#### prob2cck

Montana, PH 331
Excerpt: ... Physics 331 Problem 2 Due Thursday, Sept. 30, 2004 Elliptical Orbit s Kepler's first law states that a planet (or spacecraft) will follow an elliptical orbit about the sun. 1 In polar coordinates an elliptical orbit is given by the curve: a(1 - 2 ) r() = . (1) 1 + cos( - P ) The three parameters of the orbit are the eccentricity, a the semi-major axis, and P the angle of perihelion (the point closest the Sun). perihelion r( ) p elliptical orbit aphelion The above sketch shows an elliptical orbit with perihelion at angle P . Note that describes how "non-circular" the orbit is. Setting = 0 gives the curve r() = a, a perfect circle. The orbital parameters for Mars are: a = 2.279 108 km = 0.093 P = 234 1 This assumes the orbiting body is affected only by the gravitational pull of the Sun. We will relax this assumption in the future. 1 The problem: A spacecraft A is launched from Earth into an elliptical orbit rA () with the parameters a = 1.961 108 km = 0.25 P = 0 The question you must answer is: At ...

#### Lect_10b

Rochester, AST 111
Excerpt: ... Today in Astronomy 111: Newton's laws and orbits Center of mass Elliptical orbit s and their consistency with Newtonian mechanics Kepler's laws (Pre-)validation of Newtonian dynamics by Kepler and Tycho 2 October 2008 Astronomy 111, Fall 2008 1 Uniform circular motion and gravity Suppose a large mass M lies a distance r from a mass m. At what speed will the small mass orbit the large mass at this distance? GMm v2 ^ ^ F = - 2 r = ma = - m r r r GM v= r What is the corresponding total energy, and the angular momentum of the small mass relative to the center of its orbit? 1 GMm GMm GMm cf. virial 2 GMm E = K + U = mv - = - =- theorem 2 r 2r r 2r ^ L = r mv = zrmv sin 2 October 2008 2 ^ = zm GMr 2 Astronomy 111, Fall 2008 Center of mass What if the masses are not very different? Then both move in an orbit about their constant center of mass, defined by m r + m2 r2 rCM = 1 1 m1 + m2 Suppose the two masses are separated by displacement r, and we place the center of mass at the origin of coordinates: m ...

#### Kepler_3

Maple Springs, NATS 1740
Excerpt: ... Exercise #3 Astronomy Ranking Task: Kepler's Laws Orbital Motion Description: The figure below shows a star and five orbiting planets (A E). Note that planets A, B and C are in perfectly circular orbits. In contrast, planets D and E have more elliptical orbit s. Note that the closest and farthest distances for the elliptical orbit s of planets D and E happen to match the orbital distances of planets A, B, and C as shown in the figure. Ranking Instructions: Rank the orbital period (from longest to shortest) of the planets. Ranking Order: Longest 1 _ 2 _ 3 _ 4 _ 5 _ Shortest Or, the orbital periods of the planets would all be the same. _ (indicate with check mark). Carefully explain your reasoning for ranking this way: _ _ _ _ ...

#### AA310Homework3

Washington, AA 31008
Excerpt: ... AA310 Orbital Mechanics: Homework 3 (Due Wednesday 10/22) 1) With reference to the lecture today (10/15), show that area of pfx = b (area of pfs) a where "b" and "a" are, respectively, the semi-minor axis and the semi-major axis of the elliptic orbi ...

