RudolfDvorakand ChristophLhotka CelestialDynamics RelatedTitles Kopeikin,S.,Efroimsky,M.,Kaplan,G. RelativisticCelestialMechanics of theSolar System 892pageswith65figuresand6tables 2011 ISBN:978-3-527-40856-6 Barnes,R.(ed.) Formationand Evolutionof Exoplanets 320pageswith136figuresand5tables 2010 ISBN:978-3-527-40896-2 Szebehely,V.G.,Mark,H. AdventuresinCelestialMechanics 320pageswith86figures 1998 ISBN:978-0-471-13317-9 Rudolf Dvorak and Christoph Lhotka Celestial Dynamics Chaoticity and Dynamics of Celestial Systems WILEY-VCH Verlag GmbH & Co. KGaA TheAuthors AllbookspublishedbyWiley-VCHarecarefully produced.Nevertheless,authors,editors,and Prof.Dr.RudolfDvorak publisherdonotwarranttheinformation UniversityofVienna containedinthesebooks,includingthisbook,to DepartmentofAstronomy befreeoferrors.Readersareadvisedtokeepin Tuerkenschanzstrasse17 mindthatstatements,data,illustrations, 1180Vienna proceduraldetailsorotheritemsmay Austria inadvertentlybeinaccurate. Dr.ChristophLhotka LibraryofCongressCardNo.: UniversityofNamur appliedfor DepartmentofMathematics RempartdelaVierge8 BritishLibraryCataloguing-in-PublicationData: 5000Namur Acataloguerecordforthisbookisavailable Belgium fromtheBritishLibrary. Bibliographicinformationpublishedbythe DeutscheNationalbibliothek TheDeutscheNationalbibliothekliststhis publicationintheDeutscheNationalbibliografie; detailedbibliographicdataareavailableonthe Internetathttp://dnb.d-nb.de. ©2013WILEY-VCHVerlagGmbH&Co.KGaA, Boschstr.12,69469Weinheim,Germany Allrightsreserved(includingthoseoftranslation intootherlanguages).Nopartofthisbookmay bereproducedinanyform–byphotoprinting, microfilm,oranyothermeans–nortransmitted ortranslatedintoamachinelanguagewithout writtenpermissionfromthepublishers. Registerednames,trademarks,etc.usedinthis book,evenwhennotspecificallymarkedas such,arenottobeconsideredunprotectedby law. PrintISBN 978-3-527-40977-8 ePDFISBN 978-3-527-65188-7 ePubISBN 978-3-527-65187-0 mobiISBN 978-3-527-65186-3 oBookISBN 978-3-527-65185-6 CoverDesign Grafik-DesignSchulz, Fußgönheim Typesetting le-texpublishingservicesGmbH, Leipzig PrintingandBinding MarkonoPrintMedia PteLtd,Singapore PrintedinSingapore Printedonacid-freepaper V DedicatedtoCaroline,Sophia,Stephanie,andBarbara VII Contents Preface XI 1 Introduction:theChallengeofScience 1 2 HamiltonianMechanics 7 2.1 Hamilton’sEquationsfromHamiltonianPrinciple 10 2.2 PoissonBrackets 11 2.3 CanonicalTransformations 13 2.4 Hamilton–JacobiTheory 19 2.5 Action-AngleVariables 23 3 NumericalandAnalyticalTools 27 3.1 Mappings 27 3.1.1 SimpleExamples 28 3.1.1.1 TwistMap 28 3.1.1.2 LogisticMap 29 3.1.1.3 ArnoldCatMap 31 3.1.1.4 StandardMap 33 3.1.1.5 TheCircleMap 35 3.1.1.6 TheDissipativeStandardMap 37 3.1.2 HadjidemetriouMapping 38 3.2 Lie-SeriesNumericalIntegration 41 3.2.1 ASimpleExample 43 3.3 ChaosIndicators 48 3.3.1 LyapunovCharacteristicExponent 48 3.3.2 FastLyapunovIndicator 50 3.3.3 MeanExponentialGrowthFactorofNearbyOrbits 50 3.3.4 SmallerAlignmentIndex 51 3.3.5 SpectralAnalysisMethod 51 3.4 PerturbationTheory 52 3.4.1 Lie-TransformationMethod 55 3.4.2 Mappingmethod 64 4 TheStabilityProblem 69 4.1 ReviewonDifferentConceptsofStability 69 VIII Contents 4.2 IntegrableSystems 72 4.3 NearlyIntegrableSystems 78 4.4 ResonanceDynamics 80 4.5 KAMTheorem 86 4.6 NekhoroshevTheorem 91 4.7 TheFroeschlé–Guzzo–LegaHamiltonian 99 5 TheTwo-BodyProblem 105 5.1 FromNewtontoKepler 106 5.2 UnperturbedKeplerMotion 108 5.3 ClassificationofOrbits:Ellipses,HyperbolaeandParabolae 110 5.4 KeplerEquation 112 5.5 ComplexDescription 115 5.5.1 TheKS-Transformation 117 5.6 MotioninSpaceandtheKeplerianElements 118 5.7 AstronomicalDeterminationoftheGravitationalConstant 120 5.8 SolutionoftheKeplerEquation 120 6 TheRestrictedThree-BodyProblem 123 6.1 Set-UpandFormulation 124 6.2 EquilibriaoftheSystem 127 6.3 MotionClosetoL andL 131 4 5 6.4 MotionClosetoL ,L ,L 134 1 2 3 6.5 PotentialandtheZeroVelocityCurves 136 6.6 SpatialRestrictedThree-BodyProblem 141 6.7 TisserandCriterion 144 6.8 EllipticRestrictedThree-BodyProblem 145 6.9 DissipativeRestrictedThree-BodyProblem 146 7 TheSitnikovProblem 149 7.1 CircularCase:theMacMillanProblem 150 7.1.1 QualitativeEstimates 150 7.2 MotionofthePlanetoffthez-Axes 153 7.3 EllipticCase 157 7.3.1 NumericalResults 158 7.3.1.1 TheUnstableCenter 164 7.3.2 AnalyticalResults 165 7.3.2.1 AnalyticalSolutionoftheSitnikovProblem 166 7.3.2.2 TheLinearizedSolution 167 7.3.2.3 LinearStability 170 7.3.2.4 NonlinearSolution 172 7.4 TheVrabecMapping 176 7.5 GeneralSitnikovProblem 180 7.5.1 QualitativeEstimates 180 7.5.2 PhaseSpaceStructure 182 Contents IX 8 PlanetaryTheory 185 8.1 PlanetaryPerturbationTheory 185 8.1.1 ASimpleExample 185 8.1.2 PrinciplesofPlanetaryTheory 187 8.1.3 TheIntegrationConstants–theOsculatingElements 190 8.1.4 First-OrderPerturbation 191 8.1.5 Second-OrderPerturbation 192 8.2 EquationsofMotionfornBodies 194 8.2.1 TheVirialTheorem 195 8.2.2 ReductiontoHeliocentricCoordinates 196 8.3 LagrangeEquationsofthePlanetaryn-BodyProblem 198 8.3.1 LegendrePolynomials 198 8.3.2 DelaunayElements 200 8.4 ThePerturbingFunctioninEllipticOrbitalElements 203 8.5 ExplicitFirst-OrderPlanetaryTheoryfortheOsculatingElements 207 8.5.1 PerturbationoftheMeanLongitude 209 8.6 SmallDivisors 211 8.7 Long-TermEvolutionofOurPlanetarySystem 213 9 Resonances 215 9.1 MeanMotionResonancesinOurPlanetarySystem 215 9.1.1 The13W8ResonancebetweenVenusandEarth 215 9.1.2 The1W1MeanMotionResonance:TrojanAsteroids 218 9.2 MethodofLaplace–Lagrange 223 9.3 SecularResonances 231 9.3.1 AsteroidswithSmallInclinationsandEccentricities 231 9.3.2 CometsandAsteroidswithLargeInclinationsandEccentricities:the KozaiResonance 236 9.4 Three-BodyResonances 239 9.4.1 AsteroidsinThree-BodyResonances 239 9.4.2 ThreeMassiveCelestialBodiesinThree-BodyResonances 240 9.4.3 ApplicationtotheGalileanSatellites 245 10 LunarTheory 249 10.1 Hill’sLunarTheory 250 10.1.1 PeriodicMotion 255 10.2 ClassicalLunarTheory 261 10.2.1 SecularPart:MotionoftheNodesandthePerihelion 263 10.3 PrincipalInequalities 264 10.3.1 TheVariation 265 10.3.2 TheEvection 267 10.3.3 AnnualEquation,ParallacticInequalityandPrincipalPerturbationin Latitude 268 X Contents 11 ConcludingRemarks 271 AppendixA ImportantPersonsintheField 277 AppendixB Formulae 281 B.1 HansenCoefficients 281 B.2 LaplaceCoefficients 283 B.3 BesselFunctions 284 B.4 ExpansionsintheTwo-BodyProblem 289 Acknowledgement 293 References 295 Index 305 XI Preface The idea of writing a book about astrodynamics primarily for students and col- leagues wishing to learn about and perhaps to work in this interesting field of astronomycameintomymind(RD)alreadysometwentyyearsago.Inthosedays myconceptwasadifferentonethanthebookinyourhands:basedonaveryspe- cial topic equally interesting for nonlinear dynamics and astrodynamics (see the bookofJ.Moser1)theso-calledSitnikovProblemIwantedtopresentalltheclassical ideasofperturbationtheoryandