Dynamics of Planetary Systems

641 Pages·2023·17.287 MB·English

by  Scott Tremaine

Dynamics of Planetary Systems
PRINCETONSERIESINASTROPHYSICS
EDITEDBYDAVIDN.SPERGEL
TheoryofRotatingStars,byJean-LouisTassoul
TheoryofStellarPulsation,byJohnP.Cox
GalacticDynamics,SecondEdition,byJamesBinneyandScottTremaine
DynamicalEvolutionofGlobularClusters,byLymanS.Spitzer,Jr.
SupernovaeandNucleosynthesis:AnInvestigationoftheHistoryofMatter,
fromtheBigBangtothePresent,byDavidArnett
UnsolvedProblemsinAstrophysics,editedbyJohnN.Bahcalland
JeremiahP.Ostriker
GalacticAstronomy,byJamesBinneyandMichaelMerrifield
ActiveGalacticNuclei:FromtheCentralBlackHoletotheGalacticEnvironment,
byJulianH.Krolik
PlasmaPhysicsforAstrophysics,byRussellM.Kulsrud
ElectromagneticProcesses,byRobertJ.Gould
ConversationsonElectricandMagneticFieldsintheCosmos,byEugeneN.Parker
High-EnergyAstrophysics,byFulvioMelia
StellarSpectralClassification,byRichardO.GrayandChristopherJ.Corbally
ExoplanetAtmospheres:PhysicalProcesses,bySaraSeager
PhysicsoftheInterstellarandIntergalacticMedium,byBruceT.Draine
TheFirstGalaxiesintheUniverse,byAbrahamLoebandStevenR.Furlanetto
ExoplanetaryAtmospheres:TheoreticalConceptsandFoundations,byKevinHeng
MagneticReconnection:AModernSynthesisofTheory,Experiment,and
Observations,byMasaakiYamada
PhysicsofBinaryStarEvolution:FromStarstoX-rayBinariesandGravitational
WaveSources,byThomasM.TaurisandEdwardP.J.vandenHeuvel
DynamicsofPlanetarySystems,byScottTremaine
Dynamics of Planetary Systems
Scott Tremaine
PRINCETON UNIVERSITY PRESS
PRINCETON AND OXFORD
Copyright(cid:2)c 2023byPrincetonUniversityPress
PrincetonUniversityPressiscommittedtotheprotectionofcopyright
andtheintellectualpropertyourauthorsentrusttous.Copyright
promotestheprogressandintegrityofknowledge.Thankyoufor
supportingfreespeechandtheglobalexchangeofideasbypurchasing
anauthorizededitionofthisbook.Ifyouwishtoreproduceordistribute
anypartofitinanyform,pleaseobtainpermission.
Requestsforpermissiontoreproducematerialfromthiswork
[email protected]
PublishedbyPrincetonUniversityPress
41WilliamStreet,Princeton,NewJersey08540
99BanburyRoad,OxfordOX26JX
press.princeton.edu
AllRightsReserved
ISBN9780691207124
ISBN(pbk.)9780691207117
ISBN(e-book)9780691244228
BritishLibraryCataloging-in-PublicationDataisavailable
Editorial:IngridGnerlich&WhitneyRauenhorst
ProductionEditorial:AliParrington
Jacket/CoverDesign:WandaEspaña
Production:JacquelinePoirier
Publicity:MatthewTaylor&CharlotteCoyne
Coverimage:TheinnermostGalileansatellite,Io,castsitsshadowonto
theclouddecksofJupiterinthisVoyager1mosaic,taken4March1979,
ataplanet-spacecraftdistanceof1millionkm.PhotographbyIanRegan
(sourceimagescourtesyofNASA/JPL,viathePDSRing-MoonSystems
Node’sOPUSservice).
ThisbookhasbeencomposedinLATEX.Thepublisherwouldliketo
acknowledgetheauthorofthisvolumeforprovidingtheprint-ready
filesfromwhichthisbookwasprinted.
