0.315797046E-03 -0.939341192E-03 -0.247050092E-03 6 E 0.196912141E+04 -0.267612563E+01 0.736737863E+01 0.290341902E+00 0.209330790E+04 -0.345328799E-06 -0.120536448E-06 -0.375549847E-04 7 E 0.233437577E+04 -0.657401636E+01 0.420311956E+01 0.557807241E+00 0.235172985E+04 -0.201663299E-07 0.602572288E-08 0.865290515E-10 8 M 0.245006961E+04 -0.101434191E+02 -0.221629490E+01 0.489358182E+00 0.253489439E+04 0.159319941E-09 0.294513724E-09 -0.209371447E-10 9 E 0.301556815E+04 -0.380791529E+01 0.377943354E+01 -0.100792168E+01 0.316896572E+04 -0.198357587E-04 0.777177008E-05 0.695467900E-03 10 E 0.374603278E+04 0.449314239E+00 -0.181634050E+00 -0.544687828E+01 0.381817856E+04 0.836820880E-07 0.554950609E-06 -0.250858448E-07 11 E 0.392639723E+04 -0.251427937E+01 -0.467889747E+01 0.339669868E+00 0.404328335E+04 0.591630669E-06 0.988733667E-07 0.211313805E-04 12 E 0.429166634E+04 -0.202851218E+01 -0.348369036E+01 -0.347509462E+01 0.440123856E+04 0.550680985E-04 -0.278129591E-04 0.510070278E-03 13 E 0.465690706E+04 0.470123571E+01 -0.227081260E+01 -0.108599981E+01 0.467135775E+04 -0.651410854E-06 0.258214995E-06 -0.542507887E-07 14 M 0.475324501E+04 0.786192335E+01 -0.537508382E+01 -0.300909427E+00 0.484550014E+04 -0.103226872E-08 -0.255993218E-09 0.119412557E-09 15 E 0.536827921E+04 0.379662305E+01 -0.328388336E+01 -0.274740449E+01 0.563126205E+04 -0.142728040E-03 0.132575651E-04 -0.209784115E-04 16 E 0.609878712E+04 -0.429618832E+00 0.364185738E+00 -0.567728566E+01 0.612665386E+04 0.434700061E-08 0.791784385E-07 0.948247619E-08 17 E 0.628456545E+04 0.842705706E+00 0.140965727E+00 0.575231609E+01 0.641240619E+04 0.279647899E-03 -0.721476854E-04 0.218208326E-04 18 E 0.664982470E+04 0.398475600E+01 0.382795621E+01 0.181562615E+01 0.676305687E+04 0.127156554E-04 -0.130292333E-04 0.294044337E-03 19 E 0.701508976E+04 -0.193425817E+01 0.538762464E+01 0.113213687E+01 0.703613323E+04 0.876328455E-08 -0.711173959E-07 -0.534014864E-09 20 M 0.715537956E+04 -0.700193556E+01 0.501951903E+01 -0.726809245E+00 0.726944691E+04 0.141975958E-10 0.122317240E-10 -0.122616915E-10 21 E 0.769855741E+04 -0.408382835E+01 0.451859156E+01 -0.311261912E+01 0.825374360E+04 0.681609301E-05 -0.123860009E-04 -0.226526359E-03 22 E 0.842906556E+04 0.311929213E-01 -0.798776329E+00 -0.678957113E+01 0.845588890E+04 -0.105543591E-07 -0.545285229E-08 0.129328574E-08 23 E 0.860788784E+04 -0.196398593E+01 0.760025921E+00 0.654934093E+01 0.879051530E+04 0.154145501E-02 -0.993675657E-03 0.314517422E-03 24 E 0.897314276E+04 -0.596784546E+01 0.642003854E+00 0.335179820E+01 0.909002838E+04 0.213488487E-05 0.389710171E-04 0.322784570E-03 25 E 0.933841033E+04 -0.517878525E+01 -0.447725260E+01 -0.503260915E+00 0.935244359E+04 0.338839752E-08 -0.862290576E-08 -0.641077686E-09 26 M 0.943196541E+04 -0.429061099E+01 -0.110327887E+02 0.519502315E+00 0.952569611E+04 -0.665817146E-10 0.127239360E-09 0.205139563E-11 27 E 0.100568367E+05 0.279134585E+00 -0.507609096E+01 -0.239465151E+01 0.103636508E+05 0.264990887E-03 -0.336101940E-03 0.210896007E-04 28 E 0.107873463E+05 0.101304916E+00 0.525268341E+00 -0.559627863E+01 0.108190012E+05 -0.529404919E-07 0.114609273E-06 0.197322236E-07 29 E 0.109735515E+05 0.365446507E+01 -0.110272506E+01 0.402968730E+01 0.111342633E+05 -0.600770044E-03 0.134677423E-02 0.103512999E-03 30 E 0.113388056E+05 0.534938263E+01 -0.134877982E+01 0.480203123E+00 0.114520398E+05 0.356166374E-08 -0.341537771E-06 0.913556952E-05 31 E 0.117040773E+05 0.198887863E+01 0.510435054E+01 0.813348507E+00 0.117234276E+05 0.150867277E-06 -0.649592503E-07 0.116654276E-07 32 M 0.118330792E+05 0.929030071E+00 0.728343698E+01 -0.119041843E+01 0.119383878E+05 0.286951152E-09 -0.576699001E-09 0.913849327E-10 33 E 0.123873350E+05 0.266628540E+01 0.429488019E+01 -0.538743954E+01 0.128840872E+05 -0.226305680E-03 0.654426101E-03 0.197174285E-03 34 E 0.131178530E+05 -0.775392826E+00 -0.525055263E+00 -0.733729226E+01 0.131450338E+05 0.523441709E-06 0.262305191E-06 -0.149663920E-06 35 E 0.132990588E+05 -0.284618346E+01 0.588812096E+01 0.392886220E+01 0.134195863E+05 -0.112611480E-03 -0.638042059E-04 -0.916835910E-03 36 E 0.136642938E+05 -0.225884463E+01 0.500479013E+01 0.531451563E+01 0.137775237E+05 0.151859474E-05 0.139832617E-05 0.571629627E-05 37 E 0.140295514E+05 -0.696833051E+01 0.306835307E+01 0.441846787E+00 0.140465663E+05 -0.670134713E-06 0.110471448E-06 0.476543861E-07 38 M 0.141429837E+05 -0.101104406E+02 -0.352808569E+01 0.752515219E+00 0.142861295E+05 -0.115063851E-09 0.557652861E-10 0.872939476E-11 39 E 0.147394248E+05 -0.459158770E+01 -0.205126521E+01 -0.191320903E+01 0.151265979E+05 -0.125304649E-03 -0.533229084E-03 0.582238084E-04 40 E 0.154699400E+05 0.451640989E+00 0.115958596E+00 -0.536391242E+01 0.155448330E+05 -0.983060184E-07 0.305831160E-06 -0.422282186E-08 41 E 0.156526059E+05 0.775545382E+00 -0.510288069E+01 0.699784803E+00 0.157658361E+05 -0.169014132E-05 0.454781384E-08 0.899904396E-04 42 E 0.160178644E+05 0.755857045E+00 -0.511705197E+01 -0.670916604E+00 0.161310873E+05 -0.162038928E-06 -0.449153997E-07 -0.537029213E-05 43 E 0.163830997E+05 ================ PARENT CYCLER 5.658Gfh+ff3 ======================= Parent cycler number 158 Approximate search space (synodic periods after J2000) 7 Number of steps to walk eccentricity/inclination 27 / 27 Number of cycles 7 Total delta v over 44.96 years (km/s) 0.015256 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 1 86.7 ****** 7.435 11.855 11.855 7 128.5 5.767 5.767 7.566 7.566 13 107.7 7.753 7.753 11.502 11.502 19 80.9 6.925 6.925 8.223 8.223 25 131.5 6.730 6.730 9.113 9.113 31 90.0 6.353 6.353 11.632 11.632 37 139.2 5.552 5.552 7.096 7.096 AVERAGE 109.2 6.513 6.645 9.569 9.569 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 5954.186026 LEG E/M time start vinfx vinfy vinfz time dv dvx dvy dxz (days) (km/s) (km/s) (km/s) (days) (km/s) (km/s) (km/s) 1 E -0.148665529E+02 -0.489587058E+01 -0.555711866E+01 -0.658906708E+00 -0.186888927E+01 -0.102305459E-03 0.288618590E-05 0.525213600E-07 2 M 0.717845381E+02 -0.271548352E+01 -0.115335403E+02 0.385751427E+00 0.229357481E+03 -0.238523908E-07 -0.203087506E-07 -0.988341960E-10 3 E 0.702076308E+03 -0.255992453E+01 -0.501021274E+01 0.151010304E+01 0.870093855E+03 -0.528391834E-05 0.921658271E-05 0.521878340E-04 4 E 0.143258738E+04 -0.245340660E-01 0.576714927E+00 0.578071615E+01 0.146055535E+04 0.309523636E-05 0.736493581E-06 -0.113596858E-06 5 E 0.161904047E+04 0.527859178E+01 -0.209750253E+00 -0.233312766E+01 0.172861714E+04 0.467572845E-05 0.424199320E-05 0.172316921E-03 6 E 0.198429605E+04 0.574641074E+01 -0.176175867E+00 -0.262667192E+00 0.207561451E+04 -0.124481480E-05 0.497796900E-05 0.748039906E-05 7 E 0.234956988E+04 0.184660716E+01 0.538427341E+01 0.924457160E+00 0.236884027E+04 -0.405824083E-05 -0.107543337E-05 0.141818850E-06 8 M 0.247803914E+04 -0.290614862E+00 0.744854760E+01 -0.129286852E+01 0.258777676E+04 0.900012168E-09 -0.238038475E-08 0.127929233E-09 9 E 0.302672725E+04 0.198958269E+01 0.470392853E+01 0.552483185E+01 0.353809024E+04 -0.167170789E-03 0.621828508E-03 -0.240746393E-03 10 E 0.375724581E+04 -0.730547372E+00 -0.625945288E+00 0.747413421E+01 0.378436495E+04 -0.217760565E-07 -0.187290875E-08 -0.477397672E-08 11 E 0.393804005E+04 -0.132978384E+01 0.307655801E+01 -0.699972035E+01 0.420467615E+04 -0.115608049E-04 0.516948444E-04 -0.242623038E-04 12 E 0.430329498E+04 -0.421939476E+01 0.642587762E+01 -0.112088250E+01 0.441652299E+04 0.173360736E-06 0.273403918E-06 0.896054114E-06 231 13 E 0.466854663E+04 -0.746681951E+01 0.205236489E+01 0.383917856E+00 14 M 0.477620876E+04 -0.106151710E+02 -0.434642999E+01 0.851548030E+00 15 E 0.539517548E+04 -0.408480622E+01 -0.217261527E+01 0.546630863E+01 16 E 0.612569252E+04 0.687348079E+00 0.512234199E+00 0.710677215E+01 17 E 0.630990987E+04 0.321049583E+01 -0.551328852E+01 -0.273575540E+01 18 E 0.667515420E+04 0.354082555E+01 -0.590682982E+01 0.798645680E+00 19 E 0.704043059E+04 0.674480700E+01 -0.140763013E+01 -0.696992142E+00 20 M 0.712132233E+04 0.818396261E+01 0.573867133E+00 -0.556530800E+00 21 E 0.771375901E+04 0.538117248E+01 0.820622049E+00 0.359903862E+01 22 E 0.844426548E+04 -0.684484290E+00 0.205526859E+00 0.648436434E+01 23 E 0.862885818E+04 0.252905117E+01 0.580084513E+01 -0.224571994E+01 24 E 0.899412357E+04 0.264971943E+01 0.612945206E+01 0.810384127E+00 25 E 0.935936664E+04 -0.294789027E+01 0.596685213E+01 0.997861901E+00 26 M 0.949091467E+04 -0.849864716E+01 0.327357461E+01 -0.320729904E+00 27 E 0.100375133E+05 -0.498479507E+01 0.324518189E+01 0.235592703E+01 28 E 0.107680013E+05 0.244160426E+00 -0.648780955E+00 0.634956058E+01 29 E 0.109467754E+05 -0.163774811E+01 0.159566503E+00 -0.614524344E+01 30 E 0.113120354E+05 -0.593191903E+01 -0.135580288E+01 -0.183971962E+01 31 E 0.116773198E+05 -0.257192803E+01 -0.575443742E+01 -0.797223359E+00 32 M 0.117672907E+05 -0.725155443E+00 -0.116063116E+02 0.258330383E+00 33 E 0.123985354E+05 0.301782352E+01 -0.453401760E+01 0.116895986E+01 34 E 0.131290105E+05 -0.103344508E+00 0.522974218E+00 0.553280775E+01 35 E 0.133155151E+05 0.547951159E+01 0.635785354E+00 -0.628975368E+00 36 E 0.136807831E+05 0.547919753E+01 0.629492332E+00 -0.429913149E+00 37 E 0.140460581E+05 0.126651754E+01 0.529872236E+01 0.107208474E+01 38 M 0.141852514E+05 -0.216877853E+01 0.674557839E+01 -0.380411559E+00 39 E 0.147661630E+05 -0.114416492E+01 0.505236786E+01 0.692250413E+00 40 E 0.154965783E+05 0.168106561E-01 -0.459983888E+00 0.519066200E+01 41 E 0.156754222E+05 -0.406419274E+01 0.475126700E+00 -0.328754547E+01 42 E 0.160406902E+05 -0.521294614E+01 0.474745590E+00 -0.373139274E-01 43 E 0.164059717E+05 ================ PARENT CYCLER 5.658Gfh-ff3 ======================= Parent cycler number 159 Approximate search space (synodic periods after J2000) 21 Number of steps to walk eccentricity/inclination 9 / 9 Number of cycles 7 Total delta v over 44.92 years (km/s) 0.032760 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 1 90.2 ****** 9.447 12.095 12.095 7 143.4 7.194 7.194 7.658 7.658 13 127.2 6.739 6.739 9.865 9.865 19 90.0 5.542 5.542 10.530 10.530 25 145.9 5.456 5.456 8.204 8.204 31 98.6 7.270 7.270 11.676 11.676 37 136.9 5.306 5.306 6.784 6.784 AVERAGE 118.9 6.251 6.708 9.545 9.545 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 16873.682377 LEG E/M time start vinfx vinfy vinfz (days) (km/s) (km/s) (km/s) 1 E -0.525608468E+01 -0.937404170E+01 -0.117055366E+01 -0.518175452E-01 2 M 0.849678661E+02 -0.867720433E+01 -0.838733659E+01 0.798122757E+00 3 E 0.642097000E+03 0.210993025E+00 -0.667265296E+01 -0.324417968E+01 4 E 0.100736659E+04 0.895183150E+00 -0.914399029E-01 -0.735092501E+01 5 E 0.118869829E+04 -0.334968950E+00 0.451871423E+01 0.556498470E+01 6 E 0.155395651E+04 0.531059068E+01 0.478428110E+01 0.721021647E+00 7 E 0.228446861E+04 0.515625817E+01 0.500941493E+01 -0.269896414E+00 8 M 0.242790261E+04 0.660465367E+01 0.380231633E+01 -0.753930433E+00 9 E 0.302669529E+04 -0.121273016E+01 0.610464323E+01 -0.192830605E+01 10 E 0.339193656E+04 -0.731492185E+00 -0.663694177E-01 -0.646243061E+01 11 E 0.357513208E+04 0.112627051E+01 -0.445038124E+01 0.491046765E+01 12 E 0.394039288E+04 -0.470779309E+01 0.128673080E+00 0.481496450E+01 13 E 0.467089856E+04 -0.484710061E+01 0.462010094E+01 0.761648540E+00 14 M 0.479806553E+04 -0.986122107E+01 0.122502368E+00 0.252803820E+00 15 E 0.535700387E+04 -0.305498829E+01 -0.452829157E+01 -0.155608315E+01 16 E 0.572223906E+04 0.416187553E+00 -0.355625303E+00 -0.564668706E+01 17 E 0.590175223E+04 0.196276576E+01 0.306898552E+01 0.420197470E+01 18 E 0.626703005E+04 0.532424005E+01 -0.158507064E+01 -0.230280778E+00 19 E 0.699750284E+04 0.233728444E+01 -0.489939083E+01 -0.111497692E+01 20 M 0.708750279E+04 0.527553729E+01 -0.911277955E+01 0.737349351E-03 21 E 0.770379115E+04 0.448431218E+01 0.266869742E+01 -0.148041721E+01 22 E 0.806906691E+04 -0.215248656E+00 0.432625046E+00 -0.540478995E+01 23 E 0.825548182E+04 -0.427711668E+01 -0.274877566E+01 0.201507378E+01 24 E 0.862072846E+04 -0.418384862E+01 0.345674054E+01 -0.529595794E+00 25 E 0.935124655E+04 -0.132525304E+01 0.514867771E+01 0.122367486E+01 26 M 0.949718095E+04 -0.509028047E+01 0.634947519E+01 -0.103939640E+01 27 E 0.100405790E+05 -0.651235847E+01 0.667714919E+00 -0.283786180E+01 28 E 0.104058453E+05 -0.218279141E+00 -0.830505540E+00 -0.710659870E+01 29 E 0.105850337E+05 0.344643325E+01 -0.274789790E-01 0.639979475E+01 30 E 0.109502913E+05 -0.246874954E+01 -0.406825374E+01 0.549003409E+01 31 E 0.116808033E+05 -0.685533306E+01 -0.240939868E+01 -0.215318437E+00 32 M 0.117794424E+05 -0.694162559E+01 -0.935660818E+01 0.777934571E+00 33 E 0.123639084E+05 0.275602673E+01 -0.418572347E+01 -0.225677692E+01 34 E 0.127291699E+05 0.432847101E+00 0.240542287E+00 -0.545633084E+01 35 E 0.129127718E+05 -0.251287265E+01 0.372687494E+01 0.281825104E+01 36 E 0.132780069E+05 0.388208979E+01 0.362995534E+01 0.518069130E-03 37 E 0.140254094E+05 0.261439323E+01 0.459633763E+01 0.442698806E+00 38 M 0.141623016E+05 0.327143737E+01 0.589538834E+01 -0.754706309E+00 39 E 0.147608055E+05 -0.419674573E+01 0.300684766E+01 -0.537408659E+00 40 E 0.151260821E+05 -0.291146248E+00 -0.339549903E+00 -0.516872409E+01 41 E 0.153064634E+05 -0.135551950E+01 0.183511011E+01 0.481918692E+01 42 E 0.156717222E+05 -0.478177046E+01 -0.236044083E+01 0.856942360E-01 43 E 0.164021356E+05 ================ PARENT CYCLER 5.658gfh-fFzero======================= Parent cycler number 160 Approximate search space (synodic periods after J2000) 2 Number of steps to walk eccentricity/inclination 3 / 3 Number of cycles 7 Total delta v over 44.66 years (km/s) 0.016780 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 0.468469595E+04 0.486905377E+04 0.592845292E+04 0.616069381E+04 0.642678805E+04 0.672994566E+04 0.705256435E+04 0.721018783E+04 0.797674134E+04 0.849595143E+04 0.874574310E+04 0.911100135E+04 0.937909884E+04 0.972048609E+04 0.105926842E+05 0.107948174E+05 0.112134152E+05 0.114325792E+05 0.116908154E+05 0.121397251E+05 0.129683060E+05 0.131569862E+05 0.134287482E+05 0.137647963E+05 0.140669371E+05 0.145512257E+05 0.153431911E+05 0.155234049E+05 0.157850026E+05 0.160954824E+05 0.192064322E-07 -0.258483377E-08 0.201684843E-04 0.510683482E-06 0.447989282E-04 -0.217991884E-05 -0.544640965E-06 -0.221243085E-08 -0.681562416E-05 0.957845803E-07 0.283435293E-04 0.504592133E-06 -0.115236675E-06 -0.150246178E-09 0.148705527E-04 0.366369464E-06 0.718595074E-03 -0.112537310E-04 -0.524047160E-06 0.124263387E-07 -0.412355414E-04 0.416981446E-05 0.798156985E-05 -0.172895696E-05 -0.301580641E-05 -0.507365495E-07 -0.715218370E-05 0.169215148E-04 0.143791523E-04 0.869454312E-06 -0.124309731E-07 -0.426373792E-08 -0.914929280E-04 0.189374574E-05 0.299712987E-04 -0.136752234E-05 -0.319242588E-05 -0.104115185E-07 -0.359841223E-05 0.722142003E-07 -0.114101227E-04 0.747420535E-07 -0.123620467E-06 0.244477408E-09 -0.251958083E-04 0.764718723E-06 0.247795811E-03 0.400681131E-04 0.108952354E-06 -0.178703173E-07 -0.620601270E-04 0.350573965E-05 -0.166338071E-04 0.481702713E-05 -0.388961230E-05 0.120484928E-07 -0.227704497E-04 0.318620715E-04 0.119526716E-04 0.251947872E-04 0.101514546E-08 0.145438115E-09 -0.302693719E-04 -0.375605233E-06 0.503152418E-03 -0.832640556E-06 0.280613806E-06 0.781236164E-10 0.747680819E-06 0.158237992E-07 -0.382675644E-03 0.788129055E-05 -0.986630606E-09 0.211162887E-09 0.496257935E-03 0.128640176E-06 0.126831465E-03 -0.606682650E-03 -0.424923012E-07 0.112916313E-09 0.431195856E-03 -0.490908073E-06 -0.185538078E-03 0.650544924E-05 0.257558145E-06 0.185018521E-07 0.599913801E-03 0.559316467E-05 -0.358385469E-03 0.229335211E-07 time dv (days) 0.827750793E+01 0.196393693E+03 0.864911452E+03 0.103637967E+04 0.136767482E+04 0.166353333E+04 0.230598371E+04 0.251772151E+04 0.327140694E+04 0.341941588E+04 0.371027857E+04 0.421798504E+04 0.468997360E+04 0.488190628E+04 0.560536380E+04 0.574916604E+04 0.616109949E+04 0.637660097E+04 0.701100283E+04 0.719227181E+04 0.795948418E+04 0.809702914E+04 0.831757375E+04 0.873761136E+04 0.937313671E+04 0.983408771E+04 0.102706968E+05 0.104327236E+05 0.107749677E+05 0.112132756E+05 0.116955991E+05 0.118671123E+05 0.124333081E+05 0.127567102E+05 0.131647840E+05 0.133901173E+05 0.140459432E+05 0.142520772E+05 0.148155970E+05 0.152000384E+05 0.154343040E+05 0.157812842E+05 dvx (km/s) 0.968754846E-04 -0.406298914E-06 0.204492401E-03 0.715071324E-04 -0.869429212E-04 0.154053837E-03 0.699323273E-04 -0.200920104E-07 0.855235935E-05 0.227901481E-05 0.418081616E-04 0.209677528E-03 -0.671746731E-06 0.113640288E-08 -0.819804048E-06 0.277980762E-06 0.127688329E-03 0.271130269E-06 0.991712302E-07 0.546337441E-08 0.115381153E-04 0.540975458E-05 0.628764124E-05 -0.447748808E-05 0.710506221E-05 -0.463496495E-07 -0.104797230E-03 -0.459490932E-04 -0.128304322E-02 0.130820365E-03 0.111337850E-04 -0.122812262E-07 0.316292277E-05 -0.904791189E-05 0.453076383E-04 -0.540827263E-05 0.435126950E-05 0.886738152E-08 0.114638798E-05 -0.654131423E-05 0.715716214E-03 0.218154616E-05 dvy (km/s) 0.322504389E-05 0.166729407E-06 -0.203499668E-03 0.353626151E-04 -0.109634392E-02 0.106389276E-03 -0.145837569E-03 -0.497016647E-06 -0.142269076E-04 -0.268004157E-05 0.978012627E-04 0.214914132E-03 -0.202344061E-05 0.134440833E-08 0.205204744E-04 0.819813886E-06 -0.792022504E-04 -0.437796879E-07 -0.473035173E-06 -0.281007573E-08 -0.152996238E-04 0.341650974E-05 -0.503383941E-05 0.253405541E-05 -0.326738976E-07 0.467977517E-07 -0.977923544E-04 -0.251782553E-04 0.274923780E-03 -0.408458208E-03 0.331843732E-04 0.659615771E-07 0.541044019E-05 -0.157628458E-05 -0.317228209E-04 -0.100334394E-04 0.252073577E-06 -0.901783933E-08 0.273055197E-05 0.414105039E-05 0.162816160E-02 0.596751906E-05 dxz (km/s) 0.540227013E-05 0.862640122E-07 0.568228425E-03 0.169288019E-04 -0.594188409E-04 -0.110619199E-03 0.641704893E-05 -0.914732033E-07 -0.367104536E-03 -0.463407296E-06 -0.176353714E-04 -0.433499232E-04 0.342964342E-08 -0.114462639E-09 -0.361853028E-03 -0.609569452E-07 -0.122324150E-02 -0.154607690E-08 -0.845794221E-08 0.818370102E-10 0.461000166E-03 0.124819664E-06 -0.123879522E-04 0.264363740E-04 0.746611043E-07 0.150466902E-06 0.354499845E-03 0.886946558E-05 -0.192164402E-03 0.155716215E-04 0.126667926E-05 -0.386381535E-08 0.178223263E-04 -0.160167442E-05 0.803516143E-03 0.138100431E-06 0.253331446E-06 0.313490293E-09 -0.841485941E-07 0.217460244E-06 0.430220352E-04 -0.160315449E-07 232 5 133.2 6.506 6.506 8.950 8.950 11 93.4 6.452 6.451 11.732 11.732 17 138.9 5.326 5.326 7.038 7.038 23 120.6 6.017 6.017 11.392 11.392 29 82.0 6.661 6.661 8.365 8.365 35 131.6 6.651 6.651 9.198 9.198 41 91.7 6.169 6.168 11.486 11.486 AVERAGE 113.1 6.254 6.254 9.737 9.737 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 2004.475156 LEG E/M time start vinfx vinfy vinfz (days) (km/s) (km/s) (km/s) 1 E 0.149572716E+02 0.597303903E+01 -0.211424046E+01 -0.115989746E-02 2 E 0.695279277E+03 0.210512025E+01 0.540403413E+01 -0.248950141E+01 3 E 0.106053581E+04 -0.592443479E+00 0.313873500E+00 -0.628969670E+01 4 E 0.124568450E+04 0.340108981E+01 0.516578651E+01 0.197045416E+01 5 E 0.161093967E+04 -0.226889152E+01 0.600284115E+01 0.106989987E+01 6 M 0.174409301E+04 -0.783591200E+01 0.430292050E+01 0.428422165E+00 7 E 0.234080119E+04 -0.211185465E+01 0.605870218E+01 0.852908192E-03 8 E 0.301748286E+04 -0.596396836E+01 -0.168554118E+01 -0.184393501E+01 9 E 0.338273303E+04 0.111418125E+00 -0.689711102E+00 -0.640919773E+01 10 E 0.356149508E+04 -0.445389180E+01 0.130422151E+00 0.468467097E+01 11 E 0.392675024E+04 -0.394677076E+01 -0.506465811E+01 -0.626033363E+00 12 M 0.402014535E+04 -0.279628486E+01 -0.113851794E+02 0.442050910E+00 13 E 0.464505700E+04 -0.110836130E+01 -0.490559745E+01 -0.196142070E+01 14 E 0.537543479E+04 0.351521794E+01 0.185385334E+00 -0.410334425E+01 15 E 0.574067483E+04 0.291300641E-01 0.495316762E+00 -0.537115283E+01 16 E 0.592702072E+04 0.527415462E+01 -0.754870730E+00 0.366726355E+00 17 E 0.629229596E+04 0.142845777E+01 0.503708004E+01 0.975723987E+00 18 M 0.643124259E+04 -0.662598646E+00 0.696384224E+01 -0.775374233E+00 19 E 0.699770120E+04 0.496596692E+01 0.299233223E+01 0.485355644E-03 20 E 0.769085115E+04 -0.301452625E+01 0.483525187E+01 -0.111090606E+01 21 E 0.805611124E+04 -0.507840028E+00 -0.246768735E+00 -0.578474776E+01 22 E 0.823787977E+04 -0.129978511E+01 0.407618631E+01 0.421460991E+01 23 E 0.860315576E+04 -0.594264024E+01 0.850430357E+00 0.406335283E+00 24 M 0.872372074E+04 -0.105772281E+02 -0.414547663E+01 0.842884172E+00 25 E 0.934167657E+04 -0.488031631E+01 -0.101443891E+01 -0.476997589E+01 26 E 0.100721946E+05 0.262220714E+01 -0.251452133E+01 -0.586758970E+01 27 E 0.104374504E+05 0.651946445E+00 0.451286708E+00 -0.684723345E+01 28 E 0.106214606E+05 0.291117910E+01 -0.558635748E+01 0.221855418E+01 29 E 0.109867321E+05 0.652512203E+01 -0.112950079E+01 -0.717063732E+00 30 M 0.110687261E+05 0.829148966E+01 0.465220474E+00 -0.100527020E+01 31 E 0.117151616E+05 0.636658074E+01 -0.103859944E+01 -0.106299348E-02 32 E 0.123930468E+05 0.129199826E+01 0.599256325E+01 -0.196074002E+01 33 E 0.127582946E+05 -0.669552274E+00 0.233448066E+00 -0.641691370E+01 34 E 0.129429332E+05 0.261136025E+01 0.572471168E+01 0.213075246E+01 35 E 0.133081760E+05 -0.298948569E+01 0.585531259E+01 0.100433374E+01 36 M 0.134397597E+05 -0.857580185E+01 0.328814506E+01 0.489788031E+00 37 E 0.140343352E+05 -0.288917412E+01 0.544382877E+01 -0.104829005E-02 38 E 0.147163793E+05 -0.536551841E+01 -0.262391007E+01 -0.166172348E+01 39 E 0.150816442E+05 0.209638794E+00 -0.597797631E+00 -0.616334363E+01 40 E 0.152604129E+05 -0.486130385E+01 -0.905649858E+00 0.369968600E+01 41 E 0.156257021E+05 -0.246063543E+01 -0.560037508E+01 -0.794077362E+00 42 M 0.157174296E+05 -0.653477091E+00 -0.114582433E+02 0.456552410E+00 43 E 0.163273882E+05 ================ PARENT CYCLER 5.658gfh-fFpi3 ======================= Parent cycler number 161 Approximate search space (synodic periods after J2000) 11 Number of steps to walk eccentricity/inclination 9 / 9 Number of cycles 7 Total delta v over 44.81 years (km/s) 1.262903 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 5 161.5 6.518 6.518 11.162 11.162 11 124.0 6.480 6.480 8.880 8.880 17 172.8 6.118 6.119 8.895 8.895 23 114.0 6.097 6.115 13.033 13.033 29 182.6 8.291 8.477 7.288 7.292 35 135.8 5.693 5.624 12.048 12.048 41 128.0 7.993 7.992 9.699 9.699 AVERAGE 145.5 6.741 6.760 10.144 10.144 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 8985.275369 LEG E/M time start vinfx vinfy vinfz (days) (km/s) (km/s) (km/s) 1 E -0.264037027E+02 0.597559852E+01 -0.223968569E+01 -0.107657455E-02 2 E 0.652910760E+03 0.111757429E+01 0.338856360E+01 -0.525978440E+01 3 E 0.101816272E+04 -0.590942471E+00 0.329631389E+00 -0.632648489E+01 4 E 0.120338010E+04 -0.699575467E+00 -0.282068846E+01 0.584236959E+01 5 E 0.156862660E+04 -0.629060529E+01 -0.149886807E+01 0.815763040E+00 6 M 0.173016949E+04 -0.107115748E+02 -0.302761968E+01 0.823142051E+00 7 E 0.235120761E+04 -0.613897863E+01 0.243306216E+01 0.586034198E-03 8 E 0.302422806E+04 -0.285630868E+01 -0.560126704E+01 -0.210791203E+01 9 E 0.338945544E+04 0.606035464E+00 -0.424249998E+00 -0.659389805E+01 10 E 0.356922690E+04 0.293985876E+01 0.546930293E+01 0.189710897E+01 11 E 0.393448927E+04 0.634609375E+01 0.805103275E+00 -0.103209571E+01 12 M 0.405846492E+04 0.859241799E+01 -0.219472295E+01 -0.461673137E+00 13 E 0.467256825E+04 0.263158430E+01 -0.551708765E+01 0.895895583E-03 14 E 0.535808485E+04 0.540752196E+01 0.243359571E+01 -0.132020027E+01 15 E 0.572331607E+04 -0.202260279E+00 0.599390841E+00 -0.604807148E+01 16 E 0.590980211E+04 -0.479049732E+01 -0.243585964E+01 0.289012177E+01 17 E 0.627506413E+04 -0.477306813E+01 0.343003530E+01 0.169971339E+01 18 M 0.644789440E+04 -0.800646122E+01 0.386930650E+01 -0.217226892E+00 19 E 0.700766394E+04 0.139063000E+01 0.581418003E+01 0.730391221E-03 20 E 0.769529400E+04 -0.385805897E+01 0.745824320E+00 -0.454976886E+01 21 E 0.806051178E+04 -0.215464476E+00 -0.584527647E+00 -0.596785215E+01 22 E 0.823991760E+04 0.572492040E+01 -0.157351763E+01 0.136417185E+01 23 E 0.860517822E+04 -0.729603186E-01 -0.607821431E+01 -0.662357484E+00 24 M 0.871915484E+04 -0.326760584E+01 -0.126155569E+02 0.199087579E+00 25 E 0.937944540E+04 -0.536056878E+01 -0.661033167E+01 0.164748776E-02 26 E 0.100211823E+05 0.381437779E+01 -0.579544763E+01 -0.489088952E+01 27 E 0.103864377E+05 0.110178924E+01 0.471842484E+00 -0.841806691E+01 28 E 0.105697878E+05 -0.236416946E+01 0.239895176E+01 0.758223562E+01 time dv (days) 0.117005572E+03 0.768330584E+03 0.108830811E+04 0.136256615E+04 0.163091267E+04 0.183359924E+04 0.261824068E+04 0.326220048E+04 0.340954734E+04 0.373681756E+04 0.394075951E+04 0.438884322E+04 0.479843633E+04 0.562379802E+04 0.576862671E+04 0.603660329E+04 0.631313796E+04 0.672013649E+04 0.710167370E+04 0.793922801E+04 0.808337652E+04 0.834746257E+04 0.862124051E+04 0.884731191E+04 0.962657861E+04 0.103169160E+05 0.104650519E+05 0.107383475E+05 0.109990312E+05 0.111656915E+05 0.118168444E+05 0.126377628E+05 0.127859904E+05 0.130598109E+05 0.133279136E+05 0.135884036E+05 0.141434622E+05 0.149574541E+05 0.151549393E+05 0.153736526E+05 0.156394613E+05 0.160407077E+05 dvx (km/s) -0.110201837E-04 -0.377661086E-05 0.730102957E-06 0.190620842E-04 0.159204652E-05 -0.141638999E-08 -0.246610594E-07 0.407482148E-05 -0.116512227E-06 -0.269991257E-03 -0.256703077E-06 0.202418994E-08 -0.490183986E-04 -0.339734177E-03 -0.638636935E-04 -0.100975798E-04 0.205632834E-05 -0.906684063E-08 -0.196851804E-05 0.101355651E-04 0.131947791E-05 0.246673912E-03 0.156550548E-05 -0.487706017E-07 -0.172122865E-03 -0.306213609E-03 -0.372345759E-04 0.155235064E-05 -0.242241946E-04 -0.141791904E-07 0.163800060E-05 0.233071912E-04 0.304590524E-05 -0.137938820E-04 0.259641456E-05 0.133907865E-08 0.583249659E-06 -0.240070747E-06 0.506109864E-05 -0.752076971E-04 0.689654581E-05 0.134553042E-05 dvy (km/s) -0.490037534E-05 0.355546773E-06 -0.289733096E-05 -0.126037395E-04 -0.121594273E-05 -0.609989031E-09 0.415281413E-08 -0.653899253E-05 -0.595389753E-06 0.286004383E-03 -0.436382417E-06 -0.128833220E-08 -0.406106926E-03 -0.323078364E-03 -0.482275819E-04 0.283964448E-04 -0.525044746E-04 0.244332737E-07 -0.563298164E-05 0.256223427E-05 -0.116220515E-05 0.108985949E-03 -0.150397530E-05 -0.573368543E-07 -0.401799264E-03 0.463925170E-03 0.484117091E-05 0.123563861E-04 -0.110764337E-04 -0.797770630E-07 0.377402882E-05 -0.258010706E-04 -0.211370645E-05 0.627066861E-05 0.156214583E-05 -0.253764789E-08 0.784635365E-06 0.181131340E-04 0.227913992E-05 0.244631300E-03 -0.428869738E-05 -0.793847788E-06 dxz (km/s) 0.878575276E-07 -0.145673392E-04 -0.433758389E-06 0.346769545E-03 0.880421418E-07 0.892291802E-11 -0.395692079E-08 -0.185603447E-03 0.126433127E-06 -0.705193493E-04 0.366550424E-07 -0.327680732E-09 0.292672159E-02 -0.103223032E-02 -0.961938074E-05 0.278435951E-03 -0.359986749E-05 0.262162876E-08 -0.646448962E-07 0.921860976E-04 -0.136997757E-06 0.175517947E-02 0.631968596E-07 0.331115446E-08 -0.179294808E-03 -0.172285838E-03 -0.653428225E-05 0.182740853E-03 0.188403354E-05 0.100357266E-08 -0.204955786E-06 -0.147767243E-03 -0.796106318E-06 -0.212410011E-03 -0.413960344E-07 -0.105837201E-09 0.227479999E-06 0.529494122E-04 -0.111314890E-06 0.173538662E-02 0.577026249E-06 -0.319005522E-06 time dv (days) 0.754934667E+02 0.839189258E+03 0.104594532E+04 0.139696074E+04 0.159285803E+04 0.195995359E+04 0.245216068E+04 0.326893041E+04 0.343260059E+04 0.379934219E+04 0.395308562E+04 0.415058042E+04 0.477539574E+04 0.543113109E+04 0.579045104E+04 0.614356980E+04 0.630098867E+04 0.682853768E+04 0.711080845E+04 0.793633773E+04 0.808742265E+04 0.829470669E+04 0.862227471E+04 0.900968268E+04 0.947570593E+04 0.102476406E+05 0.104139402E+05 0.108291304E+05 dvx (km/s) -0.246079604E-03 0.372230883E-04 0.532184593E-05 -0.320872514E-03 0.199466098E-04 -0.104930467E-07 -0.628891899E-04 0.428146393E-04 0.247281351E-04 0.100164505E-03 0.377288195E-04 0.157332261E-07 -0.971597384E-05 -0.531618021E-04 0.228824318E-04 0.170692998E-04 0.226133276E-04 -0.631364159E-07 0.584547547E-04 -0.910107522E-03 -0.340265486E-03 -0.115414326E-05 -0.246615033E-03 0.247172233E-04 0.968399966E-05 -0.680172850E-03 -0.578592051E-04 0.457250632E-03 dvy (km/s) -0.105271788E-03 -0.302199862E-04 -0.365276195E-04 -0.817691504E-03 -0.233233466E-04 -0.163911797E-07 -0.304279301E-04 -0.718008523E-05 0.570950436E-05 -0.177227628E-03 -0.595545933E-06 -0.556335346E-07 0.309143403E-05 0.349535674E-04 0.186052100E-05 0.233584714E-03 0.447079442E-04 -0.285156562E-07 -0.666336205E-04 -0.100426181E-02 -0.824560854E-04 0.118854672E-03 0.737493515E-04 -0.240739696E-04 -0.428631296E-04 0.777420276E-03 -0.523445766E-04 0.321143465E-03 dxz (km/s) 0.787561792E-06 -0.447202013E-04 -0.489918001E-05 0.186385484E-03 0.563805888E-06 0.398799019E-09 -0.198773901E-05 -0.936434194E-03 0.386924904E-05 -0.302235187E-03 -0.106889493E-05 -0.498434764E-08 0.255329450E-06 0.393100351E-03 -0.121303484E-05 -0.355369839E-03 0.188140042E-05 0.119076455E-06 -0.127977036E-05 -0.197787421E-02 0.404755255E-04 0.281895068E-04 -0.152863478E-04 -0.135601183E-05 0.196464319E-05 -0.196914314E-03 -0.217268266E-04 -0.264747900E-02 233 29 E 0.109350591E+05 0.704146579E+00 0.785573530E+01 0.310736782E+01 30 M 0.111176484E+05 -0.434669752E-01 0.724135527E+01 -0.857083526E+00 31 E 0.116966714E+05 0.602550823E+01 0.162088924E+01 -0.743298163E-03 32 E 0.123788557E+05 -0.128844412E+01 0.537697708E+01 -0.214068721E+01 33 E 0.127441072E+05 -0.510203529E+00 -0.557763123E-01 -0.550060360E+01 34 E 0.129271315E+05 0.124599018E+01 -0.534043846E+01 0.159562976E+01 35 E 0.132923823E+05 -0.527569904E+01 -0.191016219E+01 0.382764752E+00 36 M 0.134281476E+05 -0.109764538E+02 -0.488738339E+01 0.879524970E+00 37 E 0.140618249E+05 -0.798043798E+01 0.182208207E+01 0.375363532E-03 38 E 0.147084062E+05 -0.317046694E+01 -0.595279981E+01 -0.461004775E+01 39 E 0.150736663E+05 0.861647546E+00 -0.683989035E+00 -0.810381330E+01 40 E 0.152532633E+05 0.343841416E+01 0.594074243E+01 0.410451626E+01 41 E 0.156184932E+05 0.766887084E+01 0.197749052E+01 -0.106874865E+01 42 M 0.157465099E+05 0.953974766E+01 -0.174607241E+01 -0.140022366E+00 43 E 0.163397702E+05 ================ PARENT CYCLER 5.658Gfh+ff3 ======================= Parent cycler number 162 Approximate search space (synodic periods after J2000) 10 Number of steps to walk eccentricity/inclination 81 / 81 Number of cycles 7 Total delta v over 44.63 years (km/s) 0.024744 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 1 100.3 ****** 11.316 7.650 7.650 7 108.6 7.968 7.968 11.021 11.021 13 114.2 5.801 5.801 8.621 8.621 19 135.2 6.254 6.254 9.132 9.132 25 87.8 6.524 6.525 12.342 12.342 31 169.3 7.270 7.270 7.521 7.521 37 110.1 7.201 7.201 11.141 11.141 AVERAGE 117.9 6.836 7.476 9.633 9.633 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 8294.078101 LEG E/M time start vinfx vinfy vinfz (days) (km/s) (km/s) (km/s) 1 E 0.506007400E+02 0.634610640E+01 0.931607198E+01 0.992710619E+00 2 M 0.150898872E+03 -0.101946648E+01 0.743985891E+01 -0.145814600E+01 3 E 0.689399847E+03 -0.497696077E+01 0.364201081E+01 0.467374451E+01 4 E 0.105467983E+04 -0.724226215E+00 -0.701741760E+00 0.766827817E+01 5 E 0.123533225E+04 -0.388439131E+01 0.564632295E+01 -0.405001287E+01 6 E 0.160059314E+04 -0.707652025E+01 0.114725743E+01 -0.345789152E+01 7 E 0.233109967E+04 -0.758882688E+01 0.240341729E+01 0.350798724E+00 8 M 0.243971930E+04 -0.100381503E+02 -0.451273052E+01 0.578704159E+00 9 E 0.300289795E+04 -0.959651454E+00 -0.560414119E+01 0.183578901E+01 10 E 0.336815839E+04 0.567500820E+00 -0.173951992E+00 0.596296951E+01 11 E 0.354889579E+04 0.785356416E+00 0.470406209E+01 -0.333789843E+01 12 E 0.391416707E+04 0.557313229E+01 0.115389055E+01 -0.113819923E+01 13 E 0.464466633E+04 0.562957677E+01 0.108473159E+01 -0.882363217E+00 14 M 0.475887528E+04 0.850404772E+01 -0.132030471E+01 -0.514735059E+00 15 E 0.536899711E+04 0.159280406E+01 0.545066664E+01 0.219584143E+01 16 E 0.573423855E+04 -0.591064286E+00 0.248524582E+00 0.606428096E+01 17 E 0.591909266E+04 0.256821189E+01 0.474025350E+01 -0.319616468E+01 18 E 0.628432180E+04 -0.300375309E+01 0.548093650E+01 -0.279552373E+00 19 E 0.701484776E+04 -0.296627535E+01 0.541176708E+01 0.101488546E+01 20 M 0.715004795E+04 -0.848575763E+01 0.335489645E+01 -0.371002221E+00 21 E 0.769594936E+04 -0.514435085E+01 -0.256692282E+01 0.306694837E+01 22 E 0.806121134E+04 0.214986638E+00 -0.675536451E+00 0.649634143E+01 23 E 0.823997402E+04 0.220082470E+00 0.814667695E+00 -0.645941932E+01 24 E 0.860522906E+04 0.365057985E+01 -0.387095170E+01 -0.376042901E+01 25 E 0.933574524E+04 -0.235386019E+01 -0.603537420E+01 -0.776643120E+00 26 M 0.942353191E+04 -0.153805302E+01 -0.122145267E+02 0.880840103E+00 27 E 0.100017942E+05 0.484208951E+01 -0.376290917E+01 0.428914469E+01 28 E 0.103670517E+05 0.676460104E+00 0.639220860E+00 0.744169710E+01 29 E 0.105516711E+05 -0.331022899E+01 0.205811311E+01 -0.615763107E+01 30 E 0.109169328E+05 -0.482039179E+00 0.718799252E+01 -0.103888039E+01 31 E 0.116474145E+05 -0.446102525E+00 0.704832415E+01 0.172489599E+01 32 M 0.118167559E+05 -0.426532278E+00 0.743266013E+01 -0.106657896E+01 33 E 0.123816464E+05 -0.506497750E+01 0.340444136E+01 0.344978359E+01 34 E 0.127469158E+05 -0.548950889E+00 -0.617309001E+00 0.694818541E+01 35 E 0.129273579E+05 -0.266473499E+01 0.363414627E+01 -0.560530584E+01 36 E 0.132926263E+05 -0.713761323E+01 0.397100620E+00 -0.737846406E+00 37 E 0.140231339E+05 -0.713599535E+01 0.940199536E+00 0.210809279E+00 38 M 0.141331861E+05 -0.945369592E+01 -0.585338442E+01 0.703815980E+00 39 E 0.147086379E+05 0.639404654E+00 -0.460236036E+01 0.273699665E+01 40 E 0.150738903E+05 0.484677020E+00 0.239202610E-01 0.535628746E+01 41 E 0.152557582E+05 -0.535395865E+00 0.276435684E+01 -0.437591698E+01 42 E 0.156210155E+05 0.512044446E+01 0.931355651E+00 -0.326212312E+00 43 E 0.163514718E+05 ================ PARENT CYCLER 5.658gfh+fFzero======================= Parent cycler number 163 Approximate search space (synodic periods after J2000) 17 Number of steps to walk eccentricity/inclination 1 / 1 Number of cycles 7 Total delta v over 44.65 years (km/s) 0.013831 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 5 128.1 7.117 7.117 9.230 9.230 11 90.6 6.255 6.253 11.837 11.837 17 128.6 6.091 6.091 7.747 7.747 23 107.1 7.501 7.501 11.422 11.422 29 93.3 5.913 5.913 8.012 8.012 35 128.9 6.876 6.876 9.453 9.453 41 90.4 5.914 5.913 11.215 11.215 AVERAGE 109.6 6.524 6.523 9.845 9.845 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 13703.935532 LEG E/M time start vinfx vinfy vinfz (days) (km/s) (km/s) (km/s) 1 E 0.293280850E+02 0.688458863E+01 -0.208682173E+00 0.906898817E-03 2 E 0.697800451E+03 0.746558098E+00 0.603459795E+01 0.328700873E+01 3 E 0.106303674E+04 -0.797466205E+00 0.211336247E+00 0.687073224E+01 4 E 0.124743509E+04 0.236619263E+01 0.609703059E+01 -0.283501836E+01 0.110062689E+05 0.112045018E+05 0.117989990E+05 0.124957362E+05 0.127715608E+05 0.129928766E+05 0.133127471E+05 0.135548831E+05 0.141588121E+05 0.149129519E+05 0.151006059E+05 0.154943150E+05 0.156376957E+05 0.159185554E+05 -0.107788998E-02 0.305263204E-06 0.469227905E-04 0.133964059E-03 0.220526304E-04 -0.410606628E-05 0.187407509E-04 0.106646579E-05 0.334421168E-04 -0.418096369E-04 -0.872428501E-05 0.360553115E-05 -0.223911656E-04 -0.560449934E-06 -0.180058856E-02 -0.657700575E-06 -0.748677733E-04 -0.374732830E-04 -0.139167501E-04 -0.294844049E-05 -0.568130894E-05 0.143171864E-05 0.306233546E-04 -0.160858208E-03 -0.506396847E-04 -0.161994314E-03 -0.442064991E-04 -0.268011497E-05 0.467487288E-01 0.482592073E-07 0.341259225E-05 -0.617939179E-04 -0.405858807E-05 -0.900672902E-05 -0.667887224E-06 0.232303736E-07 0.291735865E-05 0.100902045E-03 0.700696258E-05 0.114340115E-02 -0.186208014E-05 -0.202783376E-05 time dv (days) 0.656454597E+02 0.242444037E+03 0.930484635E+03 0.108177769E+04 0.133760530E+04 0.177591471E+04 0.234739261E+04 0.252419610E+04 0.306864483E+04 0.339526900E+04 0.379728026E+04 0.402374196E+04 0.466179767E+04 0.485039355E+04 0.561370888E+04 0.576196667E+04 0.603231369E+04 0.639390070E+04 0.703512779E+04 0.728652330E+04 0.794432751E+04 0.808802574E+04 0.847738980E+04 0.882438392E+04 0.934891324E+04 0.951027126E+04 0.100675406E+05 0.103947447E+05 0.107598703E+05 0.110265051E+05 0.116728157E+05 0.119353829E+05 0.126154188E+05 0.127739821E+05 0.130369385E+05 0.134825583E+05 0.140396417E+05 0.142195039E+05 0.147780358E+05 0.151011705E+05 0.154274291E+05 0.157305839E+05 dvx (km/s) 0.102313859E-02 0.293304874E-04 0.541178783E-03 0.916102344E-04 -0.595096077E-04 -0.449871368E-05 -0.194413384E-03 0.262514121E-04 -0.231259994E-04 0.689334869E-05 0.345902264E-03 -0.205847542E-04 -0.150355089E-03 -0.263677851E-04 -0.173645393E-03 -0.275601918E-04 -0.956547790E-04 -0.502244414E-04 0.371951843E-04 0.933744761E-04 -0.169333094E-04 -0.413864593E-04 0.186140580E-03 -0.361528307E-03 -0.280659015E-03 -0.114219208E-03 -0.662299006E-04 0.120333904E-03 0.257213570E-03 0.719590922E-05 0.227950976E-03 0.106797172E-03 0.460089554E-03 0.752032389E-04 0.121449896E-03 -0.225617852E-05 -0.199083354E-03 0.296973639E-04 0.153269358E-03 -0.222319003E-04 0.134236573E-03 -0.321842409E-04 dvy (km/s) 0.610548003E-03 -0.184432297E-04 0.358084173E-03 -0.681249524E-04 -0.129758110E-03 0.432663245E-04 0.361493357E-04 0.659651145E-04 -0.226069511E-05 -0.470135392E-04 -0.222007032E-03 -0.218105912E-05 -0.174947064E-03 -0.745870559E-04 0.142337688E-03 0.612155164E-04 0.726148471E-04 -0.320332521E-04 0.159091902E-03 0.542475446E-04 0.474417075E-04 -0.195948018E-04 -0.552716241E-04 0.221287767E-03 0.517752440E-04 0.956403451E-04 -0.980150435E-04 0.149411009E-04 0.520368424E-03 0.115809592E-03 -0.137252156E-03 -0.556321745E-04 0.253646233E-03 -0.124870334E-04 0.387861715E-04 0.252590835E-05 0.176284663E-03 0.160802767E-03 0.764886477E-04 0.593688376E-04 0.726861833E-03 -0.139296577E-04 dxz (km/s) -0.270216204E-04 -0.121756614E-04 -0.150738082E-02 -0.349692662E-05 -0.399597665E-03 0.182142386E-03 -0.205439059E-04 0.207341574E-04 -0.462466326E-04 0.534726556E-06 0.101805843E-02 -0.124248096E-04 0.772859991E-05 -0.160280075E-04 0.539048029E-03 -0.874319842E-05 0.136001609E-02 0.236471540E-04 -0.630420322E-05 -0.478643746E-04 -0.354209545E-03 -0.220217383E-05 0.306718646E-04 0.601101772E-04 -0.577041844E-05 -0.298675678E-04 0.182054708E-03 -0.132328390E-04 0.365814959E-03 0.274492442E-03 -0.195798007E-05 -0.455205760E-04 -0.846326070E-03 -0.650208222E-05 -0.100284884E-02 0.116825362E-03 -0.213768537E-04 0.137130650E-04 0.321331653E-03 0.651784242E-05 0.604615405E-05 0.675536169E-05 time dv dvx dvy dxz (days) (km/s) (km/s) (km/s) 0.129598940E+03 -0.245402191E-06 -0.352501979E-06 0.142904821E-07 0.946161125E+03 0.131393914E-04 -0.832173600E-04 0.120861567E-02 0.112020023E+04 -0.768826515E-07 -0.862374932E-07 -0.128027469E-07 0.136431384E+04 -0.580258405E-04 0.203668099E-04 0.614047056E-03 234 5 E 0.161268118E+04 -0.302521068E+01 0.636462520E+01 0.994020734E+00 6 M 0.174074225E+04 -0.861311200E+01 0.327825867E+01 0.502520543E+00 7 E 0.233599075E+04 -0.291649181E+01 0.552795117E+01 -0.820423715E-03 8 E 0.301614922E+04 -0.535296668E+01 -0.255585633E+01 0.208188395E+01 9 E 0.338141323E+04 0.208854955E+00 -0.616351017E+00 0.624652714E+01 10 E 0.356017706E+04 -0.386023522E+01 -0.511153351E+00 -0.490429637E+01 11 E 0.392543527E+04 -0.235552615E+01 -0.573929721E+01 -0.783862708E+00 12 M 0.401600735E+04 -0.961406919E+00 -0.117962873E+02 0.210773717E+00 13 E 0.465397131E+04 0.152337408E+01 -0.467115813E+01 0.358805475E+01 14 E 0.538447911E+04 -0.256469483E+01 -0.504898622E-01 0.550303699E+01 15 E 0.574973923E+04 -0.162957034E+00 0.585476913E+00 0.603370783E+01 16 E 0.593624158E+04 0.571309056E+01 0.109037598E+01 -0.175520335E+01 17 E 0.630149795E+04 0.148254958E+01 0.581721983E+01 0.102945109E+01 18 M 0.643010083E+04 -0.189964925E+01 0.751078094E+01 -0.544683298E-01 19 E 0.704315238E+04 0.294947696E+01 0.666914036E+01 -0.135630788E-02 20 E 0.770404225E+04 -0.552940170E+01 0.347626983E+01 0.333156450E+01 21 E 0.806928722E+04 -0.594113686E+00 -0.698790298E+00 0.725699952E+01 22 E 0.824963333E+04 -0.317450544E+01 0.403467992E+01 -0.548617766E+01 23 E 0.861489097E+04 -0.743807547E+01 0.941583237E+00 0.225106641E+00 24 M 0.872196470E+04 -0.977249984E+01 -0.585684076E+01 0.807686473E+00 25 E 0.933471143E+04 -0.146425995E+01 -0.591870528E+01 0.416335184E+00 26 E 0.100651967E+05 0.366419851E+01 -0.369488047E+01 0.324491833E+01 27 E 0.104304421E+05 0.489886315E+00 0.395693280E+00 0.608555441E+01 28 E 0.106147207E+05 0.316474362E+01 -0.488905072E+01 -0.113295591E+01 29 E 0.109799906E+05 0.576932133E+01 0.122328709E+01 -0.420688267E+00 30 M 0.110732557E+05 0.755999429E+01 0.247277129E+01 -0.962148036E+00 31 E 0.117187788E+05 0.664110685E+01 0.275466472E+00 -0.135084349E-02 32 E 0.123920952E+05 0.210353842E+00 0.630349505E+01 0.216273695E+01 33 E 0.127573408E+05 -0.759239038E+00 0.121273917E+00 0.662915666E+01 34 E 0.129413878E+05 0.165014118E+01 0.604940520E+01 -0.284428960E+01 35 E 0.133066385E+05 -0.373593146E+01 0.569894543E+01 0.921171890E+00 36 M 0.134355469E+05 -0.918096282E+01 0.218621534E+01 0.545372895E+00 37 E 0.140283679E+05 -0.356744957E+01 0.472005247E+01 -0.170294045E-02 38 E 0.147160816E+05 -0.452028356E+01 -0.336384545E+01 0.191307427E+01 39 E 0.150813421E+05 0.283396611E+00 -0.508723171E+00 0.593680948E+01 40 E 0.152602457E+05 -0.255706018E+01 -0.740039258E+00 -0.527549497E+01 41 E 0.156254977E+05 -0.795635357E+00 -0.578274596E+01 -0.941004247E+00 42 M 0.157158858E+05 0.155194807E+01 -0.111045633E+02 0.213532777E+00 43 E 0.163371692E+05 ================ PARENT CYCLER 5.658gfh+fFpi3 ======================= Parent cycler number 164 Approximate search space (synodic periods after J2000) 7 Number of steps to walk eccentricity/inclination 27 / 27 Number of cycles 7 Total delta v over 44.65 years (km/s) 1.566859 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 5 155.0 8.939 8.939 8.628 8.628 11 171.8 6.114 6.114 9.416 9.416 17 114.3 5.804 5.804 11.772 11.772 23 171.6 6.426 6.419 7.655 7.655 29 128.0 6.391 5.538 12.358 12.358 35 148.6 7.619 7.619 7.662 7.662 41 149.3 6.294 6.347 11.272 11.272 AVERAGE 148.4 6.798 6.683 9.823 9.823 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 5865.419269 LEG E/M time start vinfx vinfy vinfz (days) (km/s) (km/s) (km/s) 1 E -0.230150823E+01 -0.738309599E+01 0.508337536E+01 -0.169728668E-03 2 E 0.632869989E+03 -0.579411566E+01 -0.410262547E+01 0.548727464E+01 3 E 0.998126806E+03 0.544292749E+00 -0.123296545E+01 0.887387223E+01 4 E 0.117692956E+04 0.552062538E+01 0.372510607E+01 -0.593609034E+01 5 E 0.154218895E+04 0.865601186E+01 0.253660844E+00 -0.221771779E+01 6 M 0.169718593E+04 0.857952738E+01 -0.704222159E+00 -0.576909841E+00 7 E 0.232460979E+04 0.364367655E+01 -0.484719015E+01 0.189917918E-03 8 E 0.301113785E+04 0.420762519E+01 0.314799807E+01 0.296904460E+01 9 E 0.337638746E+04 -0.309116092E+00 0.536472758E+00 0.601931742E+01 10 E 0.356269400E+04 -0.421578698E+01 -0.338315204E+01 -0.285745107E+01 11 E 0.392796825E+04 -0.546182731E+01 0.228253863E+01 0.153013141E+01 12 M 0.409980893E+04 -0.917064924E+01 0.213546514E+01 0.605004361E-01 13 E 0.466453801E+04 -0.633804614E+00 0.570579849E+01 -0.470387902E-03 14 E 0.535793270E+04 -0.421699017E+01 -0.669028113E+00 0.387636275E+01 15 E 0.572319302E+04 0.160624307E-01 -0.556073634E+00 0.572508150E+01 16 E 0.590202390E+04 0.365742957E+01 0.524955585E+00 -0.446205524E+01 17 E 0.626728845E+04 0.232689513E+01 -0.522796243E+01 -0.971746262E+00 18 M 0.638158676E+04 0.175555382E+01 -0.116395752E+02 -0.976671619E-01 19 E 0.703430490E+04 -0.933872857E+00 -0.648744562E+01 0.345305324E-03 20 E 0.770941925E+04 0.564958766E+01 -0.123696033E+01 0.301866295E+01 21 E 0.807466899E+04 0.235763078E+00 0.680364791E+00 0.649034770E+01 22 E 0.826059799E+04 -0.549287970E+01 0.100572050E+01 -0.316060768E+01 23 E 0.862586805E+04 -0.215077867E+01 0.577078404E+01 0.180859422E+01 24 M 0.879741900E+04 -0.337557575E+01 0.682797595E+01 -0.761403776E+00 25 E 0.935890518E+04 0.507494990E+01 0.363595397E+01 -0.847077032E-03 26 E 0.100403265E+05 -0.317984263E+01 0.480494102E+01 0.227731189E+01 27 E 0.104055806E+05 -0.594939889E+00 -0.287699374E+00 0.616678653E+01 28 E 0.105872810E+05 0.335517742E+01 -0.515520146E+01 -0.178796027E+01 29 E 0.109525468E+05 -0.453836143E+01 -0.317260230E+01 0.814642684E-01 30 M 0.110805232E+05 -0.970710819E+01 -0.760651268E+01 0.795927679E+00 31 E 0.117178184E+05 -0.783748896E+01 -0.613420679E+00 -0.551117212E-03 32 E 0.123694670E+05 -0.102147540E+01 -0.658812406E+01 0.412702753E+01 33 E 0.127347270E+05 0.964384932E+00 -0.323989181E+00 0.779227003E+01 34 E 0.129153462E+05 0.118076018E+01 0.664669447E+01 -0.357475800E+01 35 E 0.132805976E+05 0.596394514E+01 0.472058877E+01 -0.436232455E+00 36 M 0.134292423E+05 0.683022265E+01 0.338915929E+01 -0.750198919E+00 37 E 0.140364399E+05 0.525073631E+01 -0.328171100E+01 -0.679269718E-03 38 E 0.147197702E+05 0.161032876E+01 0.295694431E+01 0.517775298E+01 39 E 0.150850217E+05 -0.507090860E+00 0.397767071E+00 0.613355412E+01 40 E 0.152707181E+05 -0.324250603E+01 -0.514381843E+01 -0.161688168E+01 41 E 0.156359627E+05 -0.608988635E+01 0.138706544E+01 