# Register now to access 7 million high quality study materials (What's Course Hero?) Course Hero is the premier provider of high quality online educational resources. With millions of study documents, online tutors, digital flashcards and free courseware, Course Hero is helping students learn more efficiently and effectively. Whether you're interested in exploring new subjects or mastering key topics for your next exam, Course Hero has the tools you need to achieve your goals.

4 Pages

### class18

Course: ASTR 18, Fall 2008
School: Maryland
Rating:

Word Count: 686

#### Document Preview

18. Class N -body Techniques, Part 1 The N -body Problem Study of the dynamics of interacting particles, usually involving mutual forces. E.g., Application Mutual Force gravity stellar dynamics, planetesimals QM molecular dynamics, solid-state physics EM plasma physics etc. etc. Stick with gravitation for now. Only a few literature references available, e.g., Aarseth, Danby (Ch. 9), etc. Generalized Newtons...

Register Now

#### Unformatted Document Excerpt

Coursehero >> Maryland >> Maryland >> ASTR 18

Course Hero has millions of student submitted documents similar to the one
below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.

Course Hero has millions of student submitted documents similar to the one below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.
18. Class N -body Techniques, Part 1 The N -body Problem Study of the dynamics of interacting particles, usually involving mutual forces. E.g., Application Mutual Force gravity stellar dynamics, planetesimals QM molecular dynamics, solid-state physics EM plasma physics etc. etc. Stick with gravitation for now. Only a few literature references available, e.g., Aarseth, Danby (Ch. 9), etc. Generalized Newtons Laws i = r j=i F ij = j=i Gmj (ri rj ) . |ri rj |3 These are 3N coupled 2nd -order ODEs. As usual, reduce to 1st -order: ri = vi (velocity), Gmj (ri rj ) vi = (acceleration). |ri rj |3 j=i This makes 6N coupled 1st -order ODEs. We know how to solve these! Key is to solve the equations eciently: 1. Solve Newtons Laws using ODE integrator. 2. Evaluate interparticle forces Fij several techniques. Typical Parameters First, need to get a feeling for the problem... What are typical problem sizes? N N 2: Jupiter and Sun, extrasolar planets. 9: Solar system. 1 N N N N N N 10100: Small stellar system. 1001000: Open cluster, rubble pile! 105 106 : Globular cluster, planetesimals. 107 108 : Cosmological volume (DM halos). 109 : Planetary rings. 1011 : Galaxy. Also have restricted problems where one or more test particles exert no gravitational forces but still feel forces due to more massive particles, e.g., Lagrange problem, comets in the Oort cloud, etc. What are typical timescales? ([T ] = [L]/[V ]) Solar system: Orbital timeevolution time (1109 yrs). Stellar system: Relaxation time ( 100s of crossing times). Globular cluster: Core collapse ( 10s of relaxation times). Galaxy: 1010 yrs (many steps). Universe: 1010 yrs (fewer steps). Often to achieve steady state over many dynamical times it seems N /t constant. = timescale and lengthscale closely coupled. E.g., crossing time for closed dynamical system. 1 Virial theorem: 2K + W = 0, K = 2 M v 2 , W = GM 2 /rg . 3/2 Crossing time = [L]/[V ] rg / v 2 1/2 rg / GM . Typically want t D /30 = cross /30. Another handy formula: 3 . G E.g., for typical asteroid, 2 g/cc so D 2.3 h. For Earth, spread out mass 4 3 r of Sun to 1 AU: = M 3 / = D 1 yr. Why? 2 r = GM/r 2 4 2 / 2 = 4 GM/r 3 = 3 G. 3/ G. D Units In MKS, G = 6.7 1011 , M = 2 1030 , r = 1.5 1011 . Often want to work in scaled units to keep values close to unity. Typically set G 1. For solar system, use masses in M , distances in AU. Then times in yr/2 and speeds in v = 30 km s1 . For galaxies, could use masses in 109 M , distances in kpc. Then times would be in 15 Myr and speeds in kpc/15 Myr. 2 Constants of motion If there are no outside forces/torques, Newtons Laws for a gravitating system imply: 1. Total energy is conserved. 2. Total angular momentum is conserved. 3. System center of mass is either stationary in the inertial frame, or moves with constant velocity. Can therefore set rg = vg 0. N = 2 problem Solved by Kepler, explained by Newton. General solution (ellipse): r = a(1 e cos ) cos e cos = 1 e cos where a = semi-major axis, e = eccentricity, = eccentric anomaly, and mean anomaly t = e sin (Keplers equation). Useful facts: if r and v are relative coordinates of two bodies, then 1 1 Gm1 m2 1 m1 m2 2 1 Gm1 m2 2 2 2 E = m1 v1 + m2 v2 = v + Mvg , 2 2 |r1 r2 | 2M 2 r where M m1 + m2 . Hence, since we can always set vg 0, v 2 GM E = , 2 r where m1 m2 /M = reduced mass. Also have Gm1 m2 E...

Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more. Course Hero has millions of course specific materials providing students with the best way to expand their education.

Below is a small sample set of documents:

Maryland - ASTR - 415
Class 18. N -body Techniques, Part 1The N -body Problem Study of the dynamics of interacting particles, usually involving mutual forces. E.g., Application Mutual Force gravity stellar dynamics, planetesimals QM molecular dynamics, solid-state physi
Maryland - ASTR - 19
Class 19. N -body Techniques, Part 2Time-integration Schemes Clearly, Newtons laws are IVP. Could use any method (Euler, RK4, etc.). But, issue is to balance accuracy vs. eciency. Typically need many particles to capture dynamics correctly (e.g.,
Maryland - ASTR - 415
Class 19. N -body Techniques, Part 2Time-integration Schemes Clearly, Newtons laws are IVP. Could use any method (Euler, RK4, etc.). But, issue is to balance accuracy vs. eciency. Typically need many particles to capture dynamics correctly (e.g.,
Maryland - ASTR - 20
Class 20. N -body Techniques, Part 3The PM Method, ContinuedThere are several distinct steps in PM process: 1. Assign particles to mesh to compute i . 2. Get boundary conditions for (0 and N +1 ). 3. Solve discretized version of Poissons equation.
Maryland - ASTR - 415
Class 20. N -body Techniques, Part 3The PM Method, ContinuedThere are several distinct steps in PM process: 1. Assign particles to mesh to compute i . 2. Get boundary conditions for (0 and N +1 ). 3. Solve discretized version of Poissons equation.
Maryland - ASTR - 21
Class 21. N -body Techniques, Part 4Tree CodesEciency can be increased by grouping particles together: Nearest particles exert greatest forces direct summation. Distant particles exert smallest forces treat in groups.Treat distant particles as
Maryland - ASTR - 415
Class 21. N -body Techniques, Part 4Tree CodesEciency can be increased by grouping particles together: Nearest particles exert greatest forces direct summation. Distant particles exert smallest forces treat in groups.Treat distant particles as
Maryland - ASTR - 22
Class 22. PDEs, Part 1 Cf. NRiC 19.Classication of PDEs A PDE is simply a dierential equation of more than one variable (so an ODE is a special case of a PDE). PDEs are usually classied into three types: 1. Hyperbolic (second or rst order in time
Maryland - ASTR - 415
Class 22. PDEs, Part 1 Cf. NRiC 19.Classication of PDEs A PDE is simply a dierential equation of more than one variable (so an ODE is a special case of a PDE). PDEs are usually classied into three types: 1. Hyperbolic (second or rst order in time
Maryland - ASTR - 23
Class 23. PDEs, Part 2Solving Hyperbolic PDEs, ContinuedUpwind dierencing In addition to amplitude errors (instability or damping), scheme may also have phase errors (dispersion) or transport errors (spurious transport of information). Upwind die
Maryland - ASTR - 415
Class 23. PDEs, Part 2Solving Hyperbolic PDEs, ContinuedUpwind dierencing In addition to amplitude errors (instability or damping), scheme may also have phase errors (dispersion) or transport errors (spurious transport of information). Upwind die
Maryland - ASTR - 24