Printedonacid-freepaper.∞
PrintedintheUnitedStatesofAmerica
10 9 8 7 6 5 4 3 2 1
frontmatter August 25, 2022 6.125x9.25
Contents
Preface xi
1 The two-body problem 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The shape of the Kepler orbit . . . . . . . . . . . . . . . . . . . . 5
1.3 Motion in the Kepler orbit . . . . . . . . . . . . . . . . . . . . . . 12
1.3.1 Orbit averages . . . . . . . . . . . . . . . . . . . . . . . . 16
1.3.2 Motion in three dimensions . . . . . . . . . . . . . . . . . 17
1.3.3 Gauss’s f and g functions . . . . . . . . . . . . . . . . . . 20
1.4 Canonical orbital elements . . . . . . . . . . . . . . . . . . . . . . 23
1.5 Units and reference frames. . . . . . . . . . . . . . . . . . . . . . 28
1.5.1 Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.5.2 Units for the solar system . . . . . . . . . . . . . . . . . . 30
1.5.3 The solar system reference frame . . . . . . . . . . . . . . 32
1.6 Orbital elements for exoplanets . . . . . . . . . . . . . . . . . . . 32
1.6.1 Radial-velocity planets . . . . . . . . . . . . . . . . . . . . 33
1.6.2 Transiting planets . . . . . . . . . . . . . . . . . . . . . . 35
1.6.3 Astrometric planets . . . . . . . . . . . . . . . . . . . . . 40
1.6.4 Imaged planets . . . . . . . . . . . . . . . . . . . . . . . . 43
1.7 Multipole expansion of a potential . . . . . . . . . . . . . . . . . 44
1.7.1 The gravitational potential of rotating fluid bodies . . . . 46
1.8 Nearly circular orbits . . . . . . . . . . . . . . . . . . . . . . . . . 50
1.8.1 Expansions for small eccentricity . . . . . . . . . . . . . . 50
1.8.2 The epicycle approximation . . . . . . . . . . . . . . . . . 53
1.8.3 Orbits and the multipole expansion. . . . . . . . . . . . . 58
1.9 Response of an orbit to an external force . . . . . . . . . . . . . . 60
1.9.1 Lagrange’s equations . . . . . . . . . . . . . . . . . . . . . 61
1.9.2 Gauss’s equations . . . . . . . . . . . . . . . . . . . . . . 65
2 Numerical orbit integration 71
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
frontmatter August 25, 2022 6.125x9.25
vi Contents
2.1.1 Order of an integrator . . . . . . . . . . . . . . . . . . . . 75
2.1.2 The Euler method . . . . . . . . . . . . . . . . . . . . . . 76
2.1.3 The modified Euler method . . . . . . . . . . . . . . . . . 81
2.1.4 Leapfrog . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
2.2 Geometric integration methods . . . . . . . . . . . . . . . . . . . 84
2.2.1 Reversible integrators . . . . . . . . . . . . . . . . . . . . 86
2.2.2 Symplectic integrators . . . . . . . . . . . . . . . . . . . . 90
2.2.3 Variable timestep . . . . . . . . . . . . . . . . . . . . . . . 93
2.3 Runge–Kutta and collocation integrators . . . . . . . . . . . . . . 96
2.3.1 Runge–Kutta methods . . . . . . . . . . . . . . . . . . . . 96
2.3.2 Collocation methods . . . . . . . . . . . . . . . . . . . . . 101
2.4 Multistep integrators . . . . . . . . . . . . . . . . . . . . . . . . . 104
2.4.1 Multistep methods for first-order differential equations . . 104
2.4.2 Multistep methods for Newtonian differential equations . 109
2.4.3 Geometric multistep methods . . . . . . . . . . . . . . . . 113
2.5 Operator splitting . . . . . . . . . . . . . . . . . . . . . . . . . . 115
2.5.1 Operator splitting for Hamiltonian systems . . . . . . . . 116
2.5.2 Composition methods . . . . . . . . . . . . . . . . . . . . 119
2.5.3 Wisdom–Holman integrators . . . . . . . . . . . . . . . . 120
2.6 Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
2.6.1 Time regularization . . . . . . . . . . . . . . . . . . . . . 122
2.6.2 Kustaanheimo–Stiefel regularization . . . . . . . . . . . . 125
2.7 Roundoff error . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
2.7.1 Floating-point numbers . . . . . . . . . . . . . . . . . . . 129
2.7.2 Floating-point arithmetic . . . . . . . . . . . . . . . . . . 129
2.7.3 Good and bad roundoff behavior . . . . . . . . . . . . . . 132
3 The three-body problem 137
3.1 The circular restricted three-body problem . . . . . . . . . . . . 138
3.1.1 The Lagrange points . . . . . . . . . . . . . . . . . . . . . 141
3.1.2 Stability of the Lagrange points . . . . . . . . . . . . . . . 147
3.1.3 Surface of section . . . . . . . . . . . . . . . . . . . . . . . 151
3.2 Co-orbital dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 155