0.112591658E+01 42 M 0.157853003E+05 -0.108185801E+02 0.273384362E+01 0.159330134E+01 43 E 0.163068324E+05 ================ PARENT CYCLER 5.658Gfh-ff3 ======================= 0.163189034E+04 0.183002952E+04 0.245841928E+04 0.326452875E+04 0.344219293E+04 0.383777330E+04 0.393902108E+04 0.411170195E+04 0.496078458E+04 0.565477160E+04 0.577771459E+04 0.604947105E+04 0.632078838E+04 0.652205856E+04 0.714228586E+04 0.794875638E+04 0.809633914E+04 0.836286320E+04 0.863095203E+04 0.885676898E+04 0.972186860E+04 0.101382457E+05 0.104580839E+05 0.107206490E+05 0.109939804E+05 0.111765394E+05 0.118197762E+05 0.126404622E+05 0.127978312E+05 0.130582681E+05 0.133259748E+05 0.135244701E+05 0.141315250E+05 0.149644587E+05 0.151081776E+05 0.155268797E+05 0.156390559E+05 0.158090783E+05 -0.915221965E-08 -0.488243862E-11 -0.614343909E-07 -0.127591775E-04 0.148439768E-06 -0.131003189E-02 0.169876957E-06 0.128709217E-09 -0.286245139E-03 0.153503720E-03 0.685101823E-06 0.772801293E-06 -0.323577530E-06 0.271597842E-09 -0.548771673E-06 -0.667809604E-07 -0.250776421E-05 -0.935206021E-05 -0.213989440E-05 -0.206671295E-07 0.274279521E-03 -0.376647531E-06 -0.568340257E-04 0.348703105E-04 -0.396892170E-04 0.234835014E-07 -0.154947365E-04 -0.236778230E-04 -0.186868648E-05 -0.266551879E-04 -0.292153421E-06 -0.102778722E-08 0.237800815E-06 -0.194962067E-05 -0.641532263E-07 -0.914097665E-03 -0.124875264E-06 -0.319380597E-09 -0.213191725E-06 -0.567903832E-10 0.396320852E-07 0.209996671E-04 -0.699664284E-08 -0.514371542E-03 -0.168429027E-06 -0.229702782E-09 0.174556815E-03 0.670073576E-04 0.103227171E-05 -0.201600957E-05 0.697196885E-06 0.280645151E-09 0.471929956E-06 -0.290243178E-04 -0.219491898E-06 -0.437806595E-05 0.218499396E-07 -0.372186469E-07 -0.640485504E-03 0.357581580E-04 0.913436087E-05 0.530816048E-04 -0.178268018E-04 -0.150683629E-07 -0.178629593E-04 -0.132645479E-04 -0.125150115E-05 0.428059255E-05 -0.349698738E-05 -0.162567515E-08 0.252699514E-07 0.179454919E-06 -0.191883515E-06 -0.324586689E-03 -0.803169618E-07 0.339422900E-10 -0.106120506E-07 -0.200950563E-11 0.101016271E-07 -0.424046334E-03 -0.112672464E-07 -0.171852679E-03 0.146286606E-07 0.315504138E-10 -0.864013279E-04 -0.219258211E-04 -0.166060512E-06 -0.226684544E-04 -0.245292100E-07 -0.216590787E-10 0.952769374E-07 0.491072936E-03 -0.459716727E-06 0.870255204E-04 0.453088537E-07 0.169534245E-08 0.159246292E-03 -0.290062065E-03 0.800572323E-05 -0.168619904E-03 -0.262455007E-05 0.226501730E-09 0.135784958E-07 0.697240171E-03 -0.386651006E-06 0.229623873E-03 0.180744996E-06 0.539407268E-10 -0.138945615E-07 -0.645048394E-04 -0.271696598E-07 -0.219160738E-03 -0.350189118E-08 0.369660167E-12 time dv (days) 0.929742164E+02 0.698616216E+03 0.102494722E+04 0.128650737E+04 0.156543850E+04 0.179129951E+04 0.242758900E+04 0.326316008E+04 0.340433344E+04 0.381108049E+04 0.395374435E+04 0.418451829E+04 0.476854721E+04 0.560996232E+04 0.575001766E+04 0.614309850E+04 0.628443320E+04 0.661656529E+04 0.713557206E+04 0.796144157E+04 0.812672911E+04 0.833730471E+04 0.865160069E+04 0.898832430E+04 0.972687269E+04 0.102740891E+05 0.104637248E+05 0.107918298E+05 0.109717432E+05 0.112780847E+05 0.118155657E+05 0.126141912E+05 0.128178118E+05 0.131600646E+05 0.133028943E+05 0.135203220E+05 0.141389395E+05 0.149206586E+05 0.151128762E+05 0.155190844E+05 0.156583633E+05 0.160721430E+05 dvx (km/s) 0.444462851E-04 0.243121220E-04 -0.308777789E-05 -0.485785299E-05 0.488429835E-05 0.849845103E-07 0.541657264E-06 -0.308206245E-04 0.309045835E-05 -0.374016520E-04 0.800534798E-05 -0.266413496E-08 0.266614199E-05 -0.774013964E-04 -0.448160629E-05 0.150329038E-03 -0.865621335E-05 -0.201812705E-04 0.417229902E-05 -0.951978766E-05 0.710527450E-06 0.247842667E-03 0.248662238E-05 -0.385659247E-07 -0.198225567E-02 -0.690947293E-04 -0.268119414E-04 -0.189122237E-04 -0.102636702E-04 0.203108031E-06 -0.311036567E-05 0.302712090E-05 0.519360439E-07 -0.174558106E-05 0.565799076E-07 -0.291094529E-09 -0.442083197E-06 0.712661114E-03 0.355793209E-06 0.115529354E-04 0.448860303E-06 -0.260298665E-02 dvy (km/s) 0.403287545E-04 -0.246409213E-04 0.245797521E-04 0.292659857E-04 0.374314289E-04 0.117758973E-06 -0.125136829E-06 -0.403230938E-04 -0.113746057E-04 0.357654999E-04 0.489705973E-05 0.121548946E-07 -0.162652435E-05 -0.488022557E-04 -0.144746396E-04 -0.146149505E-05 -0.126854681E-04 0.955275616E-05 -0.600902704E-05 0.223558610E-04 -0.872117340E-06 0.556434712E-04 0.705169647E-06 -0.960438340E-08 0.934046238E-02 -0.455730608E-05 -0.333428308E-05 -0.161996363E-04 -0.103798470E-04 0.110393707E-05 -0.506014861E-05 0.508639268E-06 -0.204741593E-06 -0.126361760E-05 -0.762568900E-07 -0.166958899E-08 -0.959400776E-07 0.141596659E-02 -0.552156080E-06 0.882542852E-06 0.324526241E-06 0.976435032E-02 dxz (km/s) -0.167238105E-04 0.731565968E-04 0.738848919E-05 0.298514352E-04 0.136474311E-05 -0.513588769E-08 -0.134438432E-07 0.707884526E-03 0.235285713E-05 0.536874746E-03 0.432312782E-06 -0.133864825E-08 0.149000328E-06 -0.418535010E-03 -0.283897806E-05 0.154737234E-05 0.515283425E-06 0.544615565E-06 0.222228327E-06 0.466698416E-03 0.133446586E-06 -0.678369849E-03 -0.369575823E-07 0.242247109E-07 0.431973067E-05 0.113299079E-03 -0.581661377E-04 0.298688524E-04 -0.121671455E-05 -0.652850429E-10 -0.181309158E-06 0.352924784E-04 -0.148508674E-05 -0.370324330E-04 0.512092200E-09 -0.219898249E-09 0.691884668E-08 0.344813257E-03 0.133359921E-06 -0.423668610E-03 -0.244815112E-07 0.328994301E+00 235 Parent cycler number 165 Approximate search space (synodic periods after J2000) 21 Number of steps to walk eccentricity/inclination 9 / 9 Number of cycles 7 Total delta v over 44.94 years (km/s) 0.046196 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 1 96.3 ****** 8.378 11.657 11.657 7 111.3 5.539 5.539 7.461 7.461 13 123.5 7.278 7.279 9.869 9.869 19 89.7 5.563 5.563 10.516 10.516 25 145.4 5.472 5.472 8.207 8.207 31 98.7 7.264 7.264 11.664 11.665 37 128.3 5.169 5.168 6.743 6.743 AVERAGE 113.3 6.047 6.380 9.445 9.446 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 16873.682377 LEG E/M time start vinfx vinfy vinfz time dv dvx dvy dxz (days) (km/s) (km/s) (km/s) (days) (km/s) (km/s) (km/s) 1 E -0.105142151E+02 -0.831075197E+01 -0.105598695E+01 -0.587821092E-01 0.393790353E+01 -0.235227078E-03 0.363089789E-03 0.991973443E-05 2 M 0.858332426E+02 -0.811303868E+01 -0.833792567E+01 0.744909965E+00 0.177965182E+03 0.196344916E-06 -0.290209261E-06 -0.189994017E-07 3 E 0.700046175E+03 -0.211888513E+01 0.236194700E+01 -0.473530755E+01 0.966681289E+03 0.196422576E-03 -0.298030689E-03 -0.177360368E-03 4 E 0.106529976E+04 0.318373699E+00 0.430780227E+00 -0.568250227E+01 0.109306323E+04 -0.406429013E-04 -0.557037212E-04 -0.131774545E-04 5 E 0.125038958E+04 0.409708048E+01 0.326398406E+01 0.186602593E+01 0.145493444E+04 -0.147256617E-03 -0.685320368E-03 0.119067190E-04 6 E 0.198090694E+04 0.392091306E+01 -0.359223737E+01 0.161998956E+01 0.208683020E+04 0.449509932E-04 0.108129638E-03 -0.296978649E-03 7 E 0.234615956E+04 0.419921525E+01 0.360985869E+01 0.131210338E+00 0.236285645E+04 -0.108122269E-03 -0.306324516E-04 0.224716026E-05 8 M 0.245747213E+04 0.521704632E+01 0.526411714E+01 -0.857382424E+00 0.254354860E+04 -0.376827241E-07 0.559881819E-07 0.108579427E-07 9 E 0.303131524E+04 -0.186749350E+01 0.622955848E+01 -0.278437518E+01 0.325412109E+04 0.258393391E-04 -0.136425158E-04 -0.587752758E-04 10 E 0.339657073E+04 -0.838055784E+00 -0.148250804E+00 -0.700278721E+01 0.342400554E+04 0.166481284E-05 -0.317582220E-06 -0.392035734E-06 11 E 0.357946948E+04 -0.503610551E+01 0.259834366E+01 0.459164565E+01 0.389358993E+04 0.854857899E-03 0.827202238E-03 -0.209092309E-03 12 E 0.430998215E+04 0.247285576E+00 0.397665468E+01 0.610177280E+01 0.458392316E+04 0.293873317E-04 -0.747480625E-03 -0.425436166E-04 13 E 0.467523683E+04 -0.509247437E+01 0.514610834E+01 0.751718515E+00 0.469376656E+04 0.724659013E-06 -0.125632235E-05 -0.770661246E-07 14 M 0.479876838E+04 -0.986560927E+01 0.681667154E-01 0.260949530E+00 0.488258337E+04 -0.221609640E-08 -0.226055751E-08 -0.454602178E-09 15 E 0.535753498E+04 -0.267478099E+01 -0.413520772E+01 -0.279124711E+01 0.560954621E+04 0.162830352E-04 0.418186225E-04 -0.790305584E-03 16 E 0.572276865E+04 0.417243065E+00 -0.348339975E+00 -0.562722656E+01 0.574969887E+04 -0.266401768E-05 -0.105463855E-04 0.896236956E-06 17 E 0.590230346E+04 0.344562201E+01 -0.371204464E+01 0.224625485E+01 0.611415474E+04 -0.385302741E-03 0.927167385E-04 0.116874567E-04 18 E 0.663282511E+04 -0.376261124E+01 -0.358420805E+01 0.193395201E+01 0.674970325E+04 0.244324021E-04 -0.204908078E-04 -0.385525267E-03 19 E 0.699806928E+04 0.231539042E+01 -0.493357127E+01 -0.111670838E+01 0.701151709E+04 -0.323554675E-05 0.971912438E-05 0.870307082E-07 20 M 0.708772135E+04 0.529188124E+01 -0.908800405E+01 -0.832105770E-02 0.721107009E+04 -0.505478721E-09 -0.591422868E-08 0.226674942E-08 21 E 0.770446504E+04 0.454225495E+01 0.275837358E+01 -0.979991664E+00 0.795650490E+04 0.577606632E-05 -0.146317811E-04 0.303526674E-03 22 E 0.806974020E+04 -0.217267089E+00 0.427690903E+00 -0.538735052E+01 0.809770084E+04 0.116757552E-04 0.991849129E-05 0.658656981E-06 23 E 0.825614444E+04 -0.894527033E+00 0.516055510E+01 0.152028859E+01 0.840224253E+04 0.237291893E-04 0.141143061E-04 -0.237159403E-03 24 E 0.898663489E+04 0.505174203E+01 0.210143253E+01 0.496831617E-02 0.909256258E+04 0.182927100E-04 -0.177525018E-04 0.847304194E-04 25 E 0.935190279E+04 -0.128429717E+01 0.517713705E+01 0.121869145E+01 0.937371560E+04 0.199463681E-04 0.400547985E-05 0.364520772E-08 26 M 0.949732148E+04 -0.509778485E+01 0.634634600E+01 -0.104207863E+01 0.977981313E+04 0.320856026E-07 0.108433291E-06 0.231836713E-06 27 E 0.100405746E+05 -0.651144578E+01 0.666265567E+00 -0.285752873E+01 0.102597333E+05 -0.212603391E-03 -0.887110573E-04 0.287980639E-03 28 E 0.104058391E+05 -0.218498242E+00 -0.830989051E+00 -0.710869866E+01 0.104327174E+05 -0.198319382E-04 -0.109337509E-04 0.389788886E-05 29 E 0.105850277E+05 -0.106135776E+01 -0.447201037E+01 0.563417109E+01 0.108041809E+05 -0.121884536E-03 0.399974336E-03 -0.479861422E-04 30 E 0.113155383E+05 -0.428135898E+01 0.206012295E+01 0.550816608E+01 0.114397258E+05 -0.346673352E-04 -0.115587020E-03 -0.987414358E-04 31 E 0.116807958E+05 -0.685166598E+01 -0.240381676E+01 -0.215904717E+00 0.116956026E+05 -0.259582411E-04 0.185325258E-03 0.243521975E-06 32 M 0.117795078E+05 -0.689250798E+01 -0.938431657E+01 0.700023350E+00 0.119510117E+05 0.112164098E-05 -0.831008355E-04 0.620323756E-06 33 E 0.123920216E+05 -0.398437476E+01 0.329777855E+01 -0.115876360E+01 0.124577679E+05 -0.217224910E-03 -0.195870507E-03 0.162770342E-03 34 E 0.127572787E+05 0.252708177E+00 0.398558042E+00 -0.529765723E+01 0.127850700E+05 0.301325892E-03 0.154395441E-03 0.536867032E-04 35 E 0.129425544E+05 0.271328949E+01 0.436236000E+01 0.746185565E+00 0.135281497E+05 0.107250540E-02 0.155210922E-03 -0.124125709E-01 36 E 0.136745485E+05 0.221128797E+01 -0.188857928E+01 0.428884033E+01 0.139046600E+05 0.197933430E-03 0.274912055E-03 0.320707638E-03 37 E 0.140398048E+05 0.305785016E+01 0.413916872E+01 0.469707582E+00 0.140590462E+05 0.182858456E-03 0.832393265E-04 0.127400988E-04 38 M 0.141680806E+05 0.295192833E+01 0.601659142E+01 -0.746970859E+00 0.142572622E+05 -0.226135206E-06 0.548868269E-06 -0.430355597E-07 39 E 0.147626242E+05 -0.429780137E+01 0.288814799E+01 -0.304757868E+00 0.149890959E+05 0.388023543E-04 0.119394518E-04 -0.618374107E-04 40 E 0.151279011E+05 -0.283528380E+00 -0.344914963E+00 -0.517265679E+01 0.151549439E+05 0.578953097E-05 -0.232437361E-05 -0.313548893E-06 41 E 0.153081868E+05 -0.454632395E+01 -0.267295086E+01 0.822919525E+00 0.154615975E+05 0.293299254E-05 0.261598633E-05 0.207601081E-03 42 E 0.160387138E+05 -0.376372858E+01 0.378830859E+01 0.112051133E+00 0.161519399E+05 0.109588397E-04 0.316172083E-04 0.172641456E-03 43 E 0.164039594E+05 ================ PARENT CYCLER 5.658Gfh-ff3 ======================= Parent cycler number 166 Approximate search space (synodic periods after J2000) 20 Number of steps to walk eccentricity/inclination 3 / 3 Number of cycles 7 Total delta v over 45.11 years (km/s) 0.004689 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 1 171.6 ****** 6.628 10.310 10.310 7 93.4 5.383 5.383 9.486 9.486 13 139.8 5.878 5.878 8.645 8.645 19 93.4 6.879 6.879 11.854 11.854 25 128.8 5.544 5.544 7.443 7.443 31 113.4 7.621 7.621 10.757 10.757 37 94.8 5.288 5.288 8.514 8.514 AVERAGE 119.3 6.099 6.174 9.573 9.573 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 16093.718352 LEG E/M time start vinfx vinfy vinfz time dv dvx dvy dxz (days) (km/s) (km/s) (km/s) (days) (km/s) (km/s) (km/s) 1 E -0.794812489E+02 -0.649042041E+01 0.240829214E+00 0.132148904E+01 -0.537425534E+02 -0.110222766E-05 -0.267165975E-04 -0.133288665E-05 2 M 0.921100546E+02 -0.102570600E+02 -0.797841944E+00 0.673773112E+00 0.182680093E+03 0.289702562E-07 0.374837990E-07 -0.673077820E-09 3 E 0.695910311E+03 0.655404819E+00 0.485332031E+01 -0.268769275E+01 0.805490913E+03 0.566225369E-04 0.427297374E-05 -0.381141859E-03 4 E 0.106117898E+04 0.510645673E+00 -0.315482512E-01 -0.554647279E+01 0.108842056E+04 -0.253555309E-05 0.105167667E-05 -0.479231070E-06 5 E 0.124278948E+04 -0.506490204E+01 0.185289679E+01 -0.159531898E-02 0.142829678E+04 0.947566998E-06 -0.136705913E-05 -0.790520212E-07 6 E 0.195627909E+04 -0.219799136E+01 -0.489210424E+01 -0.470123098E+00 0.207315540E+04 0.252546798E-06 -0.845877967E-06 0.117767399E-03 7 E 0.232151758E+04 0.465932899E+01 -0.246788252E+01 -0.108538956E+01 0.233552892E+04 -0.102881349E-05 -0.103024070E-05 -0.261813374E-07 8 M 0.241492649E+04 0.791291694E+01 -0.522202793E+01 -0.308291783E+00 0.250704440E+04 -0.175962311E-08 -0.474481106E-09 0.200570595E-09 9 E 0.302904590E+04 0.321796317E+01 0.451114189E+01 -0.148810471E+01 0.328107336E+04 -0.365953325E-05 -0.105583674E-05 -0.126453073E-03 10 E 0.339430309E+04 -0.419200550E+00 0.371729322E+00 -0.572780180E+01 0.343516596E+04 -0.119426466E-06 0.201844796E-06 -0.152582647E-06 11 E 0.358004341E+04 0.572015031E+01 -0.118299724E+01 0.590878692E+00 0.390146705E+04 -0.834856589E-06 0.698455760E-06 -0.639734056E-04 12 E 0.431055170E+04 0.416267580E+01 0.411947164E+01 0.792861469E-01 0.442378299E+04 0.614612153E-07 0.157089210E-06 -0.334152053E-05 13 E 0.467581392E+04 -0.190845401E+01 0.544396791E+01 0.112875450E+01 0.469678399E+04 -0.483048339E-07 0.201390097E-06 0.337076688E-08 14 M 0.481561438E+04 -0.700689426E+01 0.501227518E+01 -0.724022632E+00 0.504916832E+04 -0.119793169E-07 0.873075260E-09 -0.446264965E-08 15 E 0.535876308E+04 -0.611906659E+01 -0.113999906E+01 -0.282256027E+01 0.560349071E+04 -0.222790081E-05 0.202133193E-05 0.622939779E-04 16 E 0.572402820E+04 0.229556714E-01 -0.784265760E+00 -0.680635556E+01 0.575085102E+04 -0.497346229E-07 -0.731174465E-07 0.167110588E-07 17 E 0.590284698E+04 0.463170587E+00 0.541158143E+01 0.424720214E+01 0.652377988E+04 0.663513151E-03 0.687621433E-04 -0.233143114E-04 18 E 0.663335628E+04 -0.562579033E+01 0.642743488E+00 0.392173996E+01 0.675024213E+04 0.335488539E-05 0.641226376E-04 0.441566226E-03 19 E 0.699862456E+04 -0.519248511E+01 -0.448344124E+01 -0.503435401E+00 0.701264151E+04 -0.159383993E-07 0.356304531E-07 0.262591578E-08 20 M 0.709207085E+04 -0.429410800E+01 -0.110371249E+02 0.518231053E+00 0.729834837E+04 -0.395663891E-06 0.863583219E-06 0.390262971E-08 236 21 E 0.771715426E+04 -0.537465069E+01 0.153604879E+01 -0.718652504E+00 22 E 0.808239171E+04 0.848501761E-01 0.506464401E+00 -0.560319617E+01 23 E 0.826859961E+04 0.119752990E+01 -0.542049762E+01 0.194587279E+00 24 E 0.899907350E+04 0.536170484E+01 -0.133665509E+01 0.430012782E+00 25 E 0.936434506E+04 0.199990340E+01 0.510617815E+01 0.812971625E+00 26 M 0.949319431E+04 0.930639710E+00 0.728703888E+01 -0.119409996E+01 27 E 0.100473227E+05 -0.455608431E+01 0.486192845E+01 -0.322053164E+01 28 E 0.104126036E+05 -0.755489038E+00 -0.529794207E+00 -0.733963050E+01 29 E 0.105938134E+05 0.525603049E+01 0.205903342E+01 0.513482670E+01 30 E 0.113243211E+05 -0.201576049E+01 0.465939764E+01 0.570256934E+01 31 E 0.116895783E+05 -0.696825796E+01 0.305440988E+01 0.443432757E+00 32 M 0.118029309E+05 -0.101377670E+02 -0.351655933E+01 0.759114770E+00 33 E 0.124007381E+05 -0.898406308E+00 0.529673978E+01 -0.954695853E+00 34 E 0.127659944E+05 0.486783109E+00 0.135231768E+00 -0.545181905E+01 35 E 0.129487381E+05 -0.529339995E+01 -0.221332821E-01 -0.176283230E-02 36 E 0.136651537E+05 -0.348744196E+00 -0.526546991E+01 -0.337867549E+00 37 E 0.140304233E+05 0.509227607E+01 -0.109189992E+01 -0.914220162E+00 38 M 0.141252255E+05 0.824816430E+01 -0.199506694E+01 -0.687234681E+00 39 E 0.147510316E+05 0.173380256E+00 0.437317806E+01 -0.284984031E+01 40 E 0.151162977E+05 -0.458066932E+00 0.746693670E-01 -0.521975977E+01 41 E 0.153003211E+05 0.525830462E+01 0.126073102E+01 0.108218865E+00 42 E 0.160307827E+05 0.119019929E+01 0.525901093E+01 -0.128637445E+00 43 E 0.163960491E+05 ================ PARENT CYCLER 5.658Gfh+ff3 ======================= Parent cycler number 167 Approximate search space (synodic periods after J2000) 17 Number of steps to walk eccentricity/inclination 3 / 3 Number of cycles 7 Total delta v over 44.81 years (km/s) 0.054344 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 1 116.7 ****** 5.950 7.385 7.385 7 117.0 7.592 7.591 10.393 10.393 13 96.6 5.328 5.327 9.523 9.523 19 140.1 5.857 5.856 8.643 8.643 25 93.5 6.882 6.882 11.840 11.840 31 129.2 5.531 5.531 7.442 7.442 37 113.6 7.608 7.608 10.724 10.724 AVERAGE 115.2 6.466 6.392 9.421 9.421 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 13753.826276 LEG E/M time start vinfx vinfy vinfz (days) (km/s) (km/s) (km/s) 1 E 0.159366871E+02 0.357976301E+01 0.471144876E+01 0.621330897E+00 2 M 0.132609033E+03 0.228884968E+01 0.693243083E+01 -0.111251657E+01 3 E 0.690000711E+03 -0.367746199E+01 0.533836620E+01 0.353976601E+01 4 E 0.105526630E+04 -0.823902117E+00 -0.414458490E+00 0.731039406E+01 5 E 0.123702191E+04 -0.415839399E+01 -0.168978557E+01 -0.614756407E+01 6 E 0.196753273E+04 -0.163990608E+01 0.559263809E+01 -0.488484854E+01 7 E 0.233277872E+04 -0.641107987E+01 0.402658939E+01 0.560184994E+00 8 M 0.244977938E+04 -0.101479256E+02 -0.219218222E+01 0.486043928E+00 9 E 0.301526970E+04 -0.279371193E+00 -0.196841169E+01 0.509635932E+01 10 E 0.338052497E+04 0.468207582E+00 -0.193867676E+00 0.544441980E+01 11 E 0.356086788E+04 0.481823370E+01 -0.169173749E+01 -0.149969020E+01 12 E 0.429137741E+04 -0.144549560E+01 -0.209331729E+01 -0.467429070E+01 13 E 0.465662782E+04 0.470024797E+01 -0.225845286E+01 -0.108581475E+01 14 M 0.475319885E+04 0.785591586E+01 -0.537352216E+01 -0.305732262E+00 15 E 0.536863232E+04 0.273990693E+01 0.391792743E+01 0.310945254E+01 16 E 0.573388985E+04 -0.411228778E+00 0.369978186E+00 0.569006393E+01 17 E 0.591964504E+04 -0.207185851E+01 0.511935212E+01 -0.187610737E+01 18 E 0.665014654E+04 0.416669728E+01 0.408525067E+01 0.490025795E-01 19 E 0.701540989E+04 -0.191911717E+01 0.541633875E+01 0.113031499E+01 20 M 0.715547400E+04 -0.700433857E+01 0.501148130E+01 -0.719791604E+00 21 E 0.769873986E+04 -0.608739046E+01 -0.114405142E+01 0.286610921E+01 22 E 0.806400474E+04 0.240683532E-01 -0.780239795E+00 0.679897443E+01 23 E 0.824282237E+04 0.143902860E+01 -0.466939523E+01 -0.485271512E+01 24 E 0.897333290E+04 -0.365920944E+01 0.684655304E+00 -0.579680905E+01 25 E 0.933859053E+04 -0.520052724E+01 -0.447835344E+01 -0.504206083E+00 26 M 0.943205465E+04 -0.427594215E+01 -0.110283222E+02 0.521044309E+00 27 E 0.100565395E+05 -0.492934644E+01 0.149291766E+01 0.218002586E+01 28 E 0.104217743E+05 0.870115271E-01 0.500255573E+00 0.556201895E+01 29 E 0.106079672E+05 0.276490485E+01 0.464358077E+01 -0.110758486E+01 30 E 0.113384274E+05 0.533440156E+01 -0.138087401E+01 0.496033691E+00 31 E 0.117036992E+05 0.196082800E+01 0.510783316E+01 0.813191143E+00 32 M 0.118329202E+05 0.940271768E+00 0.728563554E+01 -0.119261622E+01 33 E 0.123871778E+05 -0.438418172E+01 0.462557393E+01 0.375933385E+01 34 E 0.127524598E+05 -0.754958535E+00 -0.526641343E+00 0.733596496E+01 35 E 0.129336726E+05 -0.369713418E+01 -0.253007203E+01 -0.617102929E+01 36 E 0.136641821E+05 -0.231631739E+01 0.509806143E+01 -0.517246250E+01 37 E 0.140294394E+05 -0.695129323E+01 0.306050145E+01 0.441362974E+00 38 M 0.141430307E+05 -0.100984880E+02 -0.353107839E+01 0.749210720E+00 39 E 0.147388152E+05 -0.558761779E+00 0.447776597E+01 0.286735284E+01 40 E 0.151040786E+05 0.473550678E+00 0.113019982E+00 0.533696572E+01 41 E 0.152866949E+05 0.503489775E+01 0.105529471E+01 -0.665418532E+00 42 E 0.160171968E+05 -0.298785575E+00 -0.793246547E+00 -0.510377819E+01 43 E 0.163824478E+05 ================ PARENT CYCLER 5.658Gfh+ff3 ======================= Parent cycler number 168 Approximate search space (synodic periods after J2000) 14 Number of steps to walk eccentricity/inclination 3 / 3 Number of cycles 7 Total delta v over 44.92 years (km/s) 0.008882 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 1 82.6 ****** 10.086 11.851 11.851 7 121.0 5.634 5.634 7.378 7.378 13 117.2 7.563 7.563 10.393 10.393 19 96.5 5.328 5.328 9.525 9.525 25 140.2 5.848 5.848 8.639 8.639 31 93.9 6.856 6.856 11.736 11.736 37 139.3 5.194 5.194 6.796 6.796 AVERAGE 113.0 6.071 6.644 9.474 9.474 0.783403025E+04 0.811032290E+04 0.883106451E+04 0.905751695E+04 0.938367245E+04 0.960401999E+04 0.102920609E+05 0.104868996E+05 0.112147449E+05 0.114375508E+05 0.117065812E+05 0.119822731E+05 0.125139675E+05 0.127934059E+05 0.132353043E+05 0.137783873E+05 0.140446437E+05 0.143442577E+05 0.148167795E+05 0.151439012E+05 0.155340688E+05 0.161440152E+05 -0.154200000E-05 -0.231107045E-05 -0.819575590E-05 -0.172298818E-06 -0.956236164E-06 -0.114970600E-08 0.985109401E-05 -0.179411633E-05 -0.178879208E-04 -0.129373397E-04 0.335893395E-05 -0.124835771E-07 0.353067214E-05 -0.482110071E-06 0.227240881E-05 -0.165508257E-06 -0.563834263E-07 0.576957755E-09 -0.562224228E-05 0.544812324E-05 -0.260365458E-05 0.534971418E-05 0.466192596E-06 -0.326749796E-06 -0.115314155E-04 0.127168446E-05 0.414261481E-06 0.219513316E-08 0.481133246E-04 0.176997441E-05 0.392671266E-04 -0.945628889E-05 -0.603301248E-06 -0.163026717E-07 0.100061329E-05 0.337069406E-06 -0.609391443E-06 -0.620616710E-07 -0.138196827E-06 0.481002046E-09 -0.169475088E-05 -0.424112321E-05 0.582960112E-06 0.172737041E-05 0.111066025E-03 -0.134503546E-06 -0.129830631E-03 0.114785078E-05 -0.128069091E-06 0.604728588E-09 0.880659997E-03 0.121379385E-06 0.224155195E-04 -0.386185515E-04 -0.238314619E-06 0.416524494E-09 -0.898711105E-04 -0.694639514E-07 0.171686784E-08 -0.614959508E-06 -0.120580480E-07 0.616494678E-10 0.104012469E-04 -0.108176487E-05 0.124306614E-07 0.157152354E-03 time dv (days) 0.334375390E+02 0.244087369E+03 0.752095862E+03 0.108252964E+04 0.135390364E+04 0.208075898E+04 0.235032882E+04 0.253460293E+04 0.313945649E+04 0.340757640E+04 0.369235960E+04 0.441921505E+04 0.467111347E+04 0.489474855E+04 0.544168383E+04 0.576175313E+04 0.607305036E+04 0.670493604E+04 0.703641950E+04 0.730215578E+04 0.791059349E+04 0.809082738E+04 0.856424700E+04 0.909021534E+04 0.935261015E+04 0.979425586E+04 0.101661099E+05 0.104497032E+05 0.107540592E+05 0.113932181E+05 0.117230823E+05 0.119714846E+05 0.125990414E+05 0.127995751E+05 0.131163000E+05 0.137628016E+05 0.140464781E+05 0.143694288E+05 0.148118679E+05 0.151314710E+05 0.153962702E+05 0.162253899E+05 dvx (km/s) 0.116436998E-02 0.174541939E-05 -0.245760806E-03 0.477982602E-04 0.129470966E-03 -0.130405078E-03 0.591648560E-04 -0.402664220E-07 0.123874330E-02 -0.104429786E-04 0.312242490E-04 -0.828086892E-05 0.525197614E-04 0.736261560E-06 -0.640467880E-04 -0.834369808E-05 -0.130810342E-04 0.417775148E-04 -0.440089528E-05 -0.482634514E-08 0.407607850E-04 -0.138779436E-04 -0.838885606E-05 -0.118238330E-03 -0.636347148E-05 0.278590361E-06 -0.655833177E-04 -0.874162617E-04 -0.569042060E-04 0.100575166E-04 -0.724771393E-04 0.697375242E-07 0.123584241E-02 -0.924494865E-05 0.336111216E-04 0.629397506E-04 0.184665834E-04 -0.517056556E-06 0.170370505E-03 -0.438660326E-04 0.837585628E-04 0.392629630E-03 dvy (km/s) 0.725445035E-03 -0.141313371E-05 -0.257712752E-03 0.199624519E-04 0.871816071E-04 -0.116024736E-04 -0.395468013E-04 -0.532609803E-07 0.354701174E-03 0.438190607E-04 0.591712827E-05 -0.233059841E-03 -0.253266447E-04 0.774433268E-07 0.520297529E-05 0.184157717E-04 0.137440278E-04 -0.360238708E-04 0.242350536E-04 -0.305095372E-09 0.717232710E-04 -0.210427196E-04 -0.147702143E-03 0.961673393E-03 0.136367785E-04 -0.256525203E-06 0.232113378E-03 -0.721593692E-06 0.158546593E-03 0.483800863E-04 0.405015937E-04 -0.754769341E-07 0.755910309E-04 0.108060012E-04 0.586804295E-04 0.109456707E-03 -0.345286289E-05 -0.797358629E-06 0.577232838E-04 0.491240895E-04 0.396554667E-04 0.108087699E-03 dxz (km/s) -0.787797960E-04 0.949344929E-06 0.484822980E-03 0.132578966E-04 0.381376562E-04 0.117269774E-02 -0.199359394E-05 -0.882265463E-08 -0.185373513E-04 0.488954817E-05 0.101306331E-03 0.407411650E-04 0.454874581E-05 0.476816487E-07 -0.274183018E-03 -0.360437202E-05 0.392396216E-03 0.445796111E-05 0.294445542E-06 0.209068258E-08 -0.887559023E-04 -0.480346876E-05 -0.452272060E-04 0.194530591E-03 0.100134682E-05 -0.288584021E-07 -0.843997925E-03 0.308976575E-05 0.620427982E-03 0.828637314E-05 -0.619014947E-05 0.510170992E-07 -0.118433881E-02 -0.384079243E-06 0.224739500E-04 0.375282834E-03 -0.131568639E-05 -0.997266606E-07 0.150671340E-03 0.556423073E-05 0.703420386E-05 0.949420832E-04 237 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 11413.934201 LEG E/M time start vinfx vinfy vinfz (days) (km/s) (km/s) (km/s) 1 E -0.964732407E+00 -0.920735780E+01 -0.409796626E+01 -0.408910058E+00 2 M 0.816842999E+02 -0.548630268E+01 -0.104879104E+02 0.591228760E+00 3 E 0.702527890E+03 -0.492000549E+01 0.204971690E+01 0.215930457E+01 4 E 0.106776970E+04 0.145041802E+00 0.516975560E+00 0.571793061E+01 5 E 0.125379376E+04 -0.880864572E+00 -0.515063181E+01 -0.215229793E+01 6 E 0.198430328E+04 0.528196290E+01 -0.195471489E+01 -0.250220678E+00 7 E 0.234957262E+04 0.303190049E+01 0.470883253E+01 0.612758366E+00 8 M 0.247059964E+04 0.239336569E+01 0.689278434E+01 -0.109291207E+01 9 E 0.302968247E+04 -0.372062308E+01 0.550479518E+01 0.312143118E+01 10 E 0.339494712E+04 -0.815691687E+00 -0.406966037E+00 0.726656408E+01 11 E 0.357671601E+04 0.459124713E+01 0.310554699E+01 -0.514889446E+01 12 E 0.430722668E+04 -0.208882222E+01 0.656145833E+01 -0.316564184E+01 13 E 0.467247461E+04 -0.639139529E+01 0.400526281E+01 0.560459463E+00 14 M 0.478964041E+04 -0.101477179E+02 -0.218980457E+01 0.486260857E+00 15 E 0.535518309E+04 -0.263298226E+00 -0.193045229E+01 0.510929755E+01 16 E 0.572043837E+04 0.467972450E+00 -0.193477614E+00 0.544264177E+01 17 E 0.590078247E+04 -0.523978623E+01 0.917150533E+00 0.102265677E+00 18 E 0.663130590E+04 -0.174840730E+01 -0.279615173E+01 -0.417455084E+01 19 E 0.699655623E+04 0.470118356E+01 -0.226028801E+01 -0.108589451E+01 20 M 0.709308762E+04 0.785719980E+01 -0.537573116E+01 -0.304134916E+00 21 E 0.770840604E+04 0.332336893E+01 0.456282587E+01 0.852344904E+00 22 E 0.807366467E+04 -0.411702250E+00 0.371755932E+00 0.569944685E+01 23 E 0.825942445E+04 0.567826010E+01 -0.125550107E+01 -0.556798361E+00 24 E 0.898993087E+04 0.416762511E+01 0.407242967E+01 -0.608579241E-03 25 E 0.935519460E+04 -0.192180530E+01 0.540612125E+01 0.113085302E+01 26 M 0.949537536E+04 -0.700327420E+01 0.500802276E+01 -0.714131171E+00 27 E 0.100387611E+05 -0.608622691E+01 -0.115428905E+01 0.283411435E+01 28 E 0.104040255E+05 0.255013426E-01 -0.777664735E+00 0.678681546E+01 29 E 0.105828416E+05 0.194094686E+01 0.544954764E+01 -0.371184862E+01 30 E 0.113133521E+05 -0.382080226E+01 0.666149749E+00 -0.566219636E+01 31 E 0.116786096E+05 -0.520804440E+01 -0.442950217E+01 -0.508306595E+00 32 M 0.117725071E+05 -0.415794532E+01 -0.109613660E+02 0.547464271E+00 33 E 0.123918176E+05 -0.488005650E+01 0.185143603E+01 0.794910081E+00 34 E 0.127570528E+05 0.106889922E+00 0.448614672E+00 0.526960310E+01 35 E 0.129431137E+05 -0.248701813E+00 -0.520174686E+01 -0.131538202E+00 36 E 0.136736055E+05 0.462192987E+01 -0.160419216E+01 0.179917411E+01 37 E 0.140388692E+05 0.175529560E+01 0.480896965E+01 0.874894863E+00 38 M 0.141781994E+05 0.325548863E+00 0.676270465E+01 -0.587130770E+00 39 E 0.147658057E+05 -0.355505949E+01 0.642680901E+00 0.372071195E+01 40 E 0.151310718E+05 -0.153066414E+00 -0.427918741E+00 0.518069338E+01 41 E 0.153103894E+05 0.151245155E+00 0.529053076E+01 0.269707251E-01 42 E 0.160408286E+05 -0.482084044E+01 0.214358556E+01 0.271945931E+00 43 E 0.164061080E+05 ================ PARENT CYCLER 5.658Gfh-fff3 ======================= Parent cycler number 169 Approximate search space (synodic periods after J2000) 17 Number of steps to walk eccentricity/inclination 27 / 27 Number of cycles 7 Total delta v over 44.65 years (km/s) 0.017508 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 1 91.9 ****** 12.037 7.446 7.446 8 115.0 7.947 7.947 10.402 10.402 15 96.7 5.326 5.326 9.527 9.527 22 140.1 5.852 5.852 8.645 8.645 29 93.5 6.870 6.870 11.848 11.848 36 129.1 5.536 5.536 7.437 7.437 43 114.2 7.600 7.600 10.611 10.611 AVERAGE 111.5 6.522 7.310 9.416 9.416 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 13753.826276 LEG E/M time start vinfx vinfy vinfz (days) (km/s) (km/s) (km/s) 1 E 0.573974892E+02 0.928129178E+01 0.762485905E+01 0.786175573E+00 2 M 0.149267375E+03 0.135499636E+01 0.720265400E+01 -0.131205833E+01 3 E 0.692583426E+03 -0.412201688E+01 0.538152754E+01 -0.368569502E+01 4 E 0.105785453E+04 -0.877455284E+00 -0.486373560E+00 -0.764247089E+01 5 E 0.123944686E+04 0.399648607E+01 -0.525970662E+01 0.443948559E+01 6 E 0.160468670E+04 0.998369721E+00 0.379054373E+00 0.789162077E+01 7 E 0.196993958E+04 -0.165133882E+01 0.528503224E+01 0.572330539E+01 8 E 0.233519367E+04 -0.666426367E+01 0.429342202E+01 0.557168769E+00 9 M 0.245017424E+04 -0.101499880E+02 -0.222261246E+01 0.483789620E+00 10 E 0.301504606E+04 -0.163264403E+01 -0.521389159E+01 -0.556879968E+00 11 E 0.338028541E+04 0.469408594E+00 -0.198525492E+00 -0.545471497E+01 12 E 0.356061643E+04 -0.245758856E+01 -0.437825974E+01 0.179694603E+01 13 E 0.392586731E+04 -0.255961728E+01 -0.466485922E+01 0.165893059E+00 14 E 0.429113649E+04 -0.133819253E+01 -0.182425512E+01 -0.481591316E+01 15 E 0.465638695E+04 0.470406328E+01 -0.224921037E+01 -0.108601164E+01 16 M 0.475310465E+04 0.785366357E+01 -0.538388186E+01 -0.304281161E+00 17 E 0.536851832E+04 0.230357303E+01 0.341315274E+01 -0.394967919E+01 18 E 0.573377554E+04 -0.410706967E+00 0.369393511E+00 -0.569352301E+01 19 E 0.591953223E+04 -0.213679619E+01 -0.343965622E+01 0.420061258E+01 20 E 0.628480598E+04 0.321954898E+00 -0.477846669E+00 0.579956881E+01 21 E 0.665006517E+04 0.409257545E+01 0.398607871E+01 0.116933033E+01 22 E 0.701532899E+04 -0.192386375E+01 0.540928511E+01 0.113079996E+01 23 M 0.715543405E+04 -0.700383485E+01 0.501593427E+01 -0.724724918E+00 24 E 0.769861900E+04 -0.616160072E+01 -0.113896668E+01 -0.273171694E+01 25 E 0.806388422E+04 0.225308527E-01 -0.784478399E+00 -0.680756427E+01 26 E 0.824270344E+04 0.635953784E+01 0.942418608E+00 0.249509475E+01 27 E 0.860794097E+04 0.646616270E+00 0.821796816E+00 0.680477604E+01 28 E 0.897319594E+04 -0.571221477E+01 0.643875920E+00 0.377987197E+01 29 E 0.933846401E+04 -0.518541840E+01 -0.447859866E+01 -0.503470856E+00 30 M 0.943198175E+04 -0.428849556E+01 -0.110327543E+02 0.519550010E+00 31 E 0.100568208E+05 -0.535401960E+01 0.155124123E+01 -0.695587538E+00 32 E 0.104220577E+05 0.857134459E-01 0.503004395E+00 -0.558580218E+01 33 E 0.106082597E+05 0.354806971E+01 -0.108680704E+01 0.410878542E+01 34 E 0.109735155E+05 0.511120193E+01 -0.133404722E+01 0.170790273E+01 35 E 0.113387475E+05 0.535240881E+01 -0.135442948E+01 0.408392335E+00 36 E 0.117040192E+05 0.198479045E+01 0.510402935E+01 0.813440781E+00 time dv (days) 0.114326224E+02 0.174810839E+03 0.815752852E+03 0.109567331E+04 0.187472685E+04 0.203909368E+04 0.236772667E+04 0.258241620E+04 0.327440979E+04 0.342221246E+04 0.419765008E+04 0.442775849E+04 0.469004948E+04 0.487447182E+04 0.555607350E+04 0.574748998E+04 0.622221278E+04 0.677740603E+04 0.701103594E+04 0.718538539E+04 0.796043449E+04 0.810152864E+04 0.858084727E+04 0.905933097E+04 0.937622171E+04 0.961492022E+04 0.102834883E+05 0.104308480E+05 0.112037755E+05 0.114850231E+05 0.116926942E+05 0.119335278E+05 0.125050405E+05 0.127849619E+05 0.132207006E+05 0.137831846E+05 0.140597688E+05 0.142663404E+05 0.150214920E+05 0.151579695E+05 0.155441299E+05 0.160956205E+05 dvx (km/s) -0.461589175E-06 0.175487707E-08 -0.170385594E-04 -0.125046751E-07 -0.155512275E-02 0.146918414E-08 -0.864188825E-08 -0.175942387E-08 0.135337248E-05 -0.103982090E-05 -0.163469405E-04 -0.693142510E-04 -0.118918167E-05 0.513051213E-09 0.320914856E-03 -0.194754640E-07 0.516055467E-06 0.494168407E-03 0.476118682E-07 0.875317247E-10 -0.282421069E-06 0.462153486E-09 -0.339192503E-05 -0.511602850E-09 0.112430012E-09 0.362652925E-10 -0.289280306E-05 -0.137324927E-06 0.598468466E-03 -0.452157167E-03 -0.685783030E-07 0.373432979E-10 -0.206104627E-05 -0.161794972E-08 -0.801056232E-08 -0.376674515E-06 -0.115792728E-07 -0.890842509E-11 -0.274006708E-05 -0.208381895E-06 0.100029483E-06 0.267467163E-07 dvy (km/s) 0.936537147E-06 -0.242473013E-08 -0.450147608E-04 -0.120758431E-08 0.242580635E-03 0.464773105E-08 0.923929339E-08 0.112625277E-08 0.124801105E-04 -0.369341105E-06 0.110284732E-03 -0.244520799E-04 0.851424566E-06 0.810385608E-09 0.959238051E-03 0.804174156E-07 -0.216126084E-06 -0.177799359E-05 -0.223018983E-07 0.205052502E-10 0.327208702E-08 -0.986249294E-09 0.330551696E-05 0.388187765E-09 -0.553416895E-09 -0.133328767E-10 0.145630010E-05 -0.216744142E-06 0.145370580E-03 0.560223624E-03 0.135495796E-06 -0.114109849E-09 -0.580548603E-05 -0.199476370E-07 -0.323435650E-07 -0.102950438E-05 -0.272899517E-07 0.236937984E-10 0.381944429E-05 -0.198697901E-06 0.241460303E-06 0.711311554E-07 dxz (km/s) -0.794051942E-07 0.375539492E-10 -0.635616855E-03 0.872401507E-09 -0.878594417E-04 0.302755035E-08 0.145065097E-09 0.188809258E-09 -0.255914322E-03 -0.280827852E-06 -0.595794154E-04 0.667386643E-03 0.446499123E-07 -0.870991585E-11 -0.166925642E-03 0.908342323E-08 -0.114386152E-03 -0.227066720E-04 0.410384880E-08 -0.608186431E-11 0.191005660E-04 0.252910486E-09 0.217064840E-03 0.667188199E-09 -0.626983324E-11 -0.128653828E-11 -0.725707866E-04 -0.492840381E-07 0.277516049E-04 0.126096443E-03 0.982678288E-08 -0.225738384E-12 -0.221257484E-03 0.348760047E-08 -0.186741603E-07 0.183996442E-04 0.127849199E-08 -0.466150538E-12 -0.787736551E-04 -0.453417864E-07 -0.196887957E-09 -0.485475960E-07 time dv (days) 0.711779720E+02 0.241631103E+03 0.937315067E+03 0.110688446E+04 0.148780995E+04 0.186401625E+04 0.208316835E+04 0.235244076E+04 0.253490501E+04 0.326340882E+04 0.340733506E+04 0.367384420E+04 0.404275345E+04 0.443723667E+04 0.467089460E+04 0.484541670E+04 0.562785094E+04 0.576163904E+04 0.617522385E+04 0.641264670E+04 0.676329696E+04 0.703634475E+04 0.739443543E+04 0.794334670E+04 0.809070710E+04 0.849106496E+04 0.885631435E+04 0.909008173E+04 0.935249167E+04 0.963193023E+04 0.101736966E+05 0.104499880E+05 0.107689723E+05 0.110867374E+05 0.114446763E+05 0.117233769E+05 dvx (km/s) 0.255634330E-05 -0.113141661E-07 0.172261710E-04 -0.397343296E-07 0.329948313E-04 0.309062816E-03 -0.132975786E-03 -0.330779069E-06 0.137951625E-09 -0.121048659E-04 -0.192765323E-07 0.199949108E-04 -0.102393302E-06 0.471351731E-03 0.587627297E-07 0.135348374E-09 -0.230621026E-04 -0.984411725E-08 -0.663470244E-04 0.330709360E-03 0.464739799E-05 -0.446324001E-08 0.108726435E-10 -0.214335695E-05 -0.462650393E-07 -0.170499906E-04 0.162377558E-02 0.202939730E-05 -0.146548106E-07 0.703424985E-09 -0.139652145E-05 -0.242679947E-05 0.178209438E-03 0.904424414E-05 -0.367464017E-06 -0.227083491E-05 dvy (km/s) 0.212099865E-05 0.821322981E-08 0.446555363E-04 -0.234644904E-06 0.784191979E-04 -0.954410882E-03 -0.579194218E-04 0.424351572E-07 0.214050209E-09 0.145176269E-04 0.975167043E-07 -0.107048626E-04 0.289667395E-07 0.387904258E-04 -0.322409627E-07 0.453423626E-10 0.786891128E-06 0.228062511E-07 0.344447888E-05 -0.614752666E-04 -0.467582859E-05 0.269296730E-07 -0.925728880E-11 0.231904659E-05 -0.672780598E-07 0.195066368E-04 0.465270873E-03 0.569577050E-04 0.342378354E-07 -0.159590434E-08 0.993000605E-06 -0.229160740E-06 -0.401330775E-03 0.240346305E-04 0.278727176E-05 0.101847731E-05 dxz (km/s) -0.227472344E-06 0.919851312E-09 0.730238858E-03 0.396839833E-07 0.107424799E-02 -0.300876027E-03 -0.722954850E-03 -0.303806178E-08 0.156575326E-11 -0.172251451E-03 -0.103640978E-07 -0.358661331E-03 0.969201668E-05 -0.166019943E-04 0.525803464E-08 -0.617057862E-11 -0.339769114E-03 0.444493955E-08 -0.890583840E-03 0.333874889E-04 0.162734046E-03 0.294591639E-09 -0.244173599E-11 0.646866390E-04 0.153214470E-07 0.576793258E-03 -0.122081473E-03 0.407098487E-03 0.254120183E-08 0.937245723E-11 0.109658839E-03 -0.121353518E-06 -0.109857599E-04 -0.526971836E-03 -0.826515731E-05 -0.178102684E-06 238 37 M 0.118330704E+05 0.929095292E+00 0.728259111E+01 -0.118955379E+01 38 E 0.123873597E+05 -0.456094026E+01 0.484768767E+01 -0.319921055E+01 39 E 0.127526408E+05 -0.751065550E+00 -0.529070141E+00 -0.732470519E+01 40 E 0.129338418E+05 0.434826502E+01 -0.467377148E+01 0.415165228E+01 41 E 0.132990825E+05 0.948675223E+00 0.334986626E+00 0.754948124E+01 42 E 0.136643361E+05 -0.232308019E+01 0.507953347E+01 0.517637777E+01 43 E 0.140295939E+05 -0.692275405E+01 0.310643984E+01 0.433730035E+00 44 M 0.141438024E+05 -0.996928819E+01 -0.357143606E+01 0.665173267E+00 45 E 0.147240744E+05 0.274081814E+00 -0.499597439E+01 -0.254833448E+00 46 E 0.150893577E+05 0.403787279E+00 -0.177789183E-01 -0.500573987E+01 47 E 0.152709721E+05 -0.218955983E+00 0.469510878E+01 0.125263963E+01 48 E 0.156362099E+05 -0.469476813E+00 -0.144643351E+01 0.460481260E+01 49 E 0.160014666E+05 -0.571406679E+00 -0.478891388E+01 -0.484027859E+00 50 E 0.163667334E+05 ================ PARENT CYCLER 5.658Gfh+fff3 ======================= Parent cycler number 170 Approximate search space (synodic periods after J2000) 10 Number of steps to walk eccentricity/inclination 81 / 81 Number of cycles 7 Total delta v over 44.66 years (km/s) 0.023400 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 1 99.8 ****** 11.530 7.656 7.656 8 108.7 7.969 7.969 11.009 11.009 15 113.8 5.775 5.775 8.612 8.612 22 135.1 6.269 6.269 9.135 9.135 29 87.9 6.510 6.510 12.331 12.331 36 169.2 7.251 7.251 7.520 7.520 43 110.4 7.192 7.192 11.070 11.070 AVERAGE 117.8 6.828 7.499 9.619 9.619 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 8294.078101 LEG E/M time start vinfx vinfy vinfz (days) (km/s) (km/s) (km/s) 1 E 0.516037828E+02 0.646623418E+01 0.949426948E+01 0.996869452E+00 2 M 0.151388613E+03 -0.104656910E+01 0.744051485E+01 -0.146880643E+01 3 E 0.689438292E+03 -0.542475729E+01 0.409042301E+01 0.373731281E+01 4 E 0.105471605E+04 -0.726722396E+00 -0.705017741E+00 0.768349218E+01 5 E 0.123536648E+04 -0.120655716E+00 0.164476179E+01 -0.780322391E+01 6 E 0.160061974E+04 -0.393090980E+01 0.573135132E+01 -0.391371274E+01 7 E 0.196586889E+04 -0.457995045E+01 0.640703553E+01 -0.134924036E+01 8 E 0.233112720E+04 -0.758777535E+01 0.240943579E+01 0.349955387E+00 9 M 0.243980139E+04 -0.100208355E+02 -0.452006742E+01 0.584819876E+00 10 E 0.300365116E+04 -0.742024265E+00 -0.502727067E+01 0.304126713E+01 11 E 0.336890964E+04 0.561163175E+00 -0.164269566E+00 0.590949547E+01 12 E 0.354969214E+04 -0.209501390E+01 -0.529629682E+01 -0.906713152E+00 13 E 0.391493036E+04 -0.107654055E+01 -0.172222457E+01 -0.541560980E+01 14 E 0.428018852E+04 0.343255819E+00 0.327056190E+01 -0.475073923E+01 15 E 0.464543791E+04 0.560541962E+01 0.107956397E+01 -0.877304026E+00 16 M 0.475922193E+04 0.849985427E+01 -0.128862395E+01 -0.516320858E+00 17 E 0.536916182E+04 0.133889738E+01 0.487585744E+01 0.339285974E+01 18 E 0.573440126E+04 -0.592079345E+00 0.246809825E+00 0.606399163E+01 19 E 0.591924670E+04 0.242277258E+01 0.440762544E+01 -0.373950379E+01 20 E 0.628447419E+04 0.285509558E+01 0.556571223E+01 0.103743555E+00 21 E 0.664974307E+04 0.287045577E+01 0.554816325E+01 -0.187459810E+00 22 E 0.701499297E+04 -0.296657188E+01 0.542854157E+01 0.101441636E+01 23 M 0.715005161E+04 -0.848757309E+01 0.335673642E+01 -0.374964771E+00 24 E 0.769583870E+04 -0.475523879E+01 -0.242142939E+01 0.375716909E+01 25 E 0.806110156E+04 0.214331621E+00 -0.677432903E+00 0.650481672E+01 26 E 0.823986356E+04 0.315807323E-01 0.756441682E+00 -0.647912464E+01 27 E 0.860511864E+04 -0.508462720E+01 -0.834735040E+00 -0.400323539E+01 28 E 0.897035878E+04 -0.639295116E+01 -0.126887194E+01 -0.241833564E+00 29 E 0.933563976E+04 -0.234518634E+01 -0.602308435E+01 -0.776343609E+00 30 M 0.942356055E+04 -0.152884987E+01 -0.122041969E+02 0.875590369E+00 31 E 0.100021079E+05 0.518913859E+01 -0.408727558E+01 0.345641658E+01 32 E 0.103673650E+05 0.668762090E+00 0.638727630E+00 0.741467961E+01 33 E 0.105520027E+05 0.382969374E+01 -0.514321533E+01 -0.340755806E+01 34 E 0.109172412E+05 -0.932940003E+00 -0.324218482E+00 -0.719774795E+01 35 E 0.112824990E+05 -0.536723608E+01 0.412977453E+01 -0.263872998E+01 36 E 0.116477330E+05 -0.455871324E+00 0.702976340E+01 0.171651019E+01 37 M 0.118169461E+05 -0.438198333E+00 0.743090725E+01 -0.106595785E+01 38 E 0.123817493E+05 -0.470748706E+01 0.307178989E+01 0.418256505E+01 39 E 0.127470208E+05 -0.547225359E+00 -0.617435898E+00 0.694511503E+01 40 E 0.129274575E+05 -0.298165846E+00 0.142656459E+01 -0.703941134E+01 41 E 0.132927156E+05 -0.402974955E+01 0.491635565E+01 -0.336049010E+01 42 E 0.136579710E+05 -0.411622691E+01 0.499517097E+01 -0.317052090E+01 43 E 0.140232239E+05 -0.712296370E+01 0.971992716E+00 0.206210448E+00 44 M 0.141336390E+05 -0.934423449E+01 -0.588926461E+01 0.739970468E+00 45 E 0.147199680E+05 0.155652006E+01 -0.474557303E+01 0.170766857E+00 46 E 0.150852433E+05 0.389996690E+00 0.925742397E-01 0.499100851E+01 47 E 0.152678941E+05 -0.131147974E+01 0.381022531E+01 -0.268739922E+01 48 E 0.156331404E+05 0.236244624E+00 -0.277317408E+01 -0.395188736E+01 49 E 0.159983949E+05 0.246440887E+00 -0.270763682E+01 -0.399096312E+01 50 E 0.163636497E+05 ================ PARENT CYCLER 5.658Gffh-ff3 ======================= Parent cycler number 171 Approximate search space (synodic periods after J2000) 13 Number of steps to walk eccentricity/inclination 243 / 243 Number of cycles 7 Total delta v over 44.86 years (km/s) 0.384124 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 1 96.5 ****** 11.127 10.928 10.935 8 107.6 5.272 5.533 8.527 8.527 15 134.2 6.369 6.369 9.138 9.138 22 87.8 6.525 6.524 12.356 12.356 29 169.4 7.293 7.294 7.525 7.525 36 109.2 7.221 7.221 11.319 11.319 43 126.9 5.975 5.976 7.784 7.784 AVERAGE 118.8 6.443 7.149 9.654 9.655 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 10633.970176 0.119383854E+05 0.126320980E+05 0.128015650E+05 0.131822055E+05 0.135547600E+05 0.137775660E+05 0.140467252E+05 0.144397411E+05 0.149724670E+05 0.151165998E+05 0.155229862E+05 0.158444062E+05 0.160927833E+05 0.141851624E-08 0.133956450E-04 0.546742619E-07 0.205254320E-04 0.182358525E-03 0.385696619E-05 -0.104982952E-06 -0.158777631E-09 -0.221518093E-04 -0.313730557E-05 -0.139508670E-04 0.449720428E-03 -0.973537562E-06 -0.282378144E-08 0.486587027E-04 -0.617402666E-07 0.666287806E-04 -0.722051371E-03 0.234660538E-05 0.164752625E-07 -0.653433311E-10 0.114787978E-04 0.484849220E-05 0.543038979E-06 0.187661116E-02 -0.173170740E-06 -0.507595671E-10 0.879652453E-03 -0.252939255E-08 0.928418034E-03 -0.281776822E-03 0.234583665E-04 0.745722886E-08 0.506688119E-10 -0.862850899E-04 -0.553898155E-06 0.270032754E-03 0.130558137E-03 0.160141198E-05 time dv (days) 0.665715073E+02 0.420413453E+03 0.930521612E+03 0.110891118E+04 0.149104376E+04 0.172115195E+04 0.209005671E+04 0.234742833E+04 0.252437886E+04 0.308035544E+04 0.339602701E+04 0.371770173E+04 0.407564395E+04 0.449568566E+04 0.466250552E+04 0.485071292E+04 0.562117704E+04 0.577691572E+04 0.602881495E+04 0.640136023E+04 0.672644555E+04 0.703525177E+04 0.731924561E+04 0.794421744E+04 0.808791586E+04 0.849919467E+04 0.871834309E+04 0.902515093E+04 0.934882788E+04 0.951034266E+04 0.102504827E+05 0.103950606E+05 0.106688790E+05 0.111619640E+05 0.115235535E+05 0.116731150E+05 0.119863871E+05 0.126301339E+05 0.127740863E+05 0.130552978E+05 0.134095973E+05 0.137785045E+05 0.140397862E+05 0.142567681E+05 0.149647025E+05 0.151126409E+05 0.153372909E+05 0.159034287E+05 0.162686834E+05 dvx (km/s) 0.965885552E-05 0.490383504E-08 0.997865192E-05 0.149488360E-06 -0.270568606E-04 -0.441940768E-04 0.560102210E-05 -0.143715271E-05 -0.620048872E-08 0.155847230E-04 -0.110698950E-04 0.101793271E-03 0.818141608E-03 -0.260728726E-04 0.218283537E-05 -0.583576245E-09 -0.235553160E-04 0.802239521E-06 -0.149287799E-03 0.326669785E-05 -0.944502602E-06 0.941453308E-06 0.160679021E-09 -0.259567223E-04 -0.329049743E-07 0.259021234E-03 0.186557635E-04 0.108093950E-07 -0.183675995E-06 -0.967427188E-10 0.336159774E-06 0.440770870E-06 0.866854753E-04 0.561701186E-05 0.211386858E-04 0.975564697E-06 -0.370093146E-10 0.869893888E-04 -0.900368679E-07 0.308853286E-03 0.692277413E-05 -0.240259090E-04 -0.438454582E-06 -0.244995248E-08 -0.723011117E-04 -0.871067815E-05 0.819675155E-05 -0.294595360E-03 0.521274892E-03 dvy (km/s) 0.680557665E-05 -0.100983645E-07 0.763256704E-04 -0.685113036E-06 0.308062220E-03 -0.345161759E-04 0.562078377E-05 0.917436704E-07 -0.654668718E-08 -0.182089904E-04 0.649098333E-05 -0.295816406E-04 0.363903855E-03 -0.110552313E-02 0.145139185E-04 0.193844751E-07 -0.416028789E-04 0.338538504E-06 0.952198019E-04 -0.169195714E-06 0.289764731E-06 -0.679918930E-06 -0.231863753E-09 0.229182231E-04 -0.987531294E-07 0.102411642E-03 -0.116672723E-03 -0.472090100E-07 0.149329238E-06 0.135327707E-09 -0.115567541E-04 0.199196518E-06 0.616912585E-04 -0.208043273E-03 -0.154138782E-04 0.511331894E-07 -0.430325872E-10 0.133765306E-03 -0.313363627E-06 0.794754951E-03 0.601367516E-05 -0.206864456E-04 0.124207658E-06 -0.347808608E-08 0.170079077E-04 0.110475278E-04 0.687271754E-05 0.842035741E-03 -0.123369637E-02 dxz (km/s) -0.200530275E-06 -0.378761831E-09 -0.110433919E-02 -0.990330256E-07 -0.175369832E-03 0.395415329E-03 -0.110945314E-03 -0.969838157E-07 -0.300564081E-08 -0.121215262E-03 0.246817535E-05 0.882807037E-04 -0.274962487E-04 0.449039862E-04 -0.891108159E-06 0.106863161E-08 0.869787919E-03 0.130455102E-06 0.168426531E-02 0.103148566E-03 -0.107553034E-05 -0.791899123E-07 -0.220998343E-10 -0.460323577E-03 -0.185981222E-07 0.469472093E-04 0.964234243E-03 -0.406270990E-09 -0.156096031E-07 -0.925588343E-10 -0.979232115E-04 -0.114841381E-06 0.972675831E-03 -0.116335978E-03 -0.543497280E-03 0.307203435E-08 -0.227478569E-10 -0.121485229E-02 -0.711177117E-07 -0.583850909E-04 -0.800762051E-04 0.295855840E-03 0.181681977E-07 -0.446726393E-09 -0.167883531E-03 0.988978160E-06 -0.196632445E-04 0.712927562E-04 -0.114151289E-03 239 LEG E/M time start vinfx vinfy vinfz (days) (km/s) (km/s) (km/s) 1 E 0.844863117E+01 -0.104574448E+02 0.378760141E+01 0.313653361E+00 2 M 0.104957111E+03 -0.973319971E+01 -0.493978987E+01 0.657025636E+00 3 E 0.673648532E+03 0.231237655E+00 -0.294104848E+01 -0.455382412E+01 4 E 0.103890485E+04 0.861766315E+00 0.422953757E+01 -0.327233919E+01 5 E 0.140415270E+04 0.469297114E+00 -0.408431765E-01 -0.538944899E+01 6 E 0.158556072E+04 -0.963686374E+00 -0.515161712E+01 0.241503590E+00 7 E 0.195082667E+04 -0.532323476E+00 -0.430273484E+00 -0.522953329E+01 8 E 0.231607838E+04 0.539255340E+01 0.928201830E+00 -0.820979056E+00 9 M 0.242371844E+04 0.846240800E+01 -0.903135487E+00 -0.524632685E+00 10 E 0.303031212E+04 0.207948002E+00 0.227349635E+01 -0.572931365E+01 11 E 0.339556387E+04 -0.236471181E+01 -0.423547557E+01 -0.381163388E+01 12 E 0.376081389E+04 -0.611156302E+00 0.219297872E+00 -0.614701020E+01 13 E 0.394560427E+04 0.249928844E+01 0.486238947E+01 0.324002626E+01 14 E 0.431087528E+04 0.282634048E+01 0.568245097E+01 0.266376804E+00 15 E 0.467612246E+04 -0.296217816E+01 0.554717372E+01 0.101068056E+01 16 M 0.481031802E+04 -0.849369371E+01 0.334817762E+01 -0.375801603E+00 17 E 0.535595878E+04 -0.506535233E+01 -0.253332071E+01 -0.325075575E+01 18 E 0.572122125E+04 0.153456101E+01 -0.218620148E+00 -0.636962334E+01 19 E 0.608647324E+04 0.233354271E+00 -0.696733180E+00 -0.651893252E+01 20 E 0.626523575E+04 -0.435683349E+01 -0.602629309E+00 0.483974552E+01 21 E 0.663049802E+04 -0.637732288E+01 -0.126432821E+01 -0.677307809E+00 22 E 0.699577911E+04 -0.234598909E+01 -0.603841620E+01 -0.776048227E+00 23 M 0.708355943E+04 -0.155390913E+01 -0.122253026E+02 0.886555116E+00 24 E 0.766153857E+04 0.487611874E+01 -0.384503568E+01 -0.422980479E+01 25 E 0.802679783E+04 0.269274517E+01 -0.150343612E+01 -0.686750613E+01 26 E 0.839205020E+04 0.680485842E+00 0.658646147E+00 -0.746510646E+01 27 E 0.857665137E+04 0.221063367E+01 -0.357028919E+01 0.598227386E+01 28 E 0.894191468E+04 -0.269915719E+01 0.148422667E+01 0.661587297E+01 29 E 0.930715901E+04 -0.434803954E+00 0.707165950E+01 0.173312635E+01 30 M 0.947660285E+04 -0.409395589E+00 0.743767964E+01 -0.107021521E+01 31 E 0.100415138E+05 -0.574322875E+01 0.404612623E+01 0.475276449E-01 32 E 0.104067803E+05 -0.409796794E+01 0.258342799E+01 -0.506847158E+01 33 E 0.107720319E+05 -0.575944974E+00 -0.609065997E+00 -0.697447379E+01 34 E 0.109524829E+05 -0.310026883E+01 0.410493163E+01 0.506972543E+01 35 E 0.113177403E+05 -0.415659167E+01 0.509700038E+01 0.302152222E+01 36 E 0.116829923E+05 -0.716651874E+01 0.859451273E+00 0.222148884E+00 37 M 0.117921748E+05 -0.970947846E+01 -0.578039935E+01 0.651178389E+00 38 E 0.123570296E+05 -0.294024881E+00 -0.535501192E+01 -0.302186882E+01 39 E 0.127222862E+05 0.338565708E+00 -0.174294892E+01 -0.591598077E+01 40 E 0.130875406E+05 0.630168503E+00 -0.106597314E+00 -0.613655776E+01 41 E 0.132686391E+05 -0.588688894E+00 0.179850041E+00 0.594975281E+01 42 E 0.136338956E+05 0.129641682E+00 0.397595258E+01 0.446618411E+01 43 E 0.139991168E+05 0.534387445E+01 0.259181406E+01 -0.659064743E+00 44 M 0.141260115E+05 0.764818499E+01 0.120858175E+01 -0.794320363E+00 45 E 0.147515752E+05 -0.884811766E+00 0.221036125E+01 -0.467595375E+01 46 E 0.151168252E+05 -0.120013169E+01 0.377425610E+01 -0.345639114E+01 47 E 0.154820844E+05 -0.460982930E+00 -0.992431870E-01 -0.521699172E+01 48 E 0.156648780E+05 -0.542793087E+00 0.535416726E+01 0.524801962E+00 49 E 0.160301476E+05 -0.523814593E+00 0.531987685E+01 -0.683346731E+00 50 E 0.163954138E+05 ================ PARENT CYCLER 5.658Gffh+ff3 ======================= Parent cycler number 172 Approximate search space (synodic periods after J2000) 4 Number of steps to walk eccentricity/inclination 3 / 3 Number of cycles 7 Total delta v over 44.82 years (km/s) 0.021825 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 1 135.2 ****** 6.328 8.857 8.857 8 91.5 6.670 6.670 11.845 11.845 15 130.0 5.674 5.674 7.563 7.563 22 107.7 7.739 7.739 11.509 11.509 29 80.8 6.937 6.937 8.224 8.224 36 131.7 6.721 6.721 9.103 9.103 43 90.5 6.327 6.328 11.455 11.455 AVERAGE 109.6 6.678 6.628 9.794 9.794 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 3614.293950 LEG E/M time start vinfx vinfy vinfz (days) (km/s) (km/s) (km/s) 1 E -0.429816893E+00 -0.225203219E+01 0.581530026E+01 0.107309960E+01 2 M 0.134800248E+03 -0.772452467E+01 0.429687168E+01 -0.558884675E+00 3 E 0.678841864E+03 -0.635548323E+01 -0.198296323E+01 0.760599513E-01 4 E 0.104411385E+04 -0.568966785E+01 -0.184290636E+01 0.296115491E+01 5 E 0.140935394E+04 0.125634281E+00 -0.749790396E+00 0.663305823E+01 6 E 0.158810463E+04 -0.354627719E+01 0.214292482E+00 -0.566745416E+01 7 E 0.195335839E+04 -0.667589374E+01 -0.301804130E+00 -0.289780317E+00 8 E 0.231862402E+04 -0.413223209E+01 -0.519812730E+01 -0.629629847E+00 9 M 0.241014139E+04 -0.287213412E+01 -0.114846509E+02 0.406668957E+00 10 E 0.303992929E+04 -0.452491694E+01 0.568502466E+00 0.348615183E+01 11 E 0.340515504E+04 -0.152712054E+01 0.539908134E+00 0.549454691E+01 12 E 0.377041333E+04 0.645250074E-03 0.559245555E+00 0.570090770E+01 13 E 0.395683914E+04 0.549713462E+01 -0.381628250E+00 -0.141389103E+01 14 E 0.432208699E+04 0.565725600E+01 -0.364874243E+00 -0.104165699E+00 15 E 0.468736459E+04 0.167131262E+01 0.534195930E+01 0.927641332E+00 16 M 0.481735227E+04 -0.256990130E+00 0.744837059E+01 -0.128731278E+01 17 E 0.536647939E+04 -0.567233183E+01 0.493316186E+01 0.470106875E-01 18 E 0.573175047E+04 -0.467775583E+01 0.383498665E+01 0.443970059E+01 19 E 0.609701214E+04 -0.728520902E+00 -0.621736144E+00 0.745901967E+01 20 E 0.627781358E+04 -0.144588139E+01 0.321534539E+01 -0.689616242E+01 21 E 0.664306905E+04 -0.420344381E+01 0.642588063E+01 -0.107518155E+01 22 E 0.700832008E+04 -0.745489292E+01 0.204035563E+01 0.384486327E+00 23 M 0.711603633E+04 -0.106245270E+02 -0.434135038E+01 0.852307926E+00 24 E 0.773518584E+04 -0.314177830E+01 0.559286740E+01 0.323073482E+01 25 E 0.810042267E+04 -0.130149731E+01 0.312108550E+01 0.632303028E+01 26 E 0.846568027E+04 0.688790120E+00 0.515371605E+00 0.711838691E+01 27 E 0.864990366E+04 0.326928967E+01 -0.557541476E+01 -0.256449795E+01 28 E 0.901514839E+04 0.358245251E+01 -0.594565358E+01 0.191467285E+00 29 E 0.938042470E+04 0.675500531E+01 -0.141543386E+01 -0.696946802E+00 30 M 0.946124634E+04 0.818537594E+01 0.575874763E+00 -0.555912996E+00 31 E 0.100535886E+05 0.130227799E+01 0.638332544E+01 -0.471072672E+00 time dv (days) 0.229249031E+02 0.190260824E+03 0.863581815E+03 0.114482672E+04 0.143136390E+04 0.169148785E+04 0.221015538E+04 0.233222439E+04 0.257536686E+04 0.322754806E+04 0.350879137E+04 0.378853245E+04 0.405883828E+04 0.441679696E+04 0.469625179E+04 0.507222558E+04 0.541805340E+04 0.586001701E+04 0.611328761E+04 0.642960377E+04 0.674738797E+04 0.700894616E+04 0.726851276E+04 0.771632746E+04 0.826055935E+04 0.841974038E+04 0.867161983E+04 0.916471372E+04 0.933257559E+04 0.967997080E+04 0.101072618E+05 0.104798306E+05 0.107990996E+05 0.110328395E+05 0.114346209E+05 0.116993697E+05 0.120124682E+05 0.124227758E+05 0.127880320E+05 0.131147054E+05 0.133891737E+05 0.138749416E+05 0.140181510E+05 0.144200264E+05 0.148392352E+05 0.151716141E+05 0.155095035E+05 0.157196684E+05 0.160849375E+05 dvx (km/s) -0.607747407E-04 0.197828358E-06 0.391146862E-04 -0.664194603E-04 -0.870233045E-05 0.517089835E-04 -0.637579267E-04 -0.227931457E-04 -0.274174109E-07 -0.276868951E-03 0.339522165E-04 0.196702459E-05 0.747225649E-04 0.218947024E-05 -0.210550123E-05 -0.336801165E-07 -0.647534202E-04 0.926001240E-03 0.937545476E-04 0.952151353E-03 0.179933094E-04 -0.833195442E-04 -0.811942830E-07 0.187781500E-04 -0.567439760E-04 0.701678294E-04 -0.170699228E-03 0.721487436E-05 -0.801569157E-04 0.118606628E-07 -0.198796499E-05 0.433182372E-04 -0.161823571E-04 0.244678333E-03 -0.226841759E-04 0.154743760E-04 -0.224866532E-06 -0.365823189E-03 0.799284675E-03 -0.150735342E-03 0.900851769E-03 0.540702685E-03 -0.104879441E-04 0.140363586E-05 -0.127873602E-02 0.653820917E-03 -0.110700479E-02 0.684846825E-03 -0.102357965E-02 dvy (km/s) -0.349338157E-04 0.371610485E-06 0.169461501E-04 0.521496079E-04 0.371884937E-04 0.100987941E-04 0.593960421E-03 -0.217014699E-04 0.188777034E-07 -0.916072126E-03 -0.180709875E-04 -0.151220838E-05 -0.337944601E-04 -0.184157863E-06 0.739736523E-06 0.949853492E-07 0.599610076E-04 -0.719805394E-03 0.112090739E-03 -0.897043850E-03 -0.226498293E-04 0.665207923E-04 0.156860716E-07 0.367694580E-04 0.350030500E-03 -0.704912872E-05 -0.136250102E-03 -0.522255636E-03 0.354681396E-05 -0.217456673E-08 -0.855219000E-05 -0.286961583E-04 -0.688461087E-05 0.325376253E-03 -0.376380527E-04 -0.543399379E-05 0.473904179E-07 0.203293338E-04 -0.197360038E-03 0.201945755E-03 0.220490447E-03 -0.978344808E-03 -0.221357947E-03 0.500144156E-05 -0.511129600E-03 0.361644673E-03 0.270666987E-03 0.126747627E-03 -0.195025631E-03 dxz (km/s) -0.353162105E-05 -0.100291274E-07 0.985483400E-05 0.515837736E-03 -0.209886132E-05 -0.132402199E-03 0.255619399E-05 0.220079585E-05 0.165693007E-09 0.174831783E-03 0.310208485E-03 -0.508950785E-06 0.710637958E-03 0.487744442E-05 0.136281207E-06 0.130301568E-06 -0.182320966E-03 0.355792944E-03 -0.171128316E-04 0.311280842E-03 -0.138533656E-03 -0.707833130E-05 -0.637091180E-07 -0.216777455E-04 -0.151283720E-03 0.116566108E-04 0.280398396E-04 0.218796327E-03 -0.514353028E-06 0.110677054E-07 -0.455944444E-04 0.712028916E-03 0.391697200E-05 0.317973048E-03 -0.123544976E-03 -0.723307440E-06 0.133774325E-06 -0.560834992E-04 0.156836380E-03 -0.382961608E-04 0.135065796E-03 0.231089329E-02 -0.163461620E-05 -0.598219834E-06 0.263674565E-03 -0.440683273E-03 0.188896895E-03 0.260107713E-03 -0.200830975E-03 time dv (days) 0.198546928E+02 0.249048987E+03 0.733632662E+03 0.128882471E+04 0.143616654E+04 0.186204495E+04 0.207389605E+04 0.233235162E+04 0.263056715E+04 0.315314927E+04 0.358778419E+04 0.383193385E+04 0.407006597E+04 0.443532305E+04 0.470686274E+04 0.509740710E+04 0.560755831E+04 0.597647579E+04 0.612413235E+04 0.639834789E+04 0.675994938E+04 0.702447751E+04 0.720890875E+04 0.785206162E+04 0.837071329E+04 0.850252494E+04 0.877043442E+04 0.913568957E+04 0.939254795E+04 0.955009768E+04 0.102982999E+05 dvx (km/s) 0.847735000E-04 -0.895292836E-07 0.728650040E-05 0.361207243E-04 -0.307609283E-05 0.105666842E-02 -0.191857306E-05 -0.684708506E-06 0.671386499E-07 -0.101120728E-04 -0.879152525E-03 0.110798739E-06 0.902982338E-06 -0.699665821E-07 -0.195059711E-06 0.164974509E-10 0.136463667E-05 -0.931267182E-05 -0.160173351E-06 0.793268318E-04 0.378561888E-04 0.169397162E-06 0.584680705E-09 -0.925242382E-04 -0.257821636E-03 -0.118246781E-06 0.395069384E-04 0.844667073E-06 0.101102027E-06 0.588491752E-09 -0.107779787E-05 dvy (km/s) -0.121658830E-03 -0.480197808E-07 -0.115362128E-04 -0.324167550E-04 -0.399045802E-06 0.357111529E-03 0.322900645E-05 -0.320689702E-05 -0.941583198E-07 -0.144237908E-03 0.551529046E-03 -0.459899148E-07 0.982982528E-05 0.626165229E-06 0.318928893E-07 -0.447669855E-10 0.472684699E-06 0.206799958E-04 -0.108873586E-07 0.912492215E-04 0.232502738E-04 -0.113601209E-06 0.127969824E-08 -0.569290752E-04 0.845977759E-03 -0.503391230E-06 0.232223879E-04 0.601185745E-06 0.717690604E-06 0.270626314E-08 0.759668644E-06 dxz (km/s) 0.324978399E-05 0.590139099E-08 0.628950712E-05 0.776004576E-03 0.308653815E-07 0.107498916E-03 0.325016168E-04 0.286384719E-06 -0.173674158E-08 -0.119139879E-02 -0.120167837E-03 -0.752965079E-08 0.211266909E-03 0.105390654E-03 0.121989156E-07 0.950942964E-11 0.172863954E-03 -0.164892134E-03 -0.335564203E-07 -0.916610136E-05 0.104240445E-04 0.872594709E-08 -0.318622056E-10 -0.958074836E-03 0.277598305E-03 0.968583977E-07 0.459411489E-03 -0.163707177E-03 -0.622220031E-07 -0.769086386E-11 -0.892425032E-05 240 32 E 0.104188294E+05 0.116000294E+01 0.600033178E+01 0.228051073E+01 0.106672101E+05 0.151526228E-04 0.417913057E-05 -0.358228566E-03 33 E 0.107840951E+05 -0.682632915E+00 0.205590744E+00 0.647607341E+01 0.108339361E+05 -0.160990727E-06 -0.112995876E-06 -0.271177974E-07 34 E 0.109686913E+05 0.250149014E+01 0.569510083E+01 -0.250582418E+01 0.110855763E+05 0.349238782E-04 -0.143571639E-04 -0.431401021E-03 35 E 0.113339570E+05 0.265070953E+01 0.611608043E+01 0.832380417E+00 0.114508348E+05 0.281105034E-06 -0.227780262E-07 0.294793329E-05 36 E 0.116992001E+05 -0.294286329E+01 0.595944276E+01 0.997643719E+00 0.117189515E+05 0.189733013E-06 0.198023918E-06 0.131453150E-08 37 M 0.118308760E+05 -0.849384482E+01 0.326121325E+01 -0.304543904E+00 0.120496907E+05 0.124092973E-08 -0.207034577E-08 0.143019087E-08 38 E 0.123779127E+05 -0.563019734E+01 -0.294723577E+01 -0.196232997E+00 0.125934138E+05 0.114922306E-03 0.136758115E-03 -0.107607699E-03 39 E 0.127431689E+05 -0.542348430E+01 -0.287525060E+01 0.174032486E+01 0.129659742E+05 0.350498325E-03 0.127504454E-03 0.437637811E-04 40 E 0.131084235E+05 0.248875262E+00 -0.640512129E+00 0.632718089E+01 0.131352402E+05 -0.368107129E-06 -0.915352026E-06 -0.157450415E-06 41 E 0.132872020E+05 -0.332967768E+01 -0.467456169E+00 -0.537147057E+01 0.135611471E+05 0.675814580E-03 0.198016408E-03 0.102249983E-03 42 E 0.136524621E+05 -0.605070896E+01 -0.146283630E+01 -0.114001661E+01 0.137730057E+05 -0.310086052E-04 0.666150452E-04 -0.328071937E-03 43 E 0.140177456E+05 -0.263721716E+01 -0.569558069E+01 -0.802801733E+00 0.140313148E+05 0.104102341E-05 -0.843810587E-07 0.836485315E-07 44 M 0.141082068E+05 -0.573368758E+00 -0.114327855E+02 0.412414737E+00 0.142678256E+05 0.496887746E-09 -0.709747520E-09 0.101741462E-09 45 E 0.147221254E+05 0.508569639E+01 -0.108333751E+00 -0.387023938E+00 0.147951751E+05 -0.486039816E-05 0.121243897E-04 -0.647516537E-04 46 E 0.150873743E+05 0.506703352E+01 -0.959415487E-01 0.392820398E+00 0.153357595E+05 0.101806007E-04 0.161229064E-04 -0.634884436E-04 47 E 0.154526466E+05 0.368410453E-01 0.419339794E+00 0.507264416E+01 0.154805937E+05 0.871198767E-05 0.236446941E-05 -0.541443361E-06 48 E 0.156389601E+05 0.128190717E+01 -0.531506177E+00 -0.484228343E+01 0.157667998E+05 0.455433479E-04 -0.120744459E-03 0.168092725E-04 49 E 0.160042164E+05 0.479999109E+01 -0.791638086E+00 0.132782414E+01 0.161137938E+05 -0.286432425E-05 0.158349748E-04 -0.937701679E-04 50 E 0.163694743E+05 ================ PARENT CYCLER 5.679Gfh-f3 ======================= Parent cycler number 175 Approximate search space (synodic periods after J2000) 7 Number of steps to walk eccentricity/inclination 27 / 27 Number of cycles 7 Total delta v over 44.97 years (km/s) 0.019963 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 1 100.9 ****** 5.677 11.847 11.847 6 131.3 5.609 5.609 7.636 7.636 11 107.6 7.750 7.750 11.529 11.529 16 80.1 7.011 7.011 8.294 8.294 21 130.8 6.768 6.768 9.180 9.180 26 89.7 6.387 6.387 11.649 11.649 31 138.2 5.602 5.602 7.145 7.145 AVERAGE 111.2 6.521 6.401 9.611 9.611 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 5954.266420 LEG E/M time start vinfx vinfy vinfz time dv dvx dvy dxz (days) (km/s) (km/s) (km/s) (days) (km/s) (km/s) (km/s) 1 E -0.333891608E+02 -0.295984604E+01 -0.480776263E+01 -0.590010118E+00 -0.182533172E+02 0.277463583E-04 -0.446150467E-04 0.367450871E-05 2 M 0.675164630E+02 -0.314995175E+01 -0.114117745E+02 0.437251556E+00 0.100591833E+04 0.876515122E-08 -0.139383950E-07 -0.122821227E-08 3 E 0.142751916E+04 -0.498637507E+01 0.806526997E+00 -0.252907111E+01 0.154074967E+04 -0.442481261E-05 0.325010145E-04 -0.162492804E-03 4 E 0.179277887E+04 0.113961047E-01 0.543235001E+00 -0.563996138E+01 0.182073618E+04 -0.869705586E-05 -0.999913034E-05 -0.198007072E-05 5 E 0.197916092E+04 0.544250489E+01 -0.633452310E+00 0.123490677E+01 0.209239716E+04 0.422191973E-05 -0.458985403E-04 0.609605842E-03 6 E 0.234443912E+04 0.142207826E+01 0.534555734E+01 0.928393258E+00 0.236414059E+04 0.476766723E-05 -0.687236277E-05 -0.700025260E-06 7 M 0.247578227E+04 -0.201022613E+00 0.752959314E+01 -0.125237250E+01 0.335989447E+04 -0.396436821E-08 0.823893908E-08 0.832158899E-09 8 E 0.375710430E+04 -0.510810417E+01 0.429781851E+01 -0.352885753E+01 0.399816012E+04 0.688012464E-04 -0.239107092E-04 -0.831677359E-03 9 E 0.412234040E+04 -0.735677328E+00 -0.648623079E+00 -0.747562077E+01 0.414945982E+04 -0.340415346E-06 0.938832102E-07 0.511560491E-07 10 E 0.430313654E+04 -0.301856910E+01 0.504528967E+01 0.506998513E+01 0.442001659E+04 -0.624398955E-04 -0.398730117E-04 -0.463461354E-03 11 E 0.466838669E+04 -0.746789108E+01 0.203716689E+01 0.385689843E+00 0.468452151E+04 -0.175474189E-06 0.115170659E-06 -0.906530076E-08 12 M 0.477595217E+04 -0.106487584E+02 -0.433547680E+01 0.854237362E+00 0.526198746E+04 0.563745575E-09 -0.283714490E-08 -0.738182341E-10 13 E 0.612605020E+04 -0.294932993E+01 0.530803252E+01 -0.390232635E+01 0.618083870E+04 0.138751883E-04 0.978975446E-05 0.133201815E-04 14 E 0.649130688E+04 0.674827439E+00 0.537841494E+00 -0.718271558E+01 0.651894353E+04 -0.677663777E-06 -0.351394441E-06 -0.184750803E-06 15 E 0.667555119E+04 0.325092332E+01 -0.547984369E+01 0.293581589E+01 0.679244026E+04 -0.773482080E-04 -0.633889357E-04 0.682189846E-03 16 E 0.704082954E+04 0.682598846E+01 -0.143706286E+01 -0.703174477E+00 0.705285117E+04 0.169451869E-09 -0.467753009E-06 0.375222023E-07 17 M 0.712097372E+04 0.825349595E+01 0.592309445E+00 -0.560070722E+00 0.775622432E+04 0.192170337E-07 -0.900401209E-08 0.654897495E-08 18 E 0.844441247E+04 0.655850803E+00 0.446147235E+01 -0.476616083E+01 0.862338851E+04 -0.308069458E-03 -0.103745434E-02 -0.831929939E-04 19 E 0.880966970E+04 -0.702250474E+00 0.228683084E+00 -0.653173870E+01 0.883735567E+04 0.375235398E-05 -0.124387729E-05 -0.904460229E-06 20 E 0.899424286E+04 0.254518033E+01 0.583029546E+01 0.230341324E+01 0.911112047E+04 -0.143122977E-04 0.747248733E-05 -0.237761677E-03 21 E 0.935948539E+04 -0.298307213E+01 0.599236923E+01 0.100012579E+01 0.937910853E+04 0.269186934E-05 0.338358341E-05 0.465357055E-07 22 M 0.949030629E+04 -0.855857505E+01 0.330695547E+01 -0.307326036E+00 0.982255315E+04 -0.326377381E-08 0.642060519E-08 0.334859694E-08 23 E 0.107681789E+05 -0.541062938E+01 -0.286238904E+01 -0.191032595E+01 0.109946433E+05 -0.643146332E-05 -0.270365726E-04 0.585148217E-04 24 E 0.111334442E+05 0.238573486E+00 -0.634919741E+00 -0.637890044E+01 0.111602613E+05 -0.949833913E-05 -0.587515007E-05 0.483986203E-06 25 E 0.113122248E+05 -0.501230802E+01 -0.105323797E+01 0.381858307E+01 0.114254646E+05 -0.125909670E-03 0.311955638E-03 0.186010035E-02 26 E 0.116775145E+05 -0.259658316E+01 -0.578034382E+01 -0.798189997E+00 0.116909660E+05 -0.112335471E-04 0.156216630E-05 -0.906343492E-06 27 M 0.117671907E+05 -0.726142779E+00 -0.116234525E+02 0.252442864E+00 0.127344734E+05 0.180383852E-02 -0.151422967E-02 -0.863892552E-04 28 E 0.131295607E+05 -0.541428171E+01 -0.469264723E+00 -0.130319524E+01 0.132354824E+05 0.444552695E-04 -0.974760562E-04 0.228855183E-03 29 E 0.134948081E+05 -0.108169769E+00 0.499401879E+00 -0.556154238E+01 0.135227845E+05 -0.640910756E-04 -0.133659630E-04 -0.187636773E-06 30 E 0.136813175E+05 0.549673073E+01 0.681665377E+00 0.685914634E+00 0.137835938E+05 -0.673901330E-04 0.117499382E-03 0.349664308E-03 31 E 0.140465901E+05 0.129975706E+01 0.534386731E+01 0.106714330E+01 0.140673194E+05 -0.468430389E-04 -0.762207757E-04 0.477528455E-05 32 M 0.141847856E+05 -0.214482261E+01 0.679215328E+01 -0.564634081E+00 0.148604944E+05 0.232115629E-07 -0.446090378E-08 -0.476047091E-09 33 E 0.154842256E+05 -0.530467186E+01 0.472374489E+00 -0.896119728E+00 0.157289435E+05 0.467044450E-05 0.761285157E-05 0.276952168E-04 34 E 0.158494762E+05 -0.881091019E-01 -0.487881290E+00 -0.535800573E+01 0.158763408E+05 0.389497065E-06 0.327533466E-06 -0.706157150E-07 35 E 0.160285735E+05 -0.505816366E+01 0.159789933E+01 0.127301741E+01 0.161418040E+05 -0.951494797E-07 -0.146435761E-05 -0.171633127E-04 36 E 0.163938334E+05 ================ PARENT CYCLER 5.679Gfh+f3 ======================= Parent cycler number 176 Approximate search space (synodic periods after J2000) 16 Number of steps to walk eccentricity/inclination 3 / 3 Number of cycles 7 Total delta v over 44.90 years (km/s) 0.010785 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 1 108.9 ****** 5.587 8.565 8.565 6 134.0 6.356 6.356 9.180 9.180 11 89.1 6.423 6.420 11.879 11.879 16 127.1 6.208 6.208 7.803 7.803 21 105.8 7.735 7.735 11.412 11.412 26 94.1 5.854 5.854 7.943 7.943 31 129.9 6.927 6.927 9.238 9.238 AVERAGE 112.7 6.584 6.441 9.432 9.432 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 12973.942628 LEG E/M time start vinfx vinfy vinfz time dv dvx dvy dxz (days) (km/s) (km/s) (km/s) (days) (km/s) (km/s) (km/s) 1 E -0.262776633E+02 0.543744108E+01 0.976116727E+00 -0.833145466E+00 -0.994185673E+01 0.185645727E-03 0.591046572E-05 -0.800273456E-05 2 M 0.826277139E+02 0.849113221E+01 -0.994089098E+00 -0.523154521E+00 0.778373740E+03 -0.234337577E-06 -0.571877688E-06 -0.165580764E-07 3 E 0.142060084E+04 0.158707364E+01 0.575783390E+01 0.144554895E+01 0.166897616E+04 0.603654473E-05 -0.141740419E-04 0.290112457E-03 4 E 0.178585867E+04 -0.591624350E+00 0.239323884E+00 0.613075509E+01 0.181357891E+04 0.230957982E-05 -0.444573154E-05 0.800879105E-06 5 E 0.197066029E+04 0.276266033E+01 0.544484472E+01 -0.170687142E+01 0.208388706E+04 -0.898629900E-05 0.288497090E-05 0.109188197E-03 6 E 0.233590794E+04 -0.298588425E+01 0.551897035E+01 0.101325506E+01 0.235601277E+04 0.293159934E-05 -0.126253364E-05 -0.203032317E-06 7 M 0.246994017E+04 -0.853904261E+01 0.335544647E+01 -0.325016329E+00 0.268717995E+04 0.603474704E-09 -0.517280566E-09 0.157107818E-09 241 8 E 0.374782126E+04 -0.262808554E+01 -0.174672396E+01 0.562595879E+01 9 E 0.411307604E+04 0.235109708E+00 -0.643210268E+00 0.641297649E+01 10 E 0.429185114E+04 -0.439275940E+01 -0.772131771E+00 -0.462408434E+01 11 E 0.465710847E+04 -0.250831819E+01 -0.585697389E+01 -0.790729137E+00 12 M 0.474619792E+04 -0.952724707E+00 -0.118390265E+02 0.196355898E+00 13 E 0.611580417E+04 -0.582234657E+01 -0.104978478E+01 0.184873825E+01 14 E 0.648106664E+04 -0.181848668E+00 0.607536844E+00 0.614763166E+01 15 E 0.666756267E+04 0.587360360E+01 0.125609754E+01 -0.152320379E+01 16 E 0.703281642E+04 0.157797120E+01 0.591555688E+01 0.102553010E+01 17 M 0.715989964E+04 -0.179437564E+01 0.748171875E+01 -0.130089152E+01 18 E 0.843580637E+04 -0.582790604E+01 0.338313015E+01 0.344286852E+01 19 E 0.880105670E+04 -0.605940456E+00 -0.760646075E+00 0.748582411E+01 20 E 0.898131089E+04 -0.380780065E+01 0.443921719E+01 -0.508453844E+01 21 E 0.934657471E+04 -0.765984365E+01 0.105420171E+01 0.221624736E+00 22 M 0.945236136E+04 -0.973839930E+01 -0.589438328E+01 0.805411850E+00 23 E 0.107947022E+05 -0.249276111E+01 0.424504101E+01 0.350362121E+01 24 E 0.111599677E+05 0.484892595E+00 0.383919312E+00 0.602718065E+01 25 E 0.113442168E+05 0.310567085E+01 -0.487613396E+01 -0.104539272E+01 26 E 0.117094849E+05 0.570851743E+01 0.122734464E+01 -0.415705815E+00 27 M 0.118035808E+05 0.753049818E+01 0.243958603E+01 -0.661117380E+00 28 E 0.131229890E+05 0.105772705E+00 0.633768069E+01 0.223707961E+01 29 E 0.134882338E+05 -0.774386730E+00 0.109984214E+00 0.667910964E+01 30 E 0.136722264E+05 0.155752916E+01 0.596766853E+01 -0.317958119E+01 31 E 0.140374787E+05 -0.364763278E+01 0.581879470E+01 0.908037472E+00 32 M 0.141674246E+05 -0.902536870E+01 0.195419814E+01 0.263067005E+00 33 E 0.154655918E+05 -0.224393526E+01 -0.337227035E+01 0.320297599E+01 34 E 0.158308499E+05 0.341765504E+00 -0.278475477E+00 0.512914483E+01 35 E 0.160103349E+05 0.279191111E+01 0.378885301E+01 -0.182623278E+01 36 E 0.163755851E+05 ================ PARENT CYCLER 5.751Ggf3 ======================= Parent cycler number 177 Approximate search space (synodic periods after J2000) 3 Number of steps to walk eccentricity/inclination 9 / 9 Number of cycles 7 Total delta v over 44.68 years (km/s) 0.001021 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 1 109.2 ****** 9.110 8.410 8.410 5 97.3 8.937 8.937 11.551 11.551 9 125.8 6.271 6.271 7.990 7.990 13 127.8 6.894 6.894 9.642 9.642 17 85.3 6.164 6.164 11.083 11.083 21 150.9 5.699 5.699 8.293 8.293 25 97.8 8.236 8.236 11.453 11.453 AVERAGE 113.4 7.033 7.330 9.775 9.775 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 2835.618682 LEG E/M time start vinfx vinfy vinfz (days) (km/s) (km/s) (km/s) 1 E 0.326492572E+02 0.273010275E+01 0.862364072E+01 0.108161991E+01 2 M 0.141879415E+03 -0.279802042E+01 0.761679722E+01 -0.221160319E+01 3 E 0.673495593E+03 -0.708508704E+01 0.547200993E+01 0.107417154E-02 4 E 0.160217111E+04 -0.852053669E+01 -0.587491323E+00 -0.267392498E+01 5 E 0.233268645E+04 -0.892508987E+01 0.440978904E+00 0.991883019E-01 6 M 0.242995838E+04 -0.914556348E+01 -0.701948147E+01 0.717788982E+00 7 E 0.299516706E+04 -0.194250699E+00 -0.626191477E+01 -0.223732270E-02 8 E 0.391310806E+04 0.508748470E+01 -0.368148260E+01 0.242441431E+00 9 E 0.464366027E+04 0.494813470E+01 0.382872730E+01 -0.433795359E+00 10 M 0.476948295E+04 0.752088764E+01 0.261496219E+01 -0.656732976E+00 11 E 0.536315327E+04 0.803788348E+00 0.683934622E+01 0.600750608E-03 12 E 0.628353025E+04 -0.448712826E+01 0.501997380E+01 -0.141175345E+01 13 E 0.701404597E+04 -0.426446834E+01 0.534800546E+01 0.858320140E+00 14 M 0.714180560E+04 -0.954370681E+01 0.137527196E+01 -0.100562568E-01 15 E 0.769274766E+04 -0.489183338E+01 -0.374498496E+01 0.326485328E-03 16 E 0.861025350E+04 0.607037539E+00 -0.611897673E+01 -0.558144409E+00 17 E 0.934078926E+04 0.377495739E+00 -0.606124719E+01 -0.105798425E+01 18 M 0.942606389E+04 0.329103218E+01 -0.105790348E+02 0.278814283E+00 19 E 0.100326667E+05 0.569054038E+01 0.500369263E+00 0.272633797E-02 20 E 0.109483151E+05 0.468913615E+00 0.505076355E+01 0.259975206E+01 21 E 0.116788222E+05 -0.166196425E+01 0.529369934E+01 0.130248544E+01 22 M 0.118296988E+05 -0.349939258E+01 0.733121994E+01 -0.166675888E+01 23 E 0.123707410E+05 -0.718027897E+01 0.406609091E+01 0.124750679E-02 24 E 0.132965779E+05 -0.796518373E+01 -0.186803169E+01 -0.103316298E+01 25 E 0.140270930E+05 -0.817686333E+01 -0.980018607E+00 -0.685226726E-01 26 M 0.141248993E+05 -0.787902199E+01 -0.827730368E+01 0.765105020E+00 27 E 0.147115721E+05 0.221597527E+01 -0.463410713E+01 0.152559339E-01 28 E 0.156250028E+05 0.472385489E+01 0.201723485E+01 -0.277678559E-02 29 E 0.163534036E+05 ================ PARENT CYCLER 5.751ggF3 ======================= Parent cycler number 178 Approximate search space (synodic periods after J2000) 9 Number of steps to walk eccentricity/inclination 9 / 9 Number of cycles 7 Total delta v over 44.49 years (km/s) 0.000000 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 3 106.3 11.665 11.665 8.908 8.908 7 91.7 7.871 7.871 11.860 11.860 11 119.9 5.711 5.711 7.768 7.768 15 114.3 8.025 8.025 10.452 10.452 19 93.4 5.408 5.408 9.558 9.558 23 140.5 5.715 5.715 8.946 8.946 27 89.5 7.348 7.348 12.257 12.257 AVERAGE 107.9 7.392 7.392 9.964 9.964 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 7465.514376 LEG E/M time start vinfx vinfy vinfz (days) (km/s) (km/s) (km/s) 1 E 0.106279360E+03 0.111694839E+02 0.336011004E+01 -0.150291299E-02 2 E 0.706814262E+03 0.541010940E+01 0.103364854E+02 -0.117665092E-02 3 E 0.164638383E+04 -0.374845058E+00 0.116041963E+02 0.112699011E+01 4 M 0.175264942E+04 -0.665883143E+01 0.590209968E+01 0.430213956E+00 0.398888942E+04 0.413989231E+04 0.456944671E+04 0.467047189E+04 0.547208923E+04 0.623268816E+04 0.650904105E+04 0.678079133E+04 0.705187890E+04 0.784888927E+04 0.867687159E+04 0.882809483E+04 0.909454267E+04 0.936244271E+04 0.994902748E+04 0.109079345E+05 0.111876050E+05 0.114611026E+05 0.117235993E+05 0.120014920E+05 0.133713555E+05 0.135507913E+05 0.137854546E+05 0.140569706E+05 0.149463249E+05 0.155349908E+05 0.158577727E+05 0.160870374E+05 -0.419321847E-03 -0.108790493E-05 -0.148942361E-02 0.253042751E-05 0.198802716E-10 -0.327346002E-05 -0.400847109E-06 -0.216147079E-06 0.408918172E-06 0.328325397E-09 -0.178847666E-05 -0.400732871E-06 0.728040329E-04 -0.442464252E-06 0.162362165E-09 0.782784105E-04 0.171218348E-05 -0.482709222E-05 0.132240134E-05 -0.525884769E-09 -0.233082339E-04 -0.192189506E-06 -0.350949253E-05 0.486944898E-07 -0.560699830E-07 0.308221544E-04 -0.228946980E-04 0.301575247E-04 -0.174612807E-03 -0.517579583E-06 -0.637316725E-03 -0.853964215E-06 -0.369304483E-09 0.111042641E-04 -0.854925022E-06 0.111842815E-05 -0.389187929E-06 0.218957449E-10 -0.197473764E-04 -0.122817206E-06 0.118066946E-03 -0.136491020E-06 -0.354564577E-09 0.506825404E-04 -0.375140685E-06 -0.444712947E-05 0.404169433E-06 0.119232028E-09 -0.157600097E-04 -0.346989890E-06 0.105673151E-04 -0.692355186E-06 0.114809891E-07 -0.490103650E-05 0.458812624E-04 -0.894822399E-05 -0.743376936E-04 -0.265707278E-07 -0.204925807E-03 0.207610045E-06 0.485769988E-10 0.144528469E-03 0.146964908E-06 0.225455956E-04 0.730604527E-08 -0.109171815E-09 0.291921667E-03 -0.916164227E-07 -0.380392557E-03 -0.157113127E-08 -0.169897972E-11 0.786217755E-03 -0.228127938E-06 -0.113299683E-03 0.883193436E-07 -0.186200841E-10 0.729247629E-03 -0.339252081E-07 0.234054586E-03 0.286479199E-07 -0.244459426E-07 -0.841129043E-04 0.816629695E-05 0.105222969E-03 time dv (days) 0.490337809E+02 0.317312754E+03 0.111925984E+04 0.177018964E+04 0.234727724E+04 0.263908559E+04 0.327054936E+04 0.402269089E+04 0.466253367E+04 0.485853350E+04 0.561165505E+04 0.646615918E+04 0.703320991E+04 0.722444690E+04 0.795882435E+04 0.878558208E+04 0.935358046E+04 0.951705431E+04 0.103256742E+05 0.112112977E+05 0.117014537E+05 0.119108552E+05 0.126392337E+05 0.134061551E+05 0.140417640E+05 0.142129002E+05 0.150312729E+05 0.157342629E+05 dvx (km/s) -0.552785089E-04 0.760090166E-08 -0.460352469E-06 -0.236877612E-06 0.447950083E-06 0.137255551E-08 0.481312593E-08 -0.101831419E-05 -0.826467927E-06 -0.152982691E-08 0.241628109E-07 -0.135372782E-05 0.118048557E-06 0.114496740E-11 -0.725264052E-07 -0.297298867E-05 -0.214326523E-07 0.189537160E-09 -0.744194091E-08 0.496155319E-04 0.210912132E-06 -0.231147838E-09 0.747018069E-09 0.490650195E-08 0.947455484E-08 0.249495424E-10 -0.183439611E-06 -0.917547472E-07 dvy (km/s) -0.450354520E-04 0.220522442E-07 0.137732689E-05 0.827077605E-05 -0.592600949E-05 0.491611802E-09 -0.184501356E-05 -0.340381485E-06 0.118233852E-05 -0.286932965E-08 0.819572165E-07 0.280488265E-06 0.873522252E-07 0.138414572E-09 -0.600913086E-07 0.319068821E-05 0.118699364E-06 0.235177131E-10 -0.274894025E-08 -0.691840618E-04 -0.272407186E-06 -0.129277980E-09 -0.437175836E-09 0.313165036E-07 -0.188086614E-07 -0.829493750E-10 0.109515072E-06 -0.121264789E-06 dxz (km/s) -0.458901941E-05 0.165310671E-07 -0.676344521E-07 0.186518369E-04 0.468527918E-06 -0.530817945E-10 -0.121943164E-08 -0.709809801E-08 0.499633063E-07 0.487089594E-10 -0.228691020E-09 0.384596860E-04 0.119395198E-07 -0.206869156E-11 0.569845687E-09 -0.248272580E-03 -0.940482612E-08 0.917782310E-11 -0.167698604E-10 0.253858736E-04 -0.231113537E-07 -0.503459115E-10 0.176438773E-11 0.944810361E-09 -0.632364335E-09 -0.369651125E-11 0.400060456E-09 0.255308538E-09 time dv dvx dvy dxz (days) (km/s) (km/s) (km/s) 0.202364945E+03 0.362619857E-10 0.113309612E-09 0.343849743E-10 0.951102350E+03 -0.133593309E-10 -0.156939070E-10 -0.207986717E-10 0.166232367E+04 0.206871222E-10 0.248437259E-10 0.629649403E-11 0.184352128E+04 0.108900041E-10 0.708639754E-11 0.335321478E-11 242 5 E 0.235846184E+04 -0.782389912E+00 0.780185178E+01 -0.561924177E-03 6 E 0.301011290E+04 -0.768660101E+01 0.174850954E+01 0.869222955E-03 7 E 0.393449668E+04 -0.702963455E+01 -0.352328255E+01 -0.349068915E+00 8 M 0.402618395E+04 -0.585398259E+01 -0.102958414E+02 0.616747168E+00 9 E 0.464798727E+04 0.112256834E+01 -0.559917619E+01 -0.307442319E+00 10 E 0.537853357E+04 0.542510591E+01 -0.185425791E+01 0.359742797E-02 11 E 0.629426212E+04 0.274733055E+01 0.497379072E+01 0.575423071E+00 12 M 0.641411366E+04 0.281192152E+01 0.722762178E+01 -0.439242560E+00 13 E 0.704979765E+04 0.627221555E+01 0.498508810E+01 -0.115801968E-02 14 E 0.769895824E+04 -0.268650749E+01 0.756260918E+01 -0.126338287E-02 15 E 0.862393159E+04 -0.673231557E+01 0.433100299E+01 0.559881825E+00 16 M 0.873823131E+04 -0.101828295E+02 -0.224558964E+01 0.712284553E+00 17 E 0.933116046E+04 -0.510020115E+01 0.172898317E+01 -0.632446858E-03 18 E 0.100346668E+05 -0.198246057E+01 -0.501963050E+01 -0.140539106E-02 19 E 0.109491014E+05 0.469768516E+01 -0.244840986E+01 -0.108825438E+01 20 M 0.110425115E+05 0.793912834E+01 -0.527018698E+01 -0.739628490E+00 21 E 0.116889393E+05 0.439776886E+01 -0.372274763E+01 0.568286744E-03 22 E 0.123839619E+05 0.397949781E+01 0.410250538E+01 -0.882912821E-02 23 E 0.132996863E+05 -0.226829407E+01 0.511655065E+01 0.115830037E+01 24 M 0.134401747E+05 -0.713837773E+01 0.537504066E+01 0.434575987E+00 25 E 0.140468115E+05 -0.127007633E+01 0.721073227E+01 -0.560638736E-03 26 E 0.147070626E+05 -0.733970729E+01 0.609392353E+00 0.828228249E-03 27 E 0.156293451E+05 -0.555414936E+01 -0.478425047E+01 -0.499036090E+00 28 M 0.157188730E+05 -0.469368004E+01 -0.113131887E+02 0.459933767E+00 29 E 0.163554557E+05 ================ PARENT CYCLER 5.751Ggff3 ======================= Parent cycler number 179 Approximate search space (synodic periods after J2000) 19 Number of steps to walk eccentricity/inclination 243 / 243 Number of cycles 7 Total delta v over 44.75 years (km/s) 0.003179 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 1 111.8 ****** 10.430 9.329 9.329 6 84.0 7.131 7.131 11.953 11.953 11 157.5 6.598 6.599 7.889 7.889 16 104.5 8.071 8.071 11.288 11.288 21 118.4 5.993 5.993 8.243 8.243 26 130.7 6.649 6.649 9.436 9.436 31 85.9 6.452 6.452 11.177 11.177 AVERAGE 113.3 6.816 7.332 9.902 9.902 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 15315.043060 LEG E/M time start vinfx vinfy vinfz (days) (km/s) (km/s) (km/s) 1 E 0.215017113E+02 -0.379819031E+01 0.966432689E+01 0.977345109E+00 2 M 0.133261381E+03 -0.873257329E+01 0.322630576E+01 -0.607043051E+00 3 E 0.670511674E+03 -0.680013382E+01 -0.208985644E+01 -0.122198257E-02 4 E 0.159185967E+04 -0.673152161E+01 -0.220356140E+01 -0.846119365E+00 5 E 0.195712423E+04 -0.676051318E+01 -0.221096884E+01 0.324084841E-02 6 E 0.232239569E+04 -0.313591397E+01 -0.635124152E+01 -0.823029279E+00 7 M 0.240637245E+04 -0.925934268E+00 -0.118951436E+02 0.719450364E+00 8 E 0.299297153E+04 0.554086035E+01 -0.357850443E+01 -0.108970487E-02 9 E 0.391214551E+04 0.147611144E+01 -0.170400651E+01 0.618279591E+01 10 E 0.427739835E+04 -0.488444084E+01 0.194605630E+01 0.396192762E+01 11 E 0.464264301E+04 -0.815535336E+00 0.641345437E+01 0.132014462E+01 12 M 0.480014655E+04 -0.571086853E+00 0.772902601E+01 -0.147353142E+01 13 E 0.535225346E+04 -0.550689479E+01 0.593216256E+01 -0.710461773E-03 14 E 0.627746032E+04 -0.570487773E+01 0.569959273E+01 0.603690936E+00 15 E 0.664270751E+04 -0.573152022E+01 0.569029983E+01 0.437044335E+00 16 E 0.700797556E+04 -0.797004932E+01 0.125688660E+01 0.210022926E+00 17 M 0.711251214E+04 -0.960527438E+01 -0.589217395E+01 0.668107780E+00 18 E 0.767864090E+04 -0.660144452E+00 -0.596789714E+01 -0.140777518E-02 19 E 0.859550665E+04 -0.708476195E+00 -0.213834435E+01 -0.556807553E+01 20 E 0.896076261E+04 -0.397253194E+00 0.463203307E+01 -0.380599686E+01 21 E 0.932600835E+04 0.532380969E+01 0.267842043E+01 -0.631742187E+00 22 M 0.944442360E+04 0.815236145E+01 0.107149042E+01 -0.586032786E+00 23 E 0.100436410E+05 0.159448703E+01 0.645353898E+01 0.281909099E-02 24 E 0.109630434E+05 0.185529138E+01 0.634692626E+01 -0.522883758E+00 25 E 0.113283095E+05 0.187490655E+01 0.636730645E+01 -0.411775003E+00 26 E 0.116935582E+05 -0.368884027E+01 0.545358188E+01 0.925670880E+00 27 M 0.118242393E+05 -0.915053186E+01 0.229461956E+01 -0.199573224E+00 28 E 0.123714790E+05 -0.563349008E+01 -0.310579745E+01 -0.480277719E-03 29 E 0.132900741E+05 -0.563959327E+01 -0.308337010E+01 0.423420822E+00 30 E 0.136553456E+05 -0.553125368E+01 -0.298540964E+01 0.136561194E+01 31 E 0.140206002E+05 -0.124998232E+01 -0.625437491E+01 -0.971502305E+00 32 M 0.141065103E+05 0.173407194E+01 -0.110395098E+02 0.226860804E+00 33 E 0.147247119E+05 0.505708363E+01 0.806545941E+00 0.116794764E-02 34 E 0.156380877E+05 0.510038478E+01 0.186319038E+00 0.465468496E+00 35 E 0.160033242E+05 0.512097551E+01 0.209993253E+00 0.258298114E+00 36 E 0.163685967E+05 ================ PARENT CYCLER 5.751Gfgf3 ======================= Parent cycler number 180 Approximate search space (synodic periods after J2000) 17 Number of steps to walk eccentricity/inclination 1 / 1 Number of cycles 7 Total delta v over 44.68 years (km/s) 0.166302 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 1 93.7 ****** 9.873 7.814 7.814 6 111.8 8.590 8.590 10.395 10.395 11 93.0 5.406 5.406 9.519 9.519 16 139.1 5.835 5.835 8.868 8.868 21 88.7 7.682 7.682 11.882 11.882 26 125.1 5.707 5.707 7.792 7.792 31 110.3 8.332 8.332 10.620 10.620 AVERAGE 108.8 6.925 7.346 9.556 9.556 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 13755.115013 LEG E/M time start vinfx vinfy vinfz (days) (km/s) (km/s) (km/s) 1 E 0.446784930E+02 0.747092741E+01 0.641611949E+01 0.705427396E+00 0.245620950E+04 0.328742803E+04 0.394824977E+04 0.411945445E+04 0.475756922E+04 0.564409485E+04 0.631223986E+04 0.650946626E+04 0.714717174E+04 0.796720051E+04 0.864107655E+04 0.882717068E+04 0.943668642E+04 0.103089972E+05 0.109631130E+05 0.111394757E+05 0.117931927E+05 0.126586792E+05 0.133207595E+05 0.135433029E+05 0.141458492E+05 0.150206387E+05 0.156427743E+05 0.158143604E+05 -0.108296556E-10 0.217056059E-10 0.784364674E-10 0.887866420E-11 0.211963641E-11 -0.200290760E-10 0.121523616E-10 0.809496088E-11 -0.610759508E-11 -0.103584416E-10 -0.240997038E-10 0.676398794E-11 0.376144526E-11 -0.688551156E-11 -0.422736846E-10 0.725090910E-11 0.139843095E-10 0.117357092E-10 0.195099139E-10 0.809909434E-11 0.443172654E-10 0.170629077E-11 -0.298915028E-10 -0.304189318E-10 0.987400049E-11 -0.148928432E-11 -0.357990386E-10 -0.792135571E-11 -0.817432324E-11 0.639197687E-11 -0.234118967E-11 0.636552275E-12 0.341175486E-10 0.326642253E-10 0.998643050E-10 0.307132339E-10 0.346548979E-11 -0.270162706E-10 -0.450728612E-10 0.680532250E-11 0.234780320E-11 0.162626706E-10 0.185674858E-10 -0.226844083E-11 -0.236582507E-10 0.631922804E-11 0.124202096E-10 0.280066467E-10 0.517211641E-12 0.986571637E-12 -0.647040921E-12 0.187536205E-12 -0.967273991E-12 -0.127365252E-11 0.847875230E-12 0.214433381E-11 0.202482646E-10 0.592404297E-11 -0.191916377E-11 -0.470025653E-12 0.764198210E-11 0.458492341E-12 0.194479120E-11 -0.347468673E-11 0.301796604E-11 0.541156664E-11 -0.265657238E-11 -0.443158187E-11 0.889283501E-11 0.635410408E-11 0.304201719E-11 -0.309854220E-12 time dv (days) 0.382656618E+02 0.326671486E+03 0.956129553E+03 0.171604962E+04 0.201191495E+04 0.233499220E+04 0.249436232E+04 0.317680633E+04 0.405094159E+04 0.452576472E+04 0.466626854E+04 0.497682076E+04 0.562981552E+04 0.640529684E+04 0.677055133E+04 0.702365604E+04 0.719743145E+04 0.793536331E+04 0.877448207E+04 0.921278217E+04 0.934377064E+04 0.953430621E+04 0.103746259E+05 0.110799285E+05 0.114451891E+05 0.117131604E+05 0.119446321E+05 0.126378716E+05 0.134106137E+05 0.137758796E+05 0.140334867E+05 0.141992406E+05 0.150261259E+05 0.156928732E+05 0.161129060E+05 dvx (km/s) 0.733568681E-05 0.251908371E-08 0.244511904E-06 -0.410580172E-06 0.483391085E-07 0.679145311E-07 -0.673664099E-09 -0.300000661E-06 0.575935232E-04 -0.190284538E-04 -0.625940956E-05 -0.282630864E-09 -0.149983977E-06 0.112826439E-05 0.375461282E-07 0.286057707E-06 -0.153462007E-09 0.146510194E-08 -0.625162763E-03 -0.984275533E-06 0.190788506E-07 0.236764379E-10 -0.502178570E-09 -0.442212882E-07 -0.277948154E-08 -0.116739388E-08 -0.320640475E-10 0.712594314E-08 -0.732341323E-07 -0.251931472E-05 0.184297591E-08 -0.332495227E-10 -0.652036500E-07 -0.285284542E-08 -0.164992895E-06 dvy (km/s) -0.400811677E-05 -0.243956758E-08 0.117092234E-06 0.843607078E-06 -0.137384659E-06 0.955838695E-07 0.208922244E-09 -0.702391552E-05 0.761223678E-03 -0.727150386E-04 -0.209068145E-05 -0.644807614E-09 0.130500273E-06 0.949582127E-06 0.945047365E-07 0.316067390E-06 0.177963482E-09 0.410798763E-08 -0.743547352E-03 -0.843239767E-06 -0.594308682E-08 0.280149136E-10 0.520947768E-10 0.155655781E-07 -0.382225667E-09 -0.220106219E-08 -0.369828605E-10 0.331787414E-08 0.121803278E-06 0.434576647E-05 -0.515603044E-08 0.193776433E-11 -0.485273527E-07 0.294156758E-07 0.377550724E-06 dxz (km/s) -0.522266763E-06 -0.249178285E-09 0.321972726E-08 -0.230932334E-04 -0.824771631E-10 0.484073669E-08 0.430188589E-09 -0.487813437E-08 0.165290610E-05 0.725507493E-03 0.140928563E-06 -0.999564811E-10 0.234548039E-08 0.824283823E-04 0.947773593E-06 0.190742710E-07 0.143937800E-10 -0.481060581E-10 0.825894158E-04 -0.194679817E-04 0.138094271E-08 -0.352625160E-11 -0.127591595E-10 0.286341289E-05 0.156826310E-07 0.178836468E-09 -0.355787254E-11 0.763185041E-10 0.870915870E-05 0.100938496E-03 -0.459122629E-09 -0.204595755E-11 0.857377248E-09 0.172599970E-07 -0.816843830E-05 time dv dvx dvy (days) (km/s) (km/s) 0.587261211E+02 -0.299675584E-10 -0.178591768E-09 dxz (km/s) 0.102546918E-10 243 2 M 0.138329347E+03 0.212792503E+01 0.735999751E+01 -0.153508396E+01 0.300879956E+03 -0.127409658E-10 3 E 0.680164710E+03 -0.378827838E+01 0.763269552E+01 -0.121878562E+01 0.913922258E+03 -0.291945455E-05 4 E 0.104541088E+04 -0.315326596E+01 0.799916084E+01 0.141421504E-02 0.124940662E+04 0.606791389E-11 5 E 0.197266425E+04 -0.350591143E+01 0.783009770E+01 -0.641144426E+00 0.210050706E+04 0.297444685E-05 6 E 0.233792942E+04 -0.712969775E+01 0.475870090E+01 0.552084704E+00 0.235470060E+04 -0.789523227E-10 7 M 0.244973727E+04 -0.101262658E+02 -0.229626684E+01 0.501804576E+00 0.253459075E+04 -0.938294587E-11 8 E 0.301542716E+04 -0.146106869E+01 -0.518225958E+01 0.560286015E+00 0.326744532E+04 -0.243792521E-06 9 E 0.338067087E+04 -0.188578189E+01 -0.505487747E+01 -0.246062046E-02 0.366416151E+04 0.159520827E-09 10 E 0.429515683E+04 -0.218365500E+01 -0.491470103E+01 -0.568902163E+00 0.441203321E+04 0.280185864E-05 11 E 0.466039552E+04 0.467587028E+01 -0.248620207E+01 -0.108713854E+01 0.467434707E+04 0.124694804E-08 12 M 0.475340589E+04 0.792895036E+01 -0.526006953E+01 -0.287022977E+00 0.484530621E+04 0.475694688E-10 13 E 0.536607468E+04 0.324038385E+01 0.434619284E+01 0.211481843E+01 0.561810575E+04 0.272982476E-05 14 E 0.573133709E+04 0.390650212E+01 0.433497449E+01 -0.662418107E-01 0.600618040E+04 -0.897021199E-08 15 E 0.664748144E+04 0.424020967E+01 0.396976251E+01 -0.298864406E+00 0.676071460E+04 0.637185851E-07 16 E 0.701274969E+04 -0.209937185E+01 0.532336916E+01 0.114129184E+01 0.703361601E+04 -0.336149203E-08 17 M 0.715185850E+04 -0.699306587E+01 0.532941224E+01 -0.115253035E+01 0.735605690E+04 0.152838681E-10 18 E 0.768922270E+04 -0.767197726E+01 -0.380196427E+00 0.115160533E+00 0.792664643E+04 0.336808849E-06 19 E 0.805448999E+04 -0.766400642E+01 0.294288576E+00 -0.878204117E-03 0.875641955E+04 0.204345926E-04 20 E 0.897808152E+04 -0.767063018E+01 -0.149850147E-01 -0.414641631E-01 0.910957679E+04 -0.144160800E-05 21 E 0.934334615E+04 -0.601043276E+01 -0.475451599E+01 -0.526521589E+00 0.935664703E+04 -0.829409163E-07 22 M 0.943201870E+04 -0.417706513E+01 -0.111125042E+02 0.499160621E+00 0.964490152E+04 0.121236045E-05 23 E 0.100581446E+05 -0.171617981E+01 0.803396393E+00 -0.543491372E+01 0.102261631E+05 0.618971988E-03 24 E 0.104234023E+05 0.505908331E+01 -0.609064229E+00 -0.266129978E+01 0.110911654E+05 0.816237426E-02 25 E 0.113381462E+05 0.485184726E+01 -0.124923414E+01 0.272550217E+01 0.114477281E+05 -0.131847944E-04 26 E 