Class 24. Fluid Dynamics, Part 1 The equations of uid dynamics are coupled PDEs that form an IVP (hyperbolic). Use the techniques described so far, plus additions.Fluid Dynamics in Astrophysics Whenever mean free path problem scale L in a plasm
Maryland - ASTR - 415
Class 24. Fluid Dynamics, Part 1 The equations of uid dynamics are coupled PDEs that form an IVP (hyperbolic). Use the techniques described so far, plus additions.Fluid Dynamics in Astrophysics Whenever mean free path problem scale L in a plasm
Maryland - ASTR - 415
ASTR415 Survey ResultsSpring 2007 11 respondents 1. Computer familiarity [1=Master, 5=None]: Avg = 2.7 [Skilled], Min = 1 [Master], Max = 4 [Novice] 2. Unix familiarity [1=Master, 5=None]: Avg = 3.6 [Skilled], Min = 2 [Expert], Max = 5 [None] 3. Uni
Maryland - ASTR - 415
ASTR415 Spring 2007Due May 08, 2007Term Project1) For your term project you will install and learn how to use a freely available (opensource) 3D visualization tool based on Open-GL. You will write a short report and present in class the results
Maryland - ASTR - 415
| nsttuustBj1d4tssBnn nn Xdss &amp; Xs T a ` C F U IE W P U Rs y ` EY ' W F I 3 w i w w Te T UEY TYE S I P T C W I W p m UE H U T ` y H T y UE XVeiGu(uo&amp;d(Gvtb6dQ&amp;bc(b(xc iqgGgGeb(uu6(xn(g(BxoVXcgGY FE C
Maryland - ASTR - 415
x i e r | z i g j r v fdV i n rv G} ofe } i im f XGl Xl G im hl m f X u e i} g G i} g im g f fe ov i m v i u IfX u e ofe G i m n m i Vn} } i Vn} G oVe } } X i ofw iml g ine m l g l e iV } ne } dFVG r v m v f } | zx
Maryland - ASTR - 415
4Xeex xe e7jj \$v )e `j B I Va U b V V U R I F YH Y W V U W R H c W V V F Y WH Q V I Pgp1eCd)wss`Xgpwhe`Xe1wPX)1T Ia R Y W R S U b V Y I W R U U R S Y I R U V F Y I V H b U U T I Va b Y r WH Raa Rt V F D
Maryland - ASTR - 415
ICadafWWaHH\$CHIaRp\$a\$wraWCawHa 9 i i S x i U bIxpU B @ e F k i 6 d B wP v q D A B FP PT B 8 6 qT Q v D 4 B v 4 @ D B F f D F B A @ A B F 6P k Uif'fbS7)UdRT Y rtuCaCdRR99R7\$17t\$ICdP Y `CaC9ECGR'\$i S
Maryland - ASTR - 415
A Crash Course on UNIXUNIX is an &quot;operating system&quot;.Interface between user and data stored on computer. A Windows-style interface is not required. Many flavors of UNIX (and windows interfaces).Solaris, Mandrake, RedHat (fvwm, Gnome, KDE)
Maryland - ASTR - 415
Data RepresentationsComputers store data as different variable types, e.g. integer, floating point, complex, etc. Different machines have different wordlengths, e.g. 4-byte ints on a 32-bit machine (Pentium), 8-byte ints on a 64-bit machine (Alpha).
Maryland - ASTR - 415
VisualizationVisualization is useful for:1) Data 2) Codeentry (initial conditions) debugging and performance analysis and display of results3) Interpretation Our focus will be #3. The computational astrophysicist can either:1) Develop 2) U
Maryland - ASTR - 415
Numerical Linear Algebra Probably the simplest kind of problem. Occurs in many contexts, often as part of larger problem. Symbolic manipulation packages can do linear algebra &quot;analytically&quot; (e.g. Mathematica, Maple). Numerical methods needed when:
Maryland - ASTR - 415
Nonlinear EquationsOften (most of the time?) the relevant system of equations is not linear in the unknowns. Then, cannot decompose as Ax = b. Oh well. Instead write as:(1) (2) f(x) = 0 f(x) = 0function of one variable (1-D) x = (x1,x2,.,xn
Maryland - ASTR - 415