3.2.1 Quasi-satellites . . . . . . . . . . . . . . . . . . . . . . . . 164
3.3 The hierarchical three-body problem . . . . . . . . . . . . . . . . 168
3.3.1 Lunar theory . . . . . . . . . . . . . . . . . . . . . . . . . 171
3.4 Hill’s problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
3.4.1 Periodic orbits in Hill’s problem . . . . . . . . . . . . . . 185
3.4.2 Unbound orbits in Hill’s problem . . . . . . . . . . . . . . 194
3.5 Stability of two-planet systems . . . . . . . . . . . . . . . . . . . 197
3.6 Disk-driven migration . . . . . . . . . . . . . . . . . . . . . . . . 204
4 The N-body problem 209
4.1 Reference frames and coordinate systems . . . . . . . . . . . . . 209
4.1.1 Barycentric coordinates . . . . . . . . . . . . . . . . . . . 210
frontmatter August 25, 2022 6.125x9.25
Contents vii
4.1.2 Astrocentric coordinates . . . . . . . . . . . . . . . . . . . 211
4.1.3 Jacobi coordinates . . . . . . . . . . . . . . . . . . . . . . 217
4.2 Hamiltonian perturbation theory . . . . . . . . . . . . . . . . . . 221
4.2.1 First-order perturbation theory . . . . . . . . . . . . . . . 223
4.2.2 The Poincar´e–von Zeipel method . . . . . . . . . . . . . . 226
4.2.3 Lie operator perturbation theory . . . . . . . . . . . . . . 228
4.3 The disturbing function . . . . . . . . . . . . . . . . . . . . . . . 234
4.4 Laplace coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . 241
4.4.1 Recursion relations . . . . . . . . . . . . . . . . . . . . . . 243
4.4.2 Limiting cases . . . . . . . . . . . . . . . . . . . . . . . . 245
4.4.3 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . 246
4.5 The stability of the solar system . . . . . . . . . . . . . . . . . . 247
4.5.1 Analytic results . . . . . . . . . . . . . . . . . . . . . . . . 248
4.5.2 Numerical results . . . . . . . . . . . . . . . . . . . . . . . 252
4.6 The stability of planetary systems . . . . . . . . . . . . . . . . . 256
5 Secular dynamics 261
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
5.2 Lagrange–Laplace theory . . . . . . . . . . . . . . . . . . . . . . 267
5.3 The Milankovich equations . . . . . . . . . . . . . . . . . . . . . 276
5.3.1 The Laplace surface . . . . . . . . . . . . . . . . . . . . . 281
5.3.2 Stellar flybys . . . . . . . . . . . . . . . . . . . . . . . . . 287
5.4 ZLK oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
5.4.1 Beyond the quadrupole approximation . . . . . . . . . . . 297
5.4.2 High-eccentricity migration . . . . . . . . . . . . . . . . . 301
6 Resonances 303
6.1 The pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
6.1.1 The torqued pendulum . . . . . . . . . . . . . . . . . . . 311
6.1.2 Resonances in Hamiltonian systems . . . . . . . . . . . . 312
6.2 Resonance for circular orbits . . . . . . . . . . . . . . . . . . . . 316
6.2.1 The resonance-overlap criterion for nearly circular orbits . 323
6.3 Resonance capture . . . . . . . . . . . . . . . . . . . . . . . . . . 325
6.3.1 Resonance capture in the pendulum Hamiltonian . . . . . 331
6.3.2 Resonance capture for nearly circular orbits . . . . . . . . 332
6.4 The Neptune–Pluto resonance . . . . . . . . . . . . . . . . . . . . 335
6.5 Transit timing variations . . . . . . . . . . . . . . . . . . . . . . . 342
6.6 Secular resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . 348
6.6.1 Resonance sweeping . . . . . . . . . . . . . . . . . . . . . 349
7 Planetary spins 355
7.1 Precession of planetary spins . . . . . . . . . . . . . . . . . . . . 355
7.1.1 Precession and satellites . . . . . . . . . . . . . . . . . . . 360
7.1.2 The chaotic obliquity of Mars . . . . . . . . . . . . . . . . 365
7.2 Spin-orbit resonance . . . . . . . . . . . . . . . . . . . . . . . . . 368
frontmatter August 25, 2022 6.125x9.25
viii Contents
7.2.1 The chaotic rotation of Hyperion . . . . . . . . . . . . . . 372
7.3 Andoyer variables. . . . . . . . . . . . . . . . . . . . . . . . . . . 374
7.4 Colombo’s top and Cassini states . . . . . . . . . . . . . . . . . . 379
7.5 Radiative forces on small bodies . . . . . . . . . . . . . . . . . . 388
7.5.1 Yarkovsky effect . . . . . . . . . . . . . . . . . . . . . . . 388
7.5.2 YORP effect . . . . . . . . . . . . . . . . . . . . . . . . . 391
8 Tides 397
8.1 The minimum-energy state . . . . . . . . . . . . . . . . . . . . . 398
8.2 The equilibrium tide . . . . . . . . . . . . . . . . . . . . . . . . . 402