#### WEEK_2_AERODYNAMICS_JARON_JACKIE

Purdue, AAE 450
Excerpt: ... AAE 450 Spacecraft Design Mars Ascent Delta-V Trade Study & Trajectory Jackie Jaron 25 January 2005 Aerodynamics Team Mars Ascent Team Mars Lander/Ascent Vehicle Ad Hoc Group 1 AAE 450 Spacecraft Design Delta-V Trade Study Motivation: V from Mars Parking Orbit to Elliptical Orbit (km/s) Minimize Mars Lander propellant mass 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 V from Mars Parking Orbit versus Orbit Period of Elliptical Orbit Assumptions: Lander parking orbit altitude: 350 km Burn at ellipse perigee Outcome: Want orbits with P < 1 day Investigate frequency of docking opportunities 0 1 2 3 4 5 6 Orbit Period of Elliptical Orbit (days) 7 8 Figure 1: V from Mars Parking Orbit to CTV Orbit versus Orbit Period of Elliptical Orbit 2 AAE 450 Spacecraft Design Mars Ascent Trajectory & Delta-V Mars launch V to parking orbit alt.= 350 km Preliminary Mars Ascent Trajectory 350 Defined Trajectory Calculated Trajectory 250 Altitude (km) 300 200 150 100 50 Accounted for losses/gains V350 ...

#### assignment3

George Mason, C 761
Excerpt: ... project work Here is a restatement of the problems from assignment 1. A binary star problem. For this problem, integrate the solution from t=0 to t=25. Experiment with the step size, and include how varying this parameter changes the results. velocity2 mass1 11 00 11 00 11 00 11 00 distance 11 00 11 00 11 00mass2 11 00 velocity1 Subcase I: Stable circular orbits position1 = (0, 0), velocity1 = (0, 1/ 2), mass1 = 1 position2 = (1, 0), velocity2 = (0, +1/ 2), mass2 = 1 Subcase II: Elliptical Orbit s position1 = (0, 0), velocity1 = (0, 1.2/ 2), mass1 = 1 position2 = (1, 0), velocity2 = (0, +1.2/ 2), mass2 = 1 Subcase III: Parabolic Orbits position1 = (0, 0), velocity1 = (0, 1), mass1 = 1 position2 = (1, 0), velocity2 = (0, +1), mass2 = 1 Subcase IV: Elliptical Orbit s, unequal masses position1 = (0, 0), velocity1 = (0, 1/ 2), mass1 = 1 position2 = (1, 0), velocity2 = (0, +1/ 2), mass2 = 2 ...

#### PS12

MIT, UMB 113
Excerpt: ... of that problem is a bit more involved, but the result is quite similar. For now, treat this as an optional extra part (but still do the problem) and look for the variation in the solutions. (6) Chapter 8 Page 131, Problem 64 Don't be put off by the length of the problem statement. The gist of it is that the total mechanical energy of an object of mass m in an elliptical orbit of mass M is GM m , E=- 2a where a is the semimajor axis of the ellipse. This result takes some effort to derive, and needs a result from Chapter 11. A derivation is in the Supplemental Notes Virial Theorem for Elliptic Orbits. For now, we'll take it on faith. One neat thing in the above formula is that it's valid for parabolic (a ) and hyperbolic (a < 0, which is hard to interpret physically) orbits. I've added a figure to go with this problem, in a separate document, linked from the Problem Set web page. ...

#### TTh_HW03

University of Hawaii - Hilo, ASTRO 110
Excerpt: ... is discovered in an elliptical orbit with a period of exactly one year and at perihelion it is 0.5 AU from the Sun. Using Kepler's third law, how far from the Sun is this asteroid when at aphelion? (Drawing a diagram of the orbit, including the Sun, will help.) A) 1.0 AU B) 1.5 AU C) 2.5 AU D) 2.0 AU 17. Halley's Comet returns to the Sun's vicinity every 76 years in an elliptical orbit . (See Fig. 4-22, Freedman and Kaufmann, Universe, 7th ed.) What is the semimajor axis of this orbit? A) 17.5 AU B) 0.59 AU C) 1 AU D) 50.000 AU Page 3 18. Consider a comet in a long, thin elliptical orbit with a semi-major axis of one AU. What can you say about the sidereal period of this comet? A) It will be less than one year. B) It will be one year. C) It will be more than one year. D) It is not possible to determine the comet's sidereal period without knowing the eccentricity of its orbit. 19. An object orbiting the Sun in a circle can be said to be A) always accelerating. B) moving under the action of equal and oppos ...