Statistical Description of Data Cf. NRiC, Chapter 14. Statistics provides tools for understanding data.In the wrong hands these tools can be dangerous! Apply some formula to data to compute a &quot;statistic&quot;. Find where value falls in a probability
Maryland - ASTR - 415
Modeling of Data NRiC Chapter 15. Model depends on adjustable parameters. Can be used for &quot;constrained interpolation&quot;. Basic approach:1. 2. 3. 4.Choose figure-of-merit function (e.g. 2). Adjust best-fit parameters: minimize merit function. Co
Maryland - ASTR - 415
Random Numbers NRiC Chapter 7. Frequently needed to generate initial conditions. Often used to solve problems statistically. How can a computer generate a random number? It can't! Generators are pseudo-random. Generators are deterministic: i
Maryland - ASTR - 415
Numerical Integration (Quadrature) NRiC Chapter 4. Already seen Monte Carlo integration. Can cast problem as a differential equation (DE): = is equivalent to solving for I y(b) the DE dy/dx = f(x) with the boundary condition (BC) y(a) =
Maryland - ASTR - 415
Ordinary Differential Equations (ODEs) NRiC Chapter 16. ODEs involve derivatives wrt one independent variable, e.g. time t. ODEs can always be reduced to a set of firstorder equations (involving only first derivatives).e.g. = is
Maryland - MATH - 241
MATH 241 CALCULUS III FIRST MIDTERM EXAM Instructions. Answer each question on a separate answer sheet. Show all your work. Be sure your name, section number, and problem number are on each answer sheet, and that you have copied and signed the honor
Maryland - MATH - 241
MATH 241 CALCULUS III FIRST MIDTERM EXAM SOLUTIONS (1) For this problem, u = + k, and v = + 2 + 3 k. ij i j (a) u v = 1 + 2 3 = 2. k ij (b) u v = det 1 1 1 = 5 2 + 3 k. i j 1 2 3 (c) The symmetric form of the equations are: 2 x =
Maryland - ASTR - 601
Astronomy 601 - Fall 2005 Radiative ProcessesInstructor Prof. Massimo Ricotti Oce: CSS 0213 E-mail: ricotti@astro.umd.edu Phone: (301) 405 5097 Oce hours: by appointment Class web page: http:/www.astro.umd.edu/ricotti/NEWWEB/teaching/ASTR601.html Sc
Maryland - ECE - 2003
Center for Satellite and Hybrid Communication NetworksIntegrated Security Services for Dynamic Coalition ManagementHimanshu Khurana and Vijay Bharadwaj Electrical and Computer Engineering Department, University of Maryland College Park, Maryland 2
Maryland - ECE - 2003
Proceedings of the DARPA Information Survivability Conference and Exposition (DISCEX03) 0-7695-1897-4/03 \$17.00 2003 IEEEProceedings of the DARPA Information Survivability Conference and Exposition (DISCEX03) 0-7695-1897-4/03 \$17.00 2003 IEEEPr
Maryland - ECE - 2003
Integrated Security Services for Dynamic CoalitionsHimanshu Khurana1, Serban Gavrila1, Rakeshbabu Bobba, Radostina Koleva, Anuja Sonalker, Emilian Dinu, Virgil Gligor, and John Baras Electrical and Computer Engineering Department, University of Mary
Maryland - ECE - 2003
Towards Automated Negotiation of Access Control PoliciesVijay G. Bharadwaj and John S. Baras Institute for Systems Research, University of Maryland, College Park MD 20742, USA. vgb,baras @umd.eduAbstractWe examine the problem of negotiating acces
Maryland - ECE - 2003
DYNAMIC ADAPTATION OF ACCESS CONTROL POLICIESVijay Bharadwaj and John Baras Institute for Systems Research University of Maryland College Park MD 20742ABSTRACTWe describe an architecture and algorithms for deriving an access control policy by com
