Solar System Dynamics

606 Pages·2000·10.899 MB·English

by  Carl D. Murray, Stanley F. Dermott

SolarSystemDynamics
Thesolarsystemisacomplexandfascinatingdynamicalsystem. Thisisthefirsttextbook
todescribecomprehensivelythedynamicalfeaturesoftheSolarSystemandtoprovide
students with all the necessary mathematical tools and physical models they need to
understandhowitworks.
Clearlywrittenandwellillustrated,SolarSystemDynamicsprovidesstudentswitha
completeintroductiontounderstandingtheintricateandoftenbeautifulresonantstruc-
tureofthesolarsystem. Stepbystep,itshowshowabasicknowledgeofthetwo-and
three-bodyproblemsandperturbationtheorycanbecombinedtounderstandfeaturesas
diverseasthetidalheatingofJupiter’smoonIo,theunusualrotationofSaturn’smoon
Hyperion, the origin of the Kirkwood gaps in the asteroid belt, the radial structure of
Saturn’sAring,andthelong-termstabilityofthesolarsystem. Problemsattheendof
eachchapterandafreeInternetMathematica® softwarepackage(thatincludesanima-
tions and computational tools) are provided to help students to test and develop their
understanding.
SolarSystemDynamicsprovidesstudentswithaclear,comprehensive,andauthori-
tativetextbookforcoursesonsolarsystemdynamics,planetarydynamics,andcelestial
mechanics. It also provides the necessary mathematical tools for them to tackle more
generalcoursesondynamics,dynamicalsystems,applicationsofchaostheory,andnon-
lineardynamics. Thisisabenchmarkpublicationinthefieldofplanetarydynamicsand
destinedtobecomeaclassic.
CARLMURRAYisProfessorofMathematicsandAstronomyatQueenMaryandWest-
fieldCollege,London. Inadditiontoscientificpapers,heenjoyswritingpopulararticles
onthesolarsystem. HeisanassociateeditorofthejournalsIcarusandCelestialMe-
chanics and Dynamical Astronomy as well as being a member of the Imaging Team
of the Cassini mission to Saturn. Asteroid 5598 was officially named Carlmurray in
recognitionofhiscontributiontoplanetaryscience.
STANDERMOTTisProfessorofAstronomyandchairmanoftheDepartmentofAstron-
omyattheUniversityofFlorida,Gainesville. HeiscurrentlyalsoaUniversityofFlorida
ResearchFoundationProfessorandacoinvestigatorontheCosmicDustExperimenton
theGalileospacecraft. Asteroid3647wasofficiallynamedDermottinhishonour.
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SOLAR SYSTEM DYNAMICS
CARL D. MURRAY
QueenMaryandWestfieldCollege,
UniversityofLondon
STANLEY F. DERMOTT
UniversityofFlorida,Gainesville
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CAMBRIDGE UNIVERSITY PRESS
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Cambridge University Press
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© Carl D. Murray and Stanley F. Dermott 1999
This publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without the written
permission of Cambridge University Press.
First published 1999
Reprinted 2001, 2004, 2005, 2006, 2008
A catalog record for this publication is available from the British Library
Library of Congress Cataloging in Publication data
Murray, Carl D.
Solar system dynamics / Carl D. Murray, Stanley F. Dermott.
p. cm.
ISBN 0-521-57295-9 (hc.). – ISBN 0-521-57597-4 (pbk.)
1. Solar system. 2. Celestial mechanics. I. Dermot, S. F.
II. Title.
QB500.5.M87 1999
523.2 – dc21 99-19679
CIP
ISBN 978-0-521-57295-8 Hardback
ISBN 978-0-521-57597-3 Paperback
Transferred to digital printing 2009
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Achtchenaisa´laindcechnderg,
isgelcachnu´a,
isca´ıncechard,isserbcechgna´th.
Ca´ıdcechn-e´cmais,isfaillcechn-aichnid
cofestarcechn-e´olas.
Allthatisredisbeautiful,
andallthatisnewisbright,
allthatishighislovely,allthatisfamiliarisbitter.
Theunknownishonoured,theknownisneglected,
untilallknowledgeisknown.
Anonymous,Irish,ninthcentury,TheSick-BedofCu´ Chulainn
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InMemoryof
FrankMurray
Hewasaman,takehimforallinall,
Ishallnotlookuponhislikeagain.
WilliamShakespeare,Hamlet,I,ii
and
GeraldineMurphy
Attheendwepreferredtotravelallnight,
Sleepinginsnatches
Withthevoicessinginginourears,saying
Thatthiswasallfolly.
T.S.Eliot,JourneyoftheMagi
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Contents
Preface xiii
1 StructureoftheSolarSystem 1
1.1 Introduction 1
1.2 TheBeliefinNumber 2
1.3 Kepler’sLawsofPlanetaryMotion 3
1.4 Newton’sUniversalLawofGravitation 4
1.5 TheTitius–Bode“Law” 5
1.6 ResonanceintheSolarSystem 9
1.7 ThePreferenceforCommensurability 15
1.8 RecentDevelopments 17
ExerciseQuestions 19