#### TTh_HW03_key

University of Hawaii - Hilo, ASTRO 110
Excerpt: ... elliptical orbit around the Sun. The location of the Sun relative to this ellipse is at A) the focus that is closer to the point where Mars is moving the slowest. B) one end of the major axis of the ellipse. C) the exact center of the ellipse. D) the focus that is closer to the point where Mars moves the fastest. Page 2 11. The eccentricity of a planet's orbit describes A) its tilt with respect to the plane of Earth's orbit (the ecliptic plane). B) its shape compared to that of a circle. C) its motion at any specific point in its orbit as seen from Earth, i.e., whether direct, retrograde or stationary. D) the tilt of the planet's spin axis with respect to its orbital plane. 12. In any one day, the line joining a planet to the Sun will sweep through a certain angle as seen from the Sun. At what position is the planet when this angle has its smallest value? A) perihelion B) greatest elongation C) inferior conjunction D) aphelion 13. Which of the following statements is true, according to Kepler's third law ...

#### PHY 121 Ch 12b Lecture

ASU, PHY 121
Excerpt: ... Earth A. is less than the true weight but points in the same direction. B. is less than the true weight and points in a different direction. C. is more than the true weight but points in the same direction. D. is more than the true weight and points in a different direction. 5 Geosynchronous orbits GMm mv 2 GM GM = !r= 2 = 2 2 r r v ( 2" r T ) # GMT & r=% 2( \$ 4" ' 2 1/3 = Nm kg ) 5.97 ) 10 kg ) ( 24 ) 3600s ) & ( 4" 2 ' 2 -2 24 2 1/3 # 6.67 ) 10 % \$ *11 = 4.2 ) 10 km 4 6 Energy in circular orbits GMm mv 2 = 2 r r 1 2 GMm E = mv ! 2 r 1 " GMm % GMm =\$ '! 2# r & r GMm =! 2r 7 Flying satellites A 1000 kg communication satellite needs to be boosted from an orbit 300 kg above the earth to a gesynchronous orbit 35,900 km above the Earth. a)Find the velocity v1 on the lower circular orbit and the velocity v1 at the low point on the elliptical orbit that spans the two circular orbits. How much work must the rocket motor do to transfer the satellite from the circular orbit to the elliptical orbit ? b) No ...

#### lecture09

Caltech, PHYS 001
Excerpt: ... 7 Fast Moving Projectiles - Satellites We understand that the harder we throw a ball the farther it will travel before landing The curved path of the projectile becomes wider Therefore, it is possible to have the projectile's trajectory match the curve of the earth: then it is a satellite: 8 These are the same three animations we saw lecture where we saw if we threw something harder and harder, that its trajectory would match the curve of the Earth, the object would be in orbit and therefore a satellite. 8 Circular Orbits Vertical velocity only serves to fall around the earth Horizontal velocity determines the speed a satellite travels around the Earth Gravity pulls at a right angle, so it does not speed it up or slow it down, it only changes the direction of the velocity The higher the altitude of the orbit, the longer the period (the time it takes to complete one orbit) 9 The tangential velocity will be the same as the horizontal velocity. 9 Elliptical Orbit s An ellipse is lik ...

#### ch5_notes

University of Montana, MCS 185
Excerpt: ... Chapter 5 Notes Jesse Johnson February 26, 2007 1 Introduction Western Civilizations understanding of the orbit of planets about the sun represents a pivotal point in our history. Keplers Laws of planetary motion provide a clean summary of concepts 1. Each planet moves in an elliptical orbit with the sun located at one of the foci of the ellipse. 2. The speed of a planet increases as its distance from the sun decreases such that the line from the sun to the planet sweeps out equal areas in equal times. 3. The ratio T3 is the same for all planets that orbit the sun. T is the period of the a planet and a is the semi-major axis of the ellipse. These relations motivate us to seek the equations of motion for planets. 2 2 The Equations of Motion This reduces to a one body problem because the sun can be assumed to be much more massive than the Earth. A typical approach to this problem is to use the reduced mass of the system = Mm M+m (1) To begin, consider a two-body problem, such as the Earth ...