Maryland - ECE - 2003
Center for Satellite and Hybrid Communication NetworksIntegrated Security Services for Dynamic Coalition ManagementHimanshu Khurana Electrical and Computer Engineering Department, University of Maryland College Park, Maryland 20742 DARPA DC PI Mee
Maryland - ECE - 2003
Reasoning about Joint Administration of Access Policies for Coalition ResourcesHimanshu Khurana Virgil Gligor John Linnjlinn@rsasecurity.com{hkhurana, gligor}@eng.umd.eduUniversity of Maryland College Park, MD.RSA Labs Bedford, MA.OutlineC
Maryland - ECE - 2003
Center for Satellite and Hybrid Communication NetworksIntegrated Security Services for Dynamic Coalition ManagementVirgil D. Gligor and John S. Baras Electrical and Computer Engineering Department, University of Maryland College Park, Maryland 207
Maryland - ECE - 2003
Center for Satellite and Hybrid Communication NetworksIntegrated Security Services for Dynamic Coalition ManagementHimanshu Khurana and Vijay Bharadwaj Electrical and Computer Engineering Department, University of Maryland College Park, Maryland 2
Maryland - ECE - 2006
DK6037 FLPraveen K. MurthyFujitsu Laboratories of America, Sunnyvale, California, USAShuvra S. BhattacharyyaUniversity of Maryland, College Park, USAEffective Strategies for Aggressive Memory OptimizationAlthough programming in memory restri
Maryland - ECE - 2003
about the book. . . Ranging from low-level applicationand architectureoptimizationsto high-level modeling and exploration concerns,this text/reference compiles essentialresearchon various levels of abstractionappearingin embeddedsystemsand softwarede
Maryland - ECE - 2000
Marcel Dekker Catalog: Embedded MultiprocessorsPage 1 of 1Embedded MultiprocessorsScheduling and Synchronization Sundararajan Sriram, Texas Instruments, Inc., Dallas, Texas, and Shuvra S. Bhattacharyya, University of Maryland, College Park serie
Maryland - ECE - 1996
Springer - Signals &amp; CommunicationPage 1 of 1springer.com Springer Berlin Heidelberg New YorkSignals &amp; CommunicationSoftware Synthesis from Dataflow Graphs Series: The International Series in Engineering and Computer Science, Vol. 360 Bhattach
Maryland - GEOL - 100
INTRODUCTION TO PHYSICAL GEOLOGY GEOL100 -Section: 0102 -Fall 2004Professor: Bill McDonough Office Hours: Wednesday, 9-10 AM Office: Chemistry Bldg (091), room 0229 Phone: 301-405-5561 Email: mcdonough@geol.umd.edu (preferred communication medium)
Maryland - ASTR - 120
Homework #2Chapter 3 Q25 The dividing line between the illuminated and unilluminated halves of the Moon is called the terminator. The terminator appears curved when there is a crescent or gibbous moon, but appears straight when there is a first qu
Maryland - ASTR - 120
ASTR120 Homework #3 (Hamilton) due Thursday Sept. 27 (20 Points)Finish reading Chapter 4! Now you are cleared for this homework (from Chapter 3, page 92). Try to start early if you can - there are some interesting (but tricky!) problems this week.
Maryland - ASTR - 120
ASTR120 Homework #4 (Hamilton) due Thursday Oct. 4 (30 Points)Denitely nish reading Chapters 4 and 7! These rst ve problems are from Chapter 4. W4. Do problem W4 from http:/www.astro.umd.edu/hamilton/ASTR120/webexp.html. 46. A satellite is said to