2 TheTwo-BodyProblem 22
2.1 Introduction 22
2.2 EquationsofMotion 23
2.3 OrbitalPositionandVelocity 25
2.4 TheMeanandEccentricAnomalies 32
2.5 EllipticExpansions 37
2.6 TheGuidingCentreApproximation 42
2.7 BarycentricOrbits 45
2.8 TheOrbitinSpace 48
2.9 PerturbedOrbits 54
2.10 HamiltonianFormulation 57
ExerciseQuestions 60
3 TheRestrictedThree-BodyProblem 63
3.1 Introduction 63
3.2 EquationsofMotion 64
3.3 TheJacobiIntegral 68
vii
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viii Contents
3.4 TheTisserandRelation 71
3.5 LagrangianEquilibriumPoints 74
3.6 LocationofEquilibriumPoints 77
3.7 StabilityofEquilibriumPoints 83
3.8 Motionnear L and L 95
4 5
3.9 TadpoleandHorseshoeOrbits 97
3.10 OrbitsandZero-VelocityCurves 102
3.11 TrojanAsteroidsandSatellites 107
3.12 JanusandEpimetheus 110
3.13 Hill’sEquations 115
3.14 TheEffectsofDrag 121
ExerciseQuestions 128
4 Tides,Rotation,andShape 130
4.1 Introduction 130
4.2 TheTidalBulge 131
4.3 PotentialTheory 136
4.4 TidalDeformation 140
4.5 RotationalDeformation 149
4.6 TheDarwin–RadauRelation 153
4.7 ShapesandInternalStructuresofSatellites 155
4.8 TheRocheZone 158
4.9 TidalTorques 160
4.10 SatelliteTides 166
4.11 TidalHeatingofIo 174
4.12 TidesonTitan 175
4.13 TidalEvolution 178
4.14 TheDoubleSynchronousState 183
ExerciseQuestions 186
5 Spin–OrbitCoupling 189
5.1 Introduction 189
5.2 TidalDespinning 189
5.3 ThePermanentQuadrupoleMoment 194
5.4 Spin–OrbitResonance 200
5.5 CaptureintoResonance 210
5.6 ForcedLibrations 215
5.7 SurfaceofSection 217
ExerciseQuestions 222
6 TheDisturbingFunction 225
6.1 Introduction 225
6.2 TheDisturbingFunction 226
6.3 ExpansionUsingLegendrePolynomials 228
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Contents ix
6.4 LiteralExpansioninOrbitalElements 233
6.5 LiteralExpansiontoSecondOrder 238
6.6 TermsAssociatedwithaSpecificArgument 246
6.7 UseoftheDisturbingFunction 248
6.8 Lagrange’sPlanetaryEquations 251
6.9 ClassificationofArgumentsintheDisturbingFunction 253
6.10 SampleCalculationsoftheAveragedDisturbingFunction 261
6.11 TheEffectofPlanetaryOblateness 264
ExerciseQuestions 270
7 SecularPerturbations 274
7.1 Introduction 274
7.2 SecularPerturbationsforTwoPlanets 274
7.3 JupiterandSaturn 279
7.4 FreeandForcedElements 283
7.5 Jupiter,Saturn,andaTestParticle 289
7.6 Gauss’sAveragingMethod 293
7.7 GeneralisedSecularPerturbations 299
7.8 SecularTheoryfortheSolarSystem 302
7.9 GeneralisedFreeandForcedElements 307
7.10 HirayamaFamiliesandtheIRASDustBands 309
7.11 SecularResonance 314
7.12 HigherOrderSecularTheory 317
ExerciseQuestions 318
8 ResonantPerturbations 321
8.1 Introduction 321
8.2 TheGeometryofResonance 321
8.3 ThePhysicsofResonance 326
8.4 VariationofOrbitalElements 328
8.5 ResonanceintheCircularRestrictedThree-BodyProblem 332
8.6 ThePendulumModel 334
8.7 LibrationWidth 337
8.8 TheHamiltonianApproach 341
8.9 The2:1Resonance 364
8.10 The3:1and7:4Resonances 371
8.11 AdditionalResonancesandResonanceSplitting 373
8.12 ResonantEncounters 375
8.13 TheDynamicsofCaptureandEvolutioninResonance 385
8.14 Two-BodyResonancesintheSolarSystem 387
8.15 ResonantEncountersinSatelliteSystems 390
8.16 Three-bodyResonance 394
8.17 TheLaplaceResonance 396
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x Contents
8.18 SecularandResonantMotion 399
8.19 LONGSTOPUranus 402
8.20 PulsarPlanets 405
ExerciseQuestions 406
9 ChaosandLong-TermEvolution 409
9.1 Introduction 409
9.2 SensitiveDependenceonInitialConditions 410
9.3 RegularandChaoticOrbits 413
9.4 ChaosintheCircularRestrictedProblem 421
9.5 AlgebraicMappings 428
9.6 SeparatricesandResonanceOverlap 448
9.7 TheRotationofHyperion 452
9.8 TheKirkwoodGaps 456
9.9 TheNeptune-PlutoSystem 466
9.10 TheStabilityoftheSolarSystem 469
ExerciseQuestions 471
10 PlanetaryRings 474
10.1 Introduction 474
10.2 PlanetaryRingSystems 475
10.3 ResonancesinRings 481
10.4 DensityWavesandBendingWaves 492
10.5 NarrowRingsandSharpEdges 495
10.6 TheEnckeGapandPan 512
10.7 TheFRingofSaturn 515
10.8 TheAdamsRingofNeptune 518
10.9 TheEvolutionofRings 520
10.10 TheEarth’sDustRing 522
ExerciseQuestions 524
AppendixA:SolarSystemData 526
A.1 Introduction 526
A.2 AstronomicalConstants 526
A.3 JulianDate 527
A.4 OrbitalElementsofthePlanetsandTheirVariation 529
A.5 Planets,Satellites,andRings 530
A.6 Asteroids,Centaurs,Trans-NeptunianObjects,andComets 535
AppendixB:ExpansionoftheDisturbingFunction 539
References 557
Index 577
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