0.117034190E+05 0.197884236E+01 0.529431038E+01 0.787758727E+00 0.117221885E+05 0.417238324E-07 27 M 0.118285488E+05 0.129480090E+01 0.753854418E+01 -0.148862910E+01 0.119107157E+05 -0.312456231E-10 28 E 0.123763276E+05 -0.464492954E+01 0.693878469E+01 -0.201170964E+00 0.126137436E+05 0.137115166E-06 29 E 0.127415830E+05 -0.403969877E+01 0.729181818E+01 0.530178839E-03 0.133992012E+05 0.137303761E-05 30 E 0.136678058E+05 -0.433111966E+01 0.711946077E+01 -0.421936229E+00 0.137956496E+05 0.853848174E-06 31 E 0.140330738E+05 -0.751693713E+01 0.356784323E+01 0.426421461E+00 0.140496213E+05 0.206002866E-08 32 M 0.141433904E+05 -0.995445601E+01 -0.364026481E+01 0.657598362E+00 0.142301110E+05 -0.102030236E-10 33 E 0.147215279E+05 0.169470441E+00 -0.500956247E+01 -0.402156153E+00 0.149735747E+05 0.108147539E-06 34 E 0.150868131E+05 -0.274872579E+00 -0.502747999E+01 -0.116482810E-01 0.157898016E+05 0.198347404E-04 35 E 0.159997852E+05 -0.644788399E+00 -0.498343965E+01 -0.379423599E+00 0.161130175E+05 0.217843735E-06 36 E 0.163650506E+05 ================ PARENT CYCLER 6.205gG3 ======================= Parent cycler number 184 Approximate search space (synodic periods after J2000) 20 Number of steps to walk eccentricity/inclination 1 / 1 Number of cycles 7 Total delta v over 44.93 years (km/s) 0.054682 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 2 200.8 6.589 6.589 8.676 8.676 5 205.4 7.345 7.345 4.641 4.641 8 241.7 5.708 5.755 5.931 5.931 11 157.3 6.167 6.167 9.146 9.146 14 300.6 7.030 7.030 2.866 2.866 17 182.8 5.385 5.385 9.051 9.051 20 248.2 7.296 7.296 3.104 3.104 AVERAGE 219.5 6.503 6.510 6.202 6.202 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 15627.987560 LEG E/M time start vinfx vinfy vinfz time dv dvx (days) (km/s) (km/s) (km/s) (days) (km/s) 1 E -0.351534594E+02 0.425386672E+01 -0.509052827E+01 0.211360542E-03 0.358692406E+03 0.202892947E-04 2 E 0.112321673E+04 -0.598500008E+01 -0.234426566E+01 0.144715495E+01 0.115334241E+04 -0.143381615E-05 3 M 0.132405455E+04 -0.746544313E+01 -0.435338052E+01 0.764872719E+00 0.163044312E+04 0.213696716E-08 4 E 0.231240478E+04 -0.228218767E+01 0.692447622E+01 -0.168412765E-02 0.248678955E+04 0.148944011E-06 5 E 0.347496992E+04 0.716638403E+01 0.153592592E+01 0.484341857E+00 0.353042000E+04 0.522096539E-06 6 M 0.368034058E+04 0.448452128E+01 0.866697422E+00 -0.823798510E+00 0.402059018E+04 -0.465132282E-06 7 E 0.468107470E+04 0.427161344E+00 -0.572585145E+01 -0.347142529E-03 0.485372545E+04 -0.156434876E-05 8 E 0.583207971E+04 -0.443327189E+01 0.229275331E+01 -0.286535399E+01 0.586833898E+04 0.148513146E-04 9 M 0.607380816E+04 -0.588272280E+01 0.444072634E+00 0.606222907E+00 0.629794758E+04 0.437675296E-06 10 E 0.700772239E+04 0.384827887E+01 0.481861966E+01 0.340953715E-03 0.740069150E+04 -0.306018720E-05 11 E 0.816351389E+04 0.263906331E+01 -0.522989601E+01 -0.192856786E+01 0.818710211E+04 -0.504534125E-05 12 M 0.832076870E+04 0.173783496E+01 -0.897443720E+01 -0.294882569E+00 0.873990657E+04 -0.420454194E-07 13 E 0.934305619E+04 -0.673085068E+01 -0.193291605E+01 0.142003919E-02 0.972589775E+04 -0.414119223E-06 14 E 0.105031821E+05 -0.411586265E+00 0.691172343E+01 -0.121754469E+01 0.105512764E+05 0.127925539E-05 15 M 0.108037712E+05 -0.120527848E+01 0.258585544E+01 -0.270223430E+00 0.113199820E+05 -0.104904807E-08 16 E 0.116937899E+05 0.521225621E+01 -0.146544100E+01 -0.235148424E-03 0.121301936E+05 -0.300077224E-07 17 E 0.128422207E+05 -0.407592292E+01 -0.346969122E+01 -0.584574914E+00 0.128934020E+05 -0.150432911E-07 18 M 0.130250111E+05 -0.670993609E+01 -0.603541549E+01 0.688563270E+00 0.134410894E+05 0.870671242E-10 19 E 0.140156738E+05 -0.392556078E+01 0.606809792E+01 -0.176973187E-02 0.141899965E+05 -0.482606654E-08 20 E 0.151778253E+05 0.644230107E+01 0.330171783E+01 -0.907819703E+00 0.152150589E+05 0.653020626E-08 21 M 0.154260494E+05 0.254916468E+01 0.158426094E+01 -0.791655886E+00 0.157496376E+05 0.148842786E-08 22 E 0.163777793E+05 ================ PARENT CYCLER 7.954Gg3 ======================= Parent cycler number 189 Approximate search space (synodic periods after J2000) 19 Number of steps to walk eccentricity/inclination 243 / 243 Number of cycles 7 Total delta v over 45.23 years (km/s) 0.238515 ---------EARTH TO MARS TRANSIT LEG CHARACTERISTICS BELOW------------LEG E-M transit Earth vinfm Earth vinfp Mars vinfm Mars vinfp time (days) (km/s) (km/s) (km/s) (km/s) 1 216.6 ****** 4.813 5.649 5.649 4 87.1 8.645 8.645 8.951 8.951 7 154.3 6.659 6.659 4.901 4.901 10 117.3 10.486 10.486 7.642 7.642 13 98.1 6.222 6.222 6.461 6.461 16 148.1 9.384 9.384 5.943 5.943 19 92.5 8.181 8.181 7.404 7.404 AVERAGE 130.6 8.263 7.770 6.707 6.707 -------------DATA NECESSARY TO REPRODUCE CYCLER BELOW---------------EPOCH TIME (days after J2000) 15340.806697 LEG E/M time start vinfx vinfy vinfz time dv dvx (days) (km/s) (km/s) (km/s) (days) (km/s) 1 E -0.808647114E+02 -0.277442737E+01 0.293354431E+01 0.262001813E+01 -0.483738457E+02 0.477182402E-08 2 M 0.135741060E+03 -0.478802334E+01 0.279352291E+01 0.108707881E+01 0.470437599E+03 -0.220891906E-11 3 E 0.952074082E+03 -0.821356558E+01 0.275204777E+01 0.749082452E-03 0.126578224E+04 -0.908699040E-11 4 E 0.231602262E+04 -0.511205582E+01 -0.690700529E+01 -0.948214495E+00 0.232908018E+04 -0.390595090E-10 5 M 0.240307304E+04 -0.582670333E+00 -0.869631104E+01 0.203949650E+01 0.261940417E+04 0.190984521E-01 -0.548592324E-11 -0.460334633E-05 0.224860024E-10 0.144330421E-05 -0.267080801E-09 -0.355642583E-10 -0.556846446E-06 0.168589134E-08 -0.117551284E-05 0.143515753E-08 0.407228118E-10 -0.245464544E-06 -0.153257186E-07 -0.118655338E-06 -0.511966032E-07 -0.286861869E-10 0.987865881E-07 -0.687480533E-04 -0.436693179E-04 -0.166159852E-07 -0.238421556E-05 -0.795137492E-03 0.992414399E-01 -0.544035492E-04 -0.115294729E-07 0.815464799E-10 0.151967871E-06 0.747980748E-06 0.623497358E-06 0.294225374E-08 -0.610313094E-10 0.338348168E-07 0.882331751E-05 -0.414149938E-07 -0.878814151E-11 -0.237129824E-03 0.371961743E-11 -0.138072818E-03 -0.151374404E-10 0.212201314E-11 -0.104433039E-04 0.569589798E-11 0.150107979E-03 0.253172134E-10 -0.536083452E-11 -0.519318047E-04 -0.746043961E-08 -0.190421208E-04 0.374591793E-09 0.697520756E-11 0.154174571E-03 0.134819558E-07 0.653354599E-04 -0.284626951E-08 -0.166347707E-07 -0.864268252E-04 -0.710821094E-01 0.596841507E-03 0.278911135E-08 0.177521616E-10 0.411610851E-04 -0.730256467E-10 0.107924234E-04 -0.193001616E-09 -0.411824005E-11 -0.125272421E-04 0.538866200E-07 0.190193798E-04 dvy (km/s) 0.341837829E-05 0.170176237E-04 -0.104569170E-07 0.982658941E-07 -0.920607888E-06 0.947210472E-06 -0.215962612E-05 0.228363878E-04 0.193320708E-06 0.383066317E-05 0.208845116E-05 0.296204274E-07 -0.138691643E-06 0.213181000E-06 -0.135341265E-08 0.210291542E-07 -0.256530402E-07 0.806288526E-10 0.250284286E-07 0.459710944E-07 -0.713343627E-09 dxz (km/s) 0.490227625E-05 0.859555847E-06 -0.385045825E-08 -0.139499187E-07 -0.118567636E-05 0.176044883E-06 0.124962545E-05 0.393617759E-05 0.682218002E-07 -0.273880727E-06 -0.265328373E-06 0.582406994E-09 0.100297673E-06 -0.146520610E-06 0.374458076E-10 -0.270725730E-07 0.761679232E-10 0.239939390E-10 0.633677875E-08 -0.104994517E-07 0.563437891E-10 dvy dxz (km/s) (km/s) 0.981220697E-08 0.576936261E-09 0.121027601E-10 0.611327858E-11 0.254323301E-10 0.157065451E-10 0.964384964E-10 -0.700910190E-11 0.987434540E-02 -0.211657819E+00 244 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 E E M E E M E E M E E M E E M E E 0.330445274E+04 0.467642458E+04 0.483076125E+04 0.564099595E+04 0.700155201E+04 0.711882535E+04 0.797999751E+04 0.935555387E+04 0.945368094E+04 0.103284390E+05 0.116901012E+05 0.118381568E+05 0.126546666E+05 0.140201273E+05 0.141126137E+05 0.150415663E+05 0.164388241E+05 0.535063491E+01 0.328875545E+01 0.414794469E-01 -0.230015054E+01 -0.100810192E+02 -0.635664958E+01 -0.189188301E+01 0.621653961E+01 0.628736122E+01 0.680230953E+01 -0.349660110E+01 -0.556263664E+01 -0.816205616E+01 -0.326297742E+01 0.176553208E+01 0.386854345E+01 -0.393980956E+01 0.568571550E+01 0.487243722E+01 0.102594949E+02 0.288581679E+01 -0.422219223E+01 -0.588156803E+01 0.146945144E+00 0.633758547E+00 0.649027379E+01 0.868613994E+01 0.175972245E+01 0.552571135E+00 -0.742836754E+01 -0.714578149E+01 0.173325529E+01 -0.623811846E-03 0.109582931E+01 0.523821179E+00 -0.140179674E-02 -0.945690928E-01 0.404198147E+00 0.635306378E-03 -0.220428943E+00 -0.134622303E+01 -0.213041344E-02 0.623852490E+00 0.113120501E+01 0.975506630E-03 -0.104806201E+01 0.795891805E+00 -0.146103704E-03 0.370232458E+04 0.469957508E+04 0.517105983E+04 0.599474053E+04 0.701914301E+04 0.754079971E+04 0.887410914E+04 0.937027293E+04 0.993479788E+04 0.107369377E+05 0.117123096E+05 0.124178788E+05 0.128731403E+05 0.140340003E+05 0.145863795E+05 0.162292354E+05 0.242366039E-09 -0.117333945E-09 0.162858179E-10 0.204192743E-11 0.125595897E-09 -0.111801727E-10 -0.120663857E-10 -0.476041899E-10 0.334396617E-11 -0.649779335E-11 -0.148672157E-10 0.497668142E-12 -0.816770971E-11 -0.382063636E-10 -0.194768462E-10 0.118316054E-10 0.968667224E-10 -0.214093197E-09 0.667966543E-12 0.244502209E-10 -0.767731645E-10 -0.195628221E-10 0.399622558E-11 0.647563803E-10 -0.761547995E-11 0.145696069E-10 0.417164947E-10 0.710954489E-12 0.173125685E-10 -0.163916344E-10 0.607948757E-11 -0.106477835E-11 -0.976793596E-11 0.837376251E-11 0.368590635E-11 0.149457102E-10 0.610989431E-12 -0.180841298E-11 0.164041282E-10 -0.491602295E-11 0.406763104E-11 0.140720332E-10 -0.201133985E-11 0.123383672E-11 -0.569005786E-12 -0.316333413E-11 -0.932698921E-11 0.103842619E-10 245 References 1 Aldrin, B., Byrnes, D. V., Jones, R., Davis, H., "Evolutionary Space Transportation Nock, T., Duke, M., King, R., Jacobs, M., Johnson, L., McRonald, A., Penzo, P., Plan for Mars Cycling Concepts," AIAA Paper 2001-4677, Aug. 2001. 2 Rauwolf, J., Wyszowski, C., "An Interplanetary Rapid Transit System Between Earth and Mars," Expanding the Frontiers of Space; Space Technology and Applications International Forum STAIF 2003, edited by El-Genk, M. S., Melville, NY, American Institute of Physics, 2003, pp. 1074-1086. 3 Rall, C. S., "Freefall Periodic Orbits Connecting Earth and Mars," Ph.D. Thesis, Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Oct. 1969. 4 Menning, M. D., "Freefall Periodic Orbits Connecting Earth and Venus," M.S. Thesis, Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, July 1968. 5 Explanatory Supplement to the Astronomical Almanac, edited by K. P. Seidelmann, Standish, E. M., "JPL Planetary and Lunar Ephemerides," CD-ROM, Willman-Bell Ross, S., "A Systematic Approach to the Study of Nonstop Interplanetary Round University Science Books, Mill Valley, California, 1992, pg. 704. 6 Inc., Richmond, VA, 1997. 7 Trips," Advances in Astronautical Sciences, Vol. 13, Proceedings of the 9th Annual AAS Meeting, edited by Burgess, E., AAS, New York, 1963. pp. 104-164. 8 Hollister, W. M., "Periodic Orbits for Interplanetary Flight," Journal of Spacecraft Niehoff, J., "Pathways to Mars: New Trajectory Opportunities," American and Rockets, Vol. 6, No. 4, 1969, pp. 366-369. 9 Astronautical Society, AAS Paper 86-172, July 1986. 246 10 Niehoff, J., Friedlander, A., and McAdams, J., "Earth-Mars Transport Cycler Niehoff, J., "Integrated Mars Unmanned Surface Exploration (IMUSE), A New Concepts," International Astronautical Congress, IAF Paper 91-438, Oct. 1991. 11 Strategy for the Intensive Science Exploration of Mars," Presentation to the Planetary Task Group of the National Academy of Science Space Science Board Major Directions Summer Study, Woods Hole, MA, July 1985. 12 Niehoff, J., "Manned Mars Mission Design," Steps to Mars, Joint AIAA/Planetary Byrnes, Dennis V., Longuski, James M. Aldrin, Buzz, "Cycler Orbit Between Earth Society Conference, National Academy of Sciences, July 1985. 13 and Mars," Journal of Spacecraft and Rockets, Vol. 30, No. 3, May-June 1993, pp. 334-336. 14 Chen, K.; McConaghy, T.; Okutsu, M.; Longuski, J. "A Low-Thrust Version of the McConaghy, T. T., Yam, C. H., Landau, D. F., Longuski, J. M., "Two-Synodic- Aldrin Cycler," AIAA Paper 2002-4421, Aug. 2002. 15 Period Earth-Mars Cyclers with Intermediate Earth Encounter," AAS Paper 03-509, Aug. 2003. 16 Byrnes, D. V., McConaghy, T. T., Longuski, J. M., "Analysis of Various Two McConaghy, T. T., Longuski, J. M., Byrnes, D. V., "Analysis of a Broad Class of McConaghy, T. T., Chen, J. C., Landau, D. F., Longuski, J. M., "A Powered EarthRussell, R. P., and Ocampo, C. A., "A Systematic Approach for Constructing EarthRussell, R., Ocampo, C., "A Geometric Analysis of Half and Full-Revolution Return Synodic Period Earth-Mars Cycler Trajectories," AIAA Paper 2002-4423, Aug. 2002. 17 Earth-Mars Cycler Trajectories," AIAA Paper 2002-4420, Aug. 2002. 18 Mars Cycler with Three Synodic-Period Repeat Time," AAS Paper 03-510, Aug. 2003. 19 Mars Cyclers Using Direct Return Trajectories," AAS Paper 03-145, Feb. 2003. 20 Trajectories via Planetary Flybys," AAS Paper 03-508, Aug. 2003. 247 21 Russell, R., Ocampo, C., "Construction of Idealized Free-Return Earth-Mars Cyclers Landau, D., Longuski, J., "Comparative Assessment of Human Missions to Mars," Bishop, R. H., Byrnes, D. V., Newman, D. J., Carr, E. C., Aldrin, B., "Earth-Mars using Minimax Optimization and Combinatorics," AAS Paper 04-146, Feb. 2004. 22 AAS Paper 03-513, Aug. 2003. 23 Transportation Opportunities: Promising Options for Interplanetary Transportation," Advances in the Astronautical Sciences Vol. 106, The Richard H. Battin Astrodynamics Symposium, edited by Junkins, J. L., Alfriend, K. T., Howell, K. C., Univelt Inc., San Diego, CA, 2000, pp. 117 128. 24 Chen, K. J., Landau, D. F., McConaghy, T. T., Okutsu, M., Longuski, J. M., Aldrin, B., "Preliminary Analysis and Design of Powered Earth-Mars Cycling Trajectories," AIAA Paper 2002-4422, Aug. 2002. 25 McConaghy, T. T., Russell, R. P., Longuski, J. M., "Towards a Standard Nomenclature for Earth-Mars Cycler Trajectories," Journal of Spacecraft and Rockets, (to be published). 26 Poincar , H., Les M thodes Nouvelles de la M canique C leste (French), Gauthier- Villars, Paris, 1892, 1893, 1899. New Methods of Celestial Mechanics, History of Modern Physics and Astronomy (English translation), Vol. 13, Springer Verlag, New York, Sept. 1992. 27 H non, M., "Sur les orbites interplan taires qui rencontrent deux fois la terre," H non, M., Generating Families in the Restricted Three-Body Problem, Springer Russell, R. P., and Ocampo, C. A., "Systematic Method for Constructing Earth-Mars Bulletin Astronomique (French), Vol. 3, 1968, pp. 377 402. 28 Verlag, Berlin, 1997, pp. 16 19 & 35 77. 29 Cyclers Using Free-Return Trajectories," Journal of Guidance, Control, and Dynamics, Vol. 27, No. 3, May-June 2004, pp. 321-335. 248 30 Russell, R., Ocampo, C., "A Geometric Analysis of Free-Return Trajectories Following a Gravity-Assisted Flyby," Journal of Spacecraft and Rockets, (to be published). 31 Russell, R., Ocampo, C., "Global Search for Idealized Free-Return Earth-Mars Patel, M. R., Longuski, J., M., Sims, Jon A., "Mars Free Return Trajectories," Miele, A., Wang, T., Mancuso, S., "Optimal Free-Return Trajectories for Moon Cyclers," Journal of Guidance, Control, and Dynamics, (to be published). 32 Journal of Spacecraft and Rockets, Vol. 35, No. 3, 1998, pp. 350-354. 33 Missions and Mars Missions," Journal of the Astronautical Sciences, Vol. 48, AprilSept., 2000, pp. 183-206. 34 Wolf, A. A., "Free-Return Trajectories for Mars Missions," AAS Paper 91-123, Feb. Uphoff, C. W., "The Art and Science of Lunar Gravity Assist," Advances in 1991. 35 Astronautical Sciences, Vol. 69, Orbital mechanics and mission design, edited by Teles, J., Univelt, Inc., San Diego, CA, 1989, p. 333-346. 36 Uphoff, C., Crouch, M. A., "Lunar Cycler Orbits with Alternating Semi-Monthly Transfer Windows," Journal of the Astronautical Sciences, Vol. 41, No. 2, 1993, pp. 189-205. 37 Uphoff, C., Roberts, P. H., Friedman, L. D., "Orbit Design Concepts for Jupiter Orbiter Missions," Journal of Spacecraft and Rockets, Vol. 13, No. 6, 1976, pp. 348355. 38 Prussing, J. E., Conway, B. A., Orbital Mechanics, Oxford University Press, New Prussing, J. E., "A Class of Optimal Two-Impulse Rendezvous Using Multiple- York, 1993. pp.63-80. 39 Revolution Lambert Solutions," The Journal of the Astronautical Sciences, Vol. 48, Nos. 2 and 3, April September 2000, pp. 131 148. 249 40 Battin, R. H., An Introduction to the Mathematics and Methods of Astrodynamics, Revised Edition, American Institute of Aeronautics and Astronautics, Inc., Reston, VA, 1999, pg. 241. 41 Shen, H., and Tsiotras, P., "Using Battin's Method to Obtain Multiple-Revolution Ocampo, C., Guinn, J., Breeden, J., " Rendezvous Options and Dynamics for Mars Lambert's Solutions," AAS Paper 03-568, Aug. 2003. 42 Sample Return Mission," Advances in the Astronautical Sciences Vol. 109, AAS/AIAA Astrodynamics Conference, edited by Howell, K. C. et al., Univelt Inc., San Diego, CA, 2002, pp. 1661 1680. 43 Niven, I., Mathematics of Choice or How to Count Without Counting, Random House Cohen, D. I. A., Basic Techniques of Combinatorial Theory, New York, John Wiley Ch. 3. Andrews, G. E. Encyclopedia of Mathematics and its Applications: Vol. 2, The Skiena, S. Implementing Discrete Mathematics: Combinatorics and Graph Theory Nijenhuis, A., Wilf, H. S., Combinatorial Algorithms For Computers and "VG12AD-vg12.pdf," Harwell Subroutine Library, URL: Inc., New York, 1965, pp. 21,24,59,99,104. 44 & Sons, 1978, 45 Theory of Partitions, ed. by Turan, P., Reading, MA: Addison-Wesley, 1976, Ch. 1,14. 46 with Mathematica. Reading, MA: Addison-Wesley, 1990, pg. 57. 47 Calculators, New York, Academic Press, 1978, Ch. 9. 48 http://www.cse.clrc.ac.uk/nag/hsl [cited 6 February 2004]. 49 Stevens, R., Ross, M., "Preliminary Design of Earth-Mars Cyclers Using Solar Rauwolf, G., A., Friedlander, A., L., "A Mars Cycler Architecture Utilizing Low- Sails," AAS Paper 03-244, Feb. 2003. 50 Thrust Propulsion," AIAA Paper 2002-5046, Aug. 2002. 250 51 Betts, J. H., "Survey of Numerical Methods for Trajectory Optimization," Journal of Goldberg, D. E., Genetic Algorithms in Search, Optimization, and Machine Learning, Carol, D. L., "FORTRAN Genetic Algorithm (GA) Driver," CU Aerospace, Urbana, Rogata, P., Di Sotto, E., Granziano, M., Graziani, F., "Guess Value for Interplanetary Vasile, M., "A Global Approach to Optimal Space Trajectory Design," AAS Paper "VF13AD-vf13.pdf," Harwell Subroutine Library, URL: "SNOPT%20Manual.pdf," Stanford Business Software Inc., URL: http://www.sbsiGill, P. E., Murray, W., Saunders, M. A., "SNOPT: An SQP Algorithm For Large Guidance, Control, and Dynamics, Vol. 21, No. 2, March-April 1998, pp.193-207. 52 Reading, Massachusetts, Addison-Wesley Publishing Company Inc., 1989. 53 Illinois, URL: http://cuaerospace.com/carroll/ga.html [cited 26 May 2004]. 54 Transfer Through Genetic Algorithms," AAS Paper 03-140, Feb. 2003. 55 03-141, Feb. 2003. 56 http://www.cse.clrc.ac.uk/nag/hsl [cited 25 May 2004]. 57 sol-optimize.com/asp/sol_product_snopt.htm [cited 25 May 2004]. 58 Scale Constrained Optimization," SIAM Journal of Optimization, Vol. 12, No. 4, 2002, pp. 979-1006. 59 Hull, D. G., Williamson, W. E., "Numerical Derivatives for Parameter Optimization (applied to drag coefficient interpolation)," Journal of Guidance and Control, Vol. 2, 1979, pp. 158-160. 60 Zimmer, S., and Ocampo, C. A., "Combined Long Duration Burns and Gravity Assist "ADIFOR 2.0, Automatic Differentiation of Fortran," Argonne National Laboratory Trajectories Using Analytical Gradients," AAS Paper 03-576, Aug. 2003. 61 and Rice University, URL: http://www-unix.mcs.anl.gov/autodiff/ADIFOR/ [cited 27 May 2004]. 62 Montenbruck, O., Gill, E., Satellite Orbits, Berlin, Springer, 2000, Ch. 7. 251 63 Broucke, R. A., Cefola, P. J., "On the Equinoctial Orbit Elements," Celestial Mechanics, Vol. 5, 1972, pp.303-310. 64 Dow, J. M., "Non-Singular Partial Derivatives for Synchronous Orbits," Working Paper No. 22, European Space Operations Centre, Orbit and Attitude Division, Darmstadt, May, 1975. 65 Byrnes, D. V., Bright, L., E., "Design of High-Accuracy Multiple-Flyby Trajectories Sauer, C., G., "MIDAS: Mission Design and Analysis Software for the Optimization Using Constrained Optimization," AAS Paper 95-307, Feb. 1995. 66 of Ballistic Interplanetary Trajectories," Journal of the Astronautical Sciences, Vol. 37, July-Sept., 1989, pp. 251-259. 67 Ocampo, C. A., "An Architecture for a Generalized Spacecraft Trajectory Design and Optimization System," Libration Point Orbits and Applications: Proceedings of the Conference Aiguablava, Spain 10 - 14 June 2002, edited by Masdemont, J. J. et al, ISBN 9812383638, World Scientific Pub Co., River Edge, NJ, 2003. 68 Phone conversation with Troy McConaghy, Purdue University. 252 Vita Ryan Paul Russell was born in Amarillo, Texas on July 26, 1976. He is the son of Judy Persons and Paul Russell, both of Amarillo. After graduating from Tascosa High School in May 1994, he entered Texas A&M University in the fall. He graduated first in his class in the College of Engineering with a Bachelor of Science in Aerospace Engineering in May 1999. He studied and traveled abroad throughout Central America for the spring semester of 1998. During the summers of 1995-1998 he worked as a counselor and work-crew boss at Laity Lodge Youth Camp, a Christian summer camp in the hill country of Texas. During the summer of 1999, he taught English to aspiring students in an exchange program in Central China. In December 2000, he completed a Master of Science in Aerospace Engineering at the University of Texas at Austin. He became an official Ph.D. candidate in the fall of 2002 after passing the written and oral qualifying exams in the orbital mechanics group. The results of this dissertation have been presented by the author at several AAS/AIAA meetings and subsequently published or are in-press with AIAA journals. The paper based on Chapter 3 won the best paper award at the AAS/AIAA conference in Puerto Rico in February 2002. The majority of the funding for his graduate education came from a National Defense Science and Engineering Graduate fellowship and a NASA Graduate Student Researchers Program fellowship. He married Jennifer Nell Brown of Austin, Texas, on January 24, 2004. In August 2004, they will move to Pasadena, California where Ryan will begin employment at the Jet Propulsion Laboratory. Permanent Address: 3610 Patterson, Amarillo, Texas, 79109, (806) 355-7330 This dissertation was typed by the author. 253