#### ASCI512lect2_orbits

Metropolitan State College of Denver, ASCI 512
Excerpt: ... ASCI 512 Space Launch & Mission Operations Dr. Walter Goedecke Fall 2006 1 Topics Orbital Mechanics o Kinetic and Potential Energy o Conic Sections o Parabolic Trajectories. o Circular Orbits. o Elliptical Orbit s. o Hyperbolic Orbits and Flybys o Lagrange Points 2 Orbital Mechanics Kinetic and Potential Energy An object in motion has two types of energy of importance in orbital mechanics: kinetic energy and potential energy Kinetic energy energy due to the object's motion. Potential energy energy due to the object's position. The total energy is usually designated by E = T + U where T = kinetic energy U = potential energy 3 Orbital Mechanics Kinetic and Potential Energy An object in a uniform gravitational field will have mechanical energy as: mv 2 + mgh E=T+U= 2 where m = particle mass v = object velocity g = gravitational acceleration [m/s2] h = object height This means that the object's kinetic energy is proportional to the square of an object's velocity. Also, the pot ...

#### Lecture26

Wake Forest, PHY 262
Excerpt: ... Planetary Motion Monday, April 02, 2007 10:26 AM Orbital dynamics: How to change orbits around the sun that results in changing orbits around a different planet. Hohmann transfer: Start out orbiting planet A {circular sun-centered orbit} Change into an elliptical orbit around the sun; wait until intersect orbit of planet of interest Change into new orbit around planet B Let's figure out the energy costs, neglecting the planets. lLecture26 Page 1 Last time: Went through Kepler' 1st and 3rd law for an inverse-square force law. Keep in mind out entire derivation was valid only for inverse-square law, because we used U(r) =-k/r in the EOM Once we did this, we obtained an equation for a conic section, and then we discussed orbits. Today we 1) Talk about orbital dynamics (briefly) 2) Discuss stability of circular orbits with a more general force law lLecture26 Page 2 2cd transfer: same arguments, just swap r1 and r2. Total time of transfer: 1/2 period of the transfer orbit. Note: this can be a very slow pro ...

#### Phy107Lect02

Wisconsin, PHYS 107
Excerpt: ... epicycles for accuracy. Nicolaus Copernicus 1473-1543 The heliocentric universe "Natural" explanation of retrograde motion Retrograde motion occurs when planets pass each other. Comparing Ptolemy and Copernicus Which is the better theory ? Earth-centered Occam's razor: If both theories are consistent with experiment, pick the simpler one. Sun-centered A `good' theory makes predictions Quantitative test by experiments Could not be done until Galileo Galilei built a telescope Half-illuminated Venus at 460 Illuminated more than half Venus seen with a modern amateur telescope and a replica of Galileo's telescope. Earth Galileo Galilei, 1564-1642 20 years of precise observations Brahe's exacting observations challenged both Ptolemy's and Copernicus' theories. Tycho Brahe 1546-1601 Kepler's elliptical orbit s Kepler's idea: Consider non-circular orbits No more epicycles required! Johannes Kepler 1571-1630 Circular orbit Elliptical orbit Brahe's observations favored the new model. C ...

#### PH2014_Ch12_gravity

Oklahoma State, PHYS 2014
Excerpt: ... Chapter 12: Universal Gravity What is the force that holds a man to Moon, Moon to Earth, and Earth to Sun? 1 Toward a Law of Universal Gravity Theology <1642 Earth is the center of universe Planets orbit the Sun not Earth Accurate planetary observation Three laws of planetary motions Telescopic observation of the motions of Jupiter's moons Law of universal gravitation 2 1473Nicolaus Copernicus 1543 1546Tycho Brahe 1601 1571Johannes Kepler 1630 1564Galileo 1642 1642Isaac Newton 1727 Kepler's Laws of Planetary Motions First law: The planets move in elliptical orbit s, with the sun at one focal point. Second Law: A line draw between the sun and a planet sweeps out equal areas during equal intervals of time. Third Law: The square of a planet's orbital period is proportional to the cube of the semimajoraxis length. = period a=major axis length First & Second Laws Third Law 3 a 2 3 What makes the planets move as they do? Realization: the force that holds the Earth to Sun is the ...