Maryland - ASTR - 120
ASTR120 Homework #5 (Hamilton) due Thursday Oct. 18 (20 Points)These problems are from Chapter 8. Finish reading the chapter!Do problem W9 from http:/www.astro.umd.edu/hamilton/ASTR120/webexp.html. 12. What is meant by a substances condensation t
Maryland - ASTR - 120
ASTR120 Homework #6 (Hamilton) due Thursday Oct. 25 (20 Points)Finish reading Chapters 9 and 10! These rst six problems are from Chapter 9. 30. On average, the temperature beneath the Earths crust increases at a rate of 20 C per kilometer. At what
Maryland - ASTR - 120
ASTR120 Homework #7 (Hamilton) due Thursday Nov. 1 (20 Points)Finish reading Chapters 10 and 11. These problems are from Chapter 10. 29. Temperature variations between day and night are much more severe on the Moon than on Earth. Explain why. 31. U
Maryland - ASTR - 120
ASTR120 Homework #8 (Hamilton) due Thursday Nov. 8 (20 Points)Finish reading Chapters 11 and 12! This is the last graded homework before Exam 2 on Nov. 15. These problems are from Chapter 11. 56. The Mariner 2 spacecraft detected more microwave rad
Maryland - ASTR - 120
ASTR120 Homework #9 (Hamilton) due Thursday Nov. 29 (20 Points)Finish reading Chapter 12 and 13! These problems are from Chapter 12. 49. When Saturn is at dierent points in its orbit, we see dierent aspects of its rings because the planet has a 27
Maryland - ASTR - 120
ASTR120 Homework #10 (Hamilton) due Thursday Dec. 6 (20 Points)Finish reading Chapters 14, 15, and 28! These problems are from Chapter 14. 30. At certain points in its orbit, a stellar occultation by Uranus would not reveal the existence of the rin
Maryland - ASTR - 120
ASTR120 Challenge Problem #1 (Hamilton) Optional, due before Midterm #1In this challenge problem you will work out how large a shadow moons in our Solar System cast on their parent planets. a) Draw an accurate picture and use algebra and geometry t
Maryland - ASTR - 120
ASTR120 Challenge Problem #2 (Hamilton) Optional, due before Midterm #2In this challenge problem you will work out how Mars tilt and eccentricity aect its seasons. a) For an untilted Earth on a circular orbit at 1 AU, work out the average energy hi
Maryland - GEOL - 3
Geochemistry Geophysics GeosystemsClick HereAN ELECTRONIC JOURNAL OF THE EARTH SCIENCES Published by AGU and the Geochemical SocietyG3Technical Brief Volume 7, Number 11 25 November 2006 Q11021, doi:10.1029/2006GC001352 ISSN: 1525-2027Full
Maryland - GEOL - 3
American Mineralogist, Volume 91, pages 14881498, 2006Lithium isotopic systematics of granites and pegmatites from the Black Hills, South DakotaFANG-ZHEN TENG,1,* WILLIAM F. MCDONOUGH,1 ROBERTA L. RUDNICK,1 RICHARD J. WALKER,1 AND MONA-LIZA C. SIR
Maryland - GEOL - 3
Earth and Planetary Science Letters 243 (2006) 701 710 www.elsevier.com/locate/epslDiffusion-driven extreme lithium isotopic fractionation in country rocks of the Tin Mountain pegmatiteFang-Zhen Teng , William F. McDonough, Roberta L. Rudnick, Ri
Maryland - GEOL - 3
Geochimica et Cosmochimica Acta 70 (2006) 15371547 www.elsevier.com/locate/gcaExperimental partitioning of uranium between liquid iron sulde and liquid silicate: Implications for radioactivity in the Earths coreKevin T. Wheeleraa,*, David Wal