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sampselld77810.pdf
Path: Texas >> SAMPSELLD >> 77810 Fall, 2009

Description: Copyright by Matthew Brian Sampsell 2004 The Dissertation Committee for Matthew Brian Sampsell certifies that this is the approved version of the following dissertation: BEAM EMISSION SPECTROSCOPY ON THE ALCATOR C-MOD TOKAMAK Committee: __ Kenneth...

complex.txt
Path: CSU San Bernardino >> CS >> 330 Fall, 2009

Description: Laboratory: Complexity Implement: 1. Towers of Hanoi (recursive algorithm described in Ch. 2 Budd) theoretically this is O(2^N) 2. A sort algorithm of your choice (see cs202 labs for sample code) (should be O(N^2) or O(NlogN) ) For...

cadenheadjk046.pdf
Path: Texas >> CADENHEADJ >> 046 Fall, 2009

Description: Copyright by Juliet Kathryn Cadenhead 2004 The Dissertation Committee for Juliet Kathryn Cadenhead Certifies that this is the approved version of the following dissertation: The Tripartite Self: Gender, Identity, and Power Committee: William Moor...

benjaminsmr042.pdf
Path: Texas >> BENJAMINSM >> 042 Fall, 2009

Description: Copyright by Maureen Reindl Benjamins 2004 The Dissertation Committee for Maureen Reindl Benjamins certifies that this is the approved version of the following dissertation: Religion and Preventive Health Care Use in Older Adults Committee: __ Rob...

simpsonal13317.pdf
Path: Texas >> SIMPSONAL >> 13317 Fall, 2009

Description: ...

hamiltont84490.pdf
Path: Texas >> HAMILTONT >> 84490 Fall, 2009

Description: Copyright by Tracy Chapman Hamilton 2004 The Dissertation Committee for Tracy Chapman Hamilton Certifies that this is the approved version of the following dissertation: Pleasure, Politics, and Piety: The Artistic Patronage of Marie de Brabant Comm...

kotrlaka518287.pdf
Path: Texas >> KOTRLAKA >> 518287 Fall, 2009

Description: Copyright by Kimberly Ann Kotrla 2004 The Dissertation Committee for Kimberly Ann Kotrla certifies that this is the approved version of the following dissertation: Prenatal Alcohol Consumption: A Risk-Protective Model Committee: _ Diana DiNitto, ...

harrisont86130.pdf
Path: Texas >> HARRISONT >> 86130 Fall, 2009

Description: Copyright by Tracie Culp Harrison 2004 The Dissertation Committee for Tracie Culp Harrison Certifies that this is the approved version of the following dissertation: The Meaning of Aging for Women with Childhood Onset Disabilities Committee: Alex...

brandonjc99738.pdf
Path: Texas >> BRANDONJC >> 99738 Fall, 2009

Description: Copyright By Jamie Chad Brandon 2004 The Dissertation Committee for Jamie Chad Brandon certifies that this is the approved version of the following dissertation Van Winkle\'s Mill: Mountain Modernity, Cultural Memory and Historical Archaeology in th...

MATH107A46024536.doc
Path: MD University College >> ASIA >> 2092 Fall, 2009

Description: University of Maryland University College MATH 107: College Algebra 3 semester credits Spring session 2: 2008/2009 Kunsan, Korea; M W 1830-2130 Faculty Contact Information: Toni Yoon, Collegiate Assistant Professor E-mail: ayoon@asia.umuc.edu Phon...

crawforda65881.pdf
Path: Texas >> CRAWFORDA >> 65881 Fall, 2009

Description: Copyright by Arthur Bryan Crawford 2004 The Dissertation Committee for Arthur Bryan Crawford Certifies that this is the approved version of the following dissertation: Evaluation of the Impact of Non-Uniform Neutron Radiation Fields on the Dose Rec...

achacosom07761.pdf
Path: Texas >> ACHACOSOM >> 07761 Fall, 2009

Description: Copyright by Michelle Valleau Achacoso 2002 The Dissertation Committee for Michelle Valleau Achacoso Certifies that this is the approved version of the following dissertation: \"WHAT DO YOU MEAN MY GRADE IS NOT AN A?\" AN INVESTIGATION OF ACADEMIC EN...

jarroldwl86380.pdf
Path: Texas >> JARROLDWL >> 86380 Fall, 2009

Description: @99 668 7 4 ( 1 0 ( % \" ! )6532$# (d1 d0 ( 27h ( 22 ( 7 0 ( ) 31 S ( )6 1 4 ( 2 0 )S ( ) ( 21 h#\" ( ( ( ! ! q $ )Q $ 4 V 4 v 4 3 I t VQq 4 ( r...

sharyginany026.pdf
Path: Texas >> SHARYGINAN >> 026 Fall, 2009

Description: 45 5 4 0\' )3 120)$\" \'% \' %# ! v r p a u s t\' # (# r 3 g \' p % # q1 i # 3 # # p i gf % # a1 d# \' h # e # d(# ` b % G ` Y D R G 9 \" ( % R P I GB \" D B...

goncalvesac026.pdf
Path: Texas >> GONCALVESA >> 026 Fall, 2009

Description: Copyright by Alexandre Casassola Gonalves c 2002 The Dissertation Committee for Alexandre Casassola Gonalves c Certies that this is the approved version of the following dissertation: An Application of The Continuity Method for an Equation on Line ...

zieglerkj47418.pdf
Path: Texas >> ZIEGLERKJ >> 47418 Fall, 2009

Description: Copyright By Kirk J. Ziegler 2001 The Dissertation Committee for Kirk Jeremy Ziegler Certifies that this is the approved version of the following dissertation: Chemical Equilibria and Nanocrystal Synthesis in High Temperature Supercritical Solution...

burtnerjc90760.pdf
Path: Texas >> BURTNERJC >> 90760 Fall, 2009

Description: Copyright by Jennifer Carol Burtner 2004 The Dissertation Committee for Jennifer Carol Burtner certifies that this is the approved version of the following dissertation: Travel and transgression in the Mundo Maya: Spaces of home and alterity in a G...

alvarezla07232.pdf
Path: Texas >> ALVAREZLA >> 07232 Fall, 2009

Description: ...

MATH012A46124534.doc
Path: MD University College >> ASIA >> 2092 Fall, 2009

Description: University of Maryland University College MATH 012 Intermediate Algebra 3 semester credits Spring Session 2 2008/2009 Kunsan: MTWTh 17:00-18:15 Faculty Contact Information: My e-mails are checked nightly. So if you have any conflict with class...

bonningew86532.pdf
Path: Texas >> BONNINGEW >> 86532 Fall, 2009

Description: Copyright by Erin Wells Bonning 2004 The Dissertation Committee for Erin Wells Bonning certifies that this is the approved version of the following dissertation: Computational and Astrophysical Studies of Black Hole Spacetimes Committee: Richard ...

CMIS141AA44024445.doc
Path: MD University College >> ASIA >> 2092 Fall, 2009

Description: Syllabus University of M a ryland University College - Asia Spring Session I, 2008-2009 (01/19 ~ 03/12) Osan Course: Credit: I nstructor: Homepage: CMIS141A 3 J in-Ah Jeon Fundamentals of Programming I I Mon. ~ Thu. E-mai l: 1145 ~ 1300 jeonj1sh@ya...

CMIS102AA42086692.doc
Path: MD University College >> ASIA >> 2088 Fall, 2009

Description: Syllabus University of M a ryland University College - Asia Fall Session I I, 2008-2009 (10/28 ~ 12/20) Osan Course: Credit: I nstructor: Homepage: Prerequisites: Textbook: CMIS102A 3 J in-Ah Jeon Fundamentals of Programming I Tue. & Thu. E-mai l: ...

STAT200A42186896.doc
Path: MD University College >> ASIA >> 2088 Fall, 2009

Description: UMUC, Asia STAT 200: Introductory Statistics 3 semester credits Fall session 2: 2008 Yongsan : T Th 1800-2100 FACULTY CONTACT INFORMATION: Assistant Professor: Antonia (Toni) Yoon E-mail:ayoon@asia.umuc.edu Phone #: (DSN) 723-4295; Leave message. ...

kulkarnis86095.pdf
Path: Texas >> KULKARNIS >> 86095 Fall, 2009

Description: Copyright by Shanti Joy Kulkarni 2004 The Dissertation Committee for Shanti Joy Kulkarni certifies that this is the approved version of the following dissertation: Adolescent mothers negotiating development in the context of interpersonal violence ...

chapmanbg60287.pdf
Path: Texas >> CHAPMANBG >> 60287 Fall, 2009

Description: ...

slattonkc78713.pdf
Path: Texas >> SLATTONKC >> 78713 Fall, 2009

Description: ...

michalskylo026.pdf
Path: Texas >> MICHALSKYL >> 026 Fall, 2009

Description: Copyright by Linda Oldfather Michalsky 2002 The Dissertation Committee for Linda Oldfather Michalsky Certifies that this is the approved version of the following dissertation: Evaluation of an Interactive Multimedia Program on Calcium and Folate Co...

batemanmt33508.pdf
Path: Texas >> BATEMANMT >> 33508 Fall, 2009

Description: ...

lodowskid97061.pdf
Path: Texas >> LODOWSKID >> 97061 Fall, 2009

Description: Copyright by David T. Lodowski 2004 The Dissertation Committee for David Thomas Lodowski Certifies that this is the approved version of the following dissertation: Structural Basis for the Regulation of GRK2 by G Committee: John Tesmer, Supervisor...

raichlend29983.pdf
Path: Texas >> RAICHLEND >> 29983 Fall, 2009

Description: Copyright by David Allan Raichlen 2004 The Dissertation Committee for David Allan Raichlen Certifies that this is the approved version of the following dissertation: The Relationship Between Limb Muscle Mass Distribution and the Mechanics and Energ...

perkinsjd44616.pdf
Path: Texas >> PERKINSJD >> 44616 Fall, 2009

Description: ...

mehdiabadinj026.pdf
Path: Texas >> MEHDIABADI >> 026 Fall, 2009

Description: Copyright by Natasha Jum Mehdiabadi 2002 The Dissertation Committee for Natasha Jum Mehdiabadi Certifies that this is the approved version of the following dissertation: ANT SYMBIOSES: COLONY-LEVEL EFFECTS OF ANTAGONISTIC AND MUTUALISTIC INTERACTION...

borisovasa86653.pdf
Path: Texas >> BORISOVASA >> 86653 Fall, 2009

Description: Copyright by Svetlana Alekseyevna Borisova 2004 The Dissertation Committee for Svetlana Alekseyevna Borisova certifies that this is the approved version of the following dissertation: Genetic and Biochemical Studies of the Biosynthesis and Attachme...

Abuhakema504399.pdf
Path: Texas >> ABUHAKEMA >> 504399 Fall, 2009

Description: Copyright by Ghazi M. A. Abuhakema 2004 The Dissertation Committee for Ghazi M. A. Abuhakema certifies that this is the approved version of the following dissertation: The Cultural Component of the Arabic Summer Program at Middlebury College: Fulfi...

hw03_solution.doc
Path: Penn State >> ME >> 581 Fall, 2009

Description: ME 581 - Spring 2008 HW03 Name _ 1) View the web cutter video \"wc.mov\" from the class web page. JPG images are provided in \"wc_images.zip\". Be certain to read the \"read_me.txt\" file within the ZIP. Use suitable software to digitize the location of...

oestreichj19588.pdf
Path: Texas >> OESTREICHJ >> 19588 Fall, 2009

Description: Copyright by Jrg Oestreich 2004 The Dissertation Committee for Jrg Oestreich Certifies that this is the approved version of the following dissertation: FROM ECOLOGY TO NEURAL MECHANISMS: A NEUROETHOLOGICAL APPROACH TO A NOVEL FORM OF MEMORY Commit...

evstatieve01477.pdf
Path: Texas >> EVSTATIEVE >> 01477 Fall, 2009

Description: Copyright by Evstati Georgiev Evstatiev 2004 The Dissertation Committee for Evstati Georgiev Evstatiev certifies that this is the approved version of the following dissertation: A Model for Multi-Wave BeamPlasma Interaction Committee: Philip J. M...

paschvaldesg042.pdf
Path: Texas >> PASCHVALDE >> 042 Fall, 2009

Description: Copyright by Grete Mara Pasch Valds 2004 Identifying, Selecting, and Organizing the Attributes of Web Resources by Grete Mara Pasch Valds, BSc, MSc, MLIS Dissertation Presented to the Faculty of the School of Information The University of Texas at...

alvaradocg86236.pdf
Path: Texas >> ALVARADOCG >> 86236 Fall, 2009

Description: Copyright by Cassandre Giguere Alvarado 2004 The Dissertation Committee for Cassandre Giguere Alvarado Certifies that this is the approved version of the following dissertation: EMIC PERSPECTIVES: THE FRESHMAN INTEREST GROUP PROGRAM AT THE UNIVERSI...

martinssonpj026.pdf
Path: Texas >> MARTINSSON >> 026 Fall, 2009

Description: The dissertation committee for Per-Gunnar Johan Martinsson certifies that this is the approved version of the following dissertation: Fast multiscale methods for lattice equations Committee: Gregory Rodin, Supervisor Ivo Babuka, Supervisor s Jer...

makowitza504694.pdf
Path: Texas >> MAKOWITZA >> 504694 Fall, 2009

Description: Copyright by Astrid Makowitz 2004 The Dissertation Committee for Astrid Makowitz Certifies that this is the approved version of the following dissertation: THE GENETIC ASSOCIATION BETWEEN BRITTLE DEFORMATION AND QUARTZ CEMENTATION: EXAMPLES FROM BU...

andersonmw81540.pdf
Path: Texas >> ANDERSONMW >> 81540 Fall, 2009

Description: Copyright by Matthew William Anderson 2004 The Dissertation Committee for Matthew William Anderson certifies that this is the approved version of the following dissertation: Constrained Evolution in Numerical Relativity Committee: Richard Matzner...

martinezrs39334.pdf
Path: Texas >> MARTINEZRS >> 39334 Fall, 2009

Description: Copyright by Rebecca Suzanne Martnez 2002 The Dissertation Committee for Rebecca Suzanne Martnez Certifies that this is the approved version of the following dissertation: A COMPARISON OF LEARNING DISABILITY SUBTYPES IN MIDDLE SCHOOL: SELF-CONCEPT, ...

elshayebta87380.pdf
Path: Texas >> ELSHAYEBTA >> 87380 Fall, 2009

Description: Copyright by Tarek Abu Serie Elshayeb 2004 The Dissertation Committee for Tarek Abu Serie Elshayeb Certifies that this is the approved version of the following dissertation: Integrated Sequence Stratigraphy, Depositional Environments, Diagenesis, a...

cowmeadowr17589.pdf
Path: Texas >> COWMEADOWR >> 17589 Fall, 2009

Description: Copyright by Roshani Barbara Cowmeadow 2004 The Dissertation Committee for Roshani Barbara Cowmeadow Certifies that this is the approved version of the following dissertation: Molecular mechanisms of alcohol tolerance in the fruit fly. Committee: ...

schougaardsb029.pdf
Path: Texas >> SCHOUGAARD >> 029 Fall, 2009

Description: Copyright by Steen Brian Schougaard 2002 The Dissertation Committee for Steen Brian Schougaard certifies that this is the approved version of the following dissertation: DEVELOPMENT AND STUDY OF HIGH-TC SUPERCONDUCTOR CONDUCTIVE POLYMER ASSEMBLIES ...

kordoskyma87090.pdf
Path: Texas >> KORDOSKYMA >> 87090 Fall, 2009

Description: BAA \"@ 87 4 1 ) # % # ! 9565320(\' ! ) u ) $fdvFD 7 ! q n 5XatWs r 1 63Q6\"fn 7 p D ! ) p 6XFgf\" FD 7 h ! p n m ) l # 5d5$q6o66\"p1 s ! ! I I \"$G5PQ y kPc3\'ji g hf e d v y y x v ...

metcalfets016-x.pdf
Path: Texas >> METCALFETS >> 016 Fall, 2009

Description: u { y su } m {grYVHtAr s { u { ugVR{7 s{ ~ us y } s Vgroz67toVc u ~ u{ ~ } |x{ m n s ~ Vz\"HUo\'6UVrrwpVo% u ~ u{ ~ } |u{ yx s v pu s q p n m V\"zrr6Ugrz6%wH6trXoPl k h h f fd jige e he g w e r EyEE t t e w r t r p syx...

bocknackbm84986.pdf
Path: Texas >> BOCKNACKBM >> 84986 Fall, 2009

Description: Copyright by Brian Matthew Bocknack 2004 The Dissertation Committee for Brian Matthew Bocknack Certifies that this is the approved version of the following dissertation: Electrophilic Trapping of Enolates in Tandem Reaction Processes and (1,3-Diket...

mahdjoubid26824.pdf
Path: Texas >> MAHDJOUBID >> 26824 Fall, 2009

Description: Copyright by Darius Mahdjoubi 2004 The Dissertation Committee for Darius Mahdjoubi certifies that this is the approved version of the following dissertation: Knowledge, Innovation and Entrepreneurship: Business Plans, Capital, Technology and Growth...

vanderveenaa029.pdf
Path: Texas >> VANDERVEEN >> 029 Fall, 2009

Description: Copyright by Arthur Alvin VanderVeen, Jr. 2002 The Dissertation Committee for Arthur Alvin VanderVeen, Jr. certifies that this is the approved version of the following dissertation: Other Minds, Other Worlds: Pragmatism, Hermeneutics, and Construct...

crabtreejc17037.pdf
Path: Texas >> CRABTREEJC >> 17037 Fall, 2009

Description: ...

steubingdm73657.pdf
Path: Texas >> STEUBINGDM >> 73657 Fall, 2009

Description: ...

johnsonhl692102.pdf
Path: Texas >> JOHNSONHL >> 692102 Fall, 2009

Description: Copyright by Helen Louise Johnson 2004 The Dissertation Committee for Helen Louise Johnson certifies that this is the approved version of the following dissertation CONSEQUENCES OF HIGH-STAKES TESTING: CRITICAL PERSPECTIVES OF TEACHERS AND STUDENTS...

quintopozosd022.pdf
Path: Texas >> QUINTOPOZO >> 022 Fall, 2009

Description: Copyright by David Gilbert Quinto-Pozos 2002 The Dissertation Committee for David Gilbert Quinto-Pozos Certifies that this is the approved version of the following dissertation: Contact Between Mexican Sign Language and American Sign Language in Tw...

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