#### 2history4s

Augustana, AS 311
Excerpt: ... f the Earth is: C=(360/) X D L=length of shadow Geocentric Solar System Earth is at center of Solar System Developed between ~200BC (Hipparchus) and ~200AD (Ptolemy) Retrograde Motion and Epicycles Heliocentric Solar System Sun at center of Solar System First proposed by Aristarchus (~300BC) More comprehensive model developed by Copernicus (~1500 AD) Uraniborg Tycho, Kepler and the Motions of the Planets Tycho Brahe carefully observed the planets for 20 years at Uraniborg (15761597 AD) Planets move in elliptical orbit s and follow mathematical laws Galileo's Observations Galileo and the Telescope Made many important observations starting in 1610, including: Galileo's writings were condemned by the church, but presented hard evidence Careful observation and theorizing by Copernicus, Brahe, Kepler and Galileo disproved the seeming obvious and incontrovertible geocentric model. Newton and Gravity Why do the planets move? Isaac Newton used ...

#### L2LogicIntro

St. Thomas, PHIL 115
Excerpt: ... (equivalently), in which the truth of the premises guarantees the truth of the conclusion or for which the assertion of the premises and the denial of the conclusion would be a contradiction true premises a sound argument must have true premises as well as a valid form premises that are in some sense prior to the conclusion this avoids triviality two forms of priority premises better known than the conclusion e.g., Planets move in elliptical orbit s, so they must move under the influence of an inverse-square law. here, the observed fact is better known than the scientific law premises state cause of conclusion e.g., Planets move under the influence of an inverse-square law and , so they must move in elliptical orbit s. here the scientific law gives the cause of the observed fact ...

#### Lecture15a

Texas Tech, PHYS 5306
Excerpt: ... Planetary Orbits Planetary orbits in terms of ellipse geometry. In the figure, e Compute major & minor axes (2a & 2b) as in text. Get (recall k = GmM): a ()/[1 - e2] = (k)/(2|E|) (depends only on energy E) b ()/[1 - e2] = ()/(2m|E|) a[1 - e2] (a) (Depends on both energy E & angular momentum ) Apsidal distances rmin & rmax (or r1 & r2): rmin = a(1- e) = ()/(1 + e), rmax = a(1+ e) = ()/(1 - e) Orbit eqtn is: r = a(1- e2)/[1 + e cos( - )] Planetary orbits = ellipses, sun at one focus: Fig: For a general central force, Period of elliptical orbit : = [(2m)/()] A (A = ellipse area) (1) Analytic geometry: Area of ellipse: A ab (2) In terms of k, E & , we just had: a = (k)/(2|E|); b = ()/(2mE) (3) (1), (2), (3) = k(m/2)|E|-(3/2) Alternatively: b = (a) ; [ 2/(mk)] 2 = [(42m)/(k)] a3 The square of the period is proportional to cube of semimajor axis of the elliptic orbit Kepler's Third Law Kepler's Third Law 2 = [(42m)/(k)] a3 The square of period is proportional ...