8.2.1 Love numbers . . . . . . . . . . . . . . . . . . . . . . . . . 404
8.3 Tidal friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406
8.4 Spin and orbit evolution . . . . . . . . . . . . . . . . . . . . . . . 411
8.4.1 Semimajor axis migration . . . . . . . . . . . . . . . . . . 412
8.4.2 Spinup and spindown . . . . . . . . . . . . . . . . . . . . 416
8.4.3 Eccentricity damping. . . . . . . . . . . . . . . . . . . . . 418
8.5 Non-equilibrium tides . . . . . . . . . . . . . . . . . . . . . . . . 422
8.5.1 Planets on high-eccentricity orbits . . . . . . . . . . . . . 423
8.5.2 Resonance locking . . . . . . . . . . . . . . . . . . . . . . 425
8.6 Tidal disruption . . . . . . . . . . . . . . . . . . . . . . . . . . . 425
8.6.1 The Roche limit . . . . . . . . . . . . . . . . . . . . . . . 426
8.6.2 Tidal disruption of regolith . . . . . . . . . . . . . . . . . 428
8.6.3 Tidal disruption of rigid bodies . . . . . . . . . . . . . . . 429
9 Planet-crossing orbits 433
9.1 Local structure of a planetesimal disk . . . . . . . . . . . . . . . 434
9.2 Disk-planet interactions . . . . . . . . . . . . . . . . . . . . . . . 440
9.2.1 Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . 441
9.2.2 Gravitational stirring . . . . . . . . . . . . . . . . . . . . 444
9.3 Evolution of high-eccentricity orbits . . . . . . . . . . . . . . . . 451
9.4 The Galactic tidal field. . . . . . . . . . . . . . . . . . . . . . . . 460
9.5 The Oort cloud . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467
9.6 The trans-Neptunian belt . . . . . . . . . . . . . . . . . . . . . . 475
9.7 Earth-crossing asteroids . . . . . . . . . . . . . . . . . . . . . . . 480
A Physical, astronomical and solar-system constants 483
B Mathematical background 491
B.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491
B.2 Coordinate systems . . . . . . . . . . . . . . . . . . . . . . . . . . 493
B.3 Vector calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495
B.4 Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499
B.5 Spherical trigonometry . . . . . . . . . . . . . . . . . . . . . . . . 499
B.6 Euler angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501
B.7 Calculus of variations . . . . . . . . . . . . . . . . . . . . . . . . 504
frontmatter August 25, 2022 6.125x9.25
Contents ix
C Special functions 505
C.1 Kronecker delta and permutation symbol . . . . . . . . . . . . . 505
C.2 Delta function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506
C.3 Gamma function . . . . . . . . . . . . . . . . . . . . . . . . . . . 507
C.4 Elliptic integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 508
C.5 Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509
C.6 Legendre functions . . . . . . . . . . . . . . . . . . . . . . . . . . 511
C.7 Spherical harmonics . . . . . . . . . . . . . . . . . . . . . . . . . 512
C.8 Vector spherical harmonics . . . . . . . . . . . . . . . . . . . . . 514
D Lagrangian and Hamiltonian dynamics 517
D.1 Hamilton’s equations . . . . . . . . . . . . . . . . . . . . . . . . . 519
D.2 Rotating reference frame . . . . . . . . . . . . . . . . . . . . . . . 520
D.3 Poisson brackets . . . . . . . . . . . . . . . . . . . . . . . . . . . 522
D.4 The propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523
D.5 Symplectic maps . . . . . . . . . . . . . . . . . . . . . . . . . . . 525
D.6 Canonical transformations and coordinates . . . . . . . . . . . . 526
D.7 Angle-action variables . . . . . . . . . . . . . . . . . . . . . . . . 530
D.8 Integrable and non-integrable systems . . . . . . . . . . . . . . . 531
D.9 The averaging principle . . . . . . . . . . . . . . . . . . . . . . . 535
D.10Adiabatic invariants . . . . . . . . . . . . . . . . . . . . . . . . . 536
D.11Rigid bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537
E Hill and Delaunay variables 541
E.1 Hill variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541
E.2 Delaunay variables . . . . . . . . . . . . . . . . . . . . . . . . . . 542
F The standard map 545
F.1 Resonance overlap . . . . . . . . . . . . . . . . . . . . . . . . . . 546
G Hill stability 549
H The Yarkovsky effect 555
I Tidal response of rigid bodies 561
I.1 Tidal disruption of a rigid body . . . . . . . . . . . . . . . . . . . 566
J Relativistic effects 569
J.1 The Einstein–Infeld–Hoffmann equations . . . . . . . . . . . . . . 573
Problems 575
References 599
Index 613