#### EP Chap13

Kansas State, PHYS 213
Excerpt: ... hen the asteroid is 10 Earth radii from Earth's center. Neglecting the effects of Earth's atmosphere on the asteroid, find the asteroid's speed vf when it reaches Earth's surface. Special case circular orbits Newton' s 2nd law ( v is orbital speed) Mm v2 GM =m v= 2 r r r Consider a single orbital motion ( x = 2r ) Speed v is constant for a circular orbit : F = ma G GM T r Square both sides x = vt 2r = 4 2 T2 = GM 3 r v r 2 Elliptical orbit s 4 2 3 T2 = a GM Kepler's Laws For an elliptical orbit : a = semimajor axis of ellipse 1.THE LAW OF ORBITS: All planets move in elliptical orbit s, with the Sun at one focus 2. THE LAW OF AREAS: A line that connects a planet to the Sun sweeps out equal areas in the p plane of the p planet's orbit in equal time intervals; q ; that is, the rate dA/dt at which it sweeps out area A is constant. 3. LAW OF PERIODS: The square of the period of a planet is proportional to the cube of the semi-major axis of its orbit. m M a Keple ...

#### review1_notes

Pittsburgh, A 88
Excerpt: ... anetary motions, especially Mars, which enabled Kepler to develop his 3 laws of planetary motion. - Kepler - 1571-1630 AD - he showed that the Sun-centered (heliocentric) model of the Universe worked better if the planets were in elliptical orbit s about the Sun instead of circular orbits. - Lippershey - Dutch lens maker often credited as the rst to build a refracting telescope (but not for astronomy). - Galilei (Galileo) - 1564-1642 AD - he did important experiments which helped reveal the laws of falling bodies; he also made observations with a telescope (he put together the rst refracting telescope used for astronomical observation) which supported the heliocentric theory. - Hooke - great scientist of his time, but he often criticized Newton. - Halley - prompted Newton to publish his great work, Principia; discovered Halleys Comet. - Newton - 1642-1727 AD - great synthesizer of experiments and observations who put forth the Theory of Gravity (for example, he explained Keplers empirical 3rd ...

#### 9-1

University of Illinois, Urbana Champaign, GEO 116
Excerpt: ... Geol 116 The Planet Class 9-1 Lecture Mar 14, 2005 Orbits Objectives Keplers Laws: aphelion, perihelion, orbital semimajor axis, orbital period, ecliptic plane, orbital inclination, retrograde, prograde Tidal heating: synchronous rotation, libration Notes What is Keplers First Law? What are the implications? Keplers First Law states that planets move in elliptical orbit s, with the Sun at one focus. One implication of this law is that the distance between a planet and the Sun changes as the planet orbit around the Sun. The maximum distance is reached when the planet is at the aphelion of its orbit, and the minimum at the perihelion of the orbit (in the case of the Moon orbiting the Earth, these points are apogee and perigee, respectively). A commonly used parameter, the orbital semimajor axis, equals to the average of the aphelion and perihelion distances. The eccentricity measures the extent of the departure of an ellipse (such as a planetary orbit) from a circle. It is a dimensionles ...

#### Chapter1B.99yrs

Youngstown, ASTRO 1504
Excerpt: ... His mother was accused of witchcraft, and Kepler narrowly kept her from being 10 burned at the stake. Kepler's Laws of Planetary Motion Empirical Laws they fit the data. 1. The planets revolve about the Sun in elliptical orbit s with the Sun at one of the foci. 2. A line between the Sun and a planet sweeps out equal areas in equal times. 3. T2 is proportional to r3 or (period)2 (distance)3 Laws 1 and 3 are a consequence of the inverse-square nature of gravity while 2 comes from angular momentum conservation. 11 Kepler's First Law Planets travel along elliptical orbit s, with the Sun at one of the two focii This extremely important result finally broke down the notion that the planets had to move in circles very clean, elegant solution! (no more epicycles) 12 13 a=SEMI-MAJOR AXIS b=SEMI-MINOR AXIS (or, 1/2 of the major axis) FOCII The eccentricity e of the orbit is a measure of how squashed or circular the ellipse is: (e=0 is a circle) ...

#### Lecture16a

Texas Tech, PHYS 5306
Excerpt: ... Brief Discussion! Apsidal Angles & Precession Particle undergoing bounded, non-circular motion in a central force field Always have r1 < r < r2 V(r) vs r curve Only 2 apsidal distances exist for bounded, noncircular motion. Possible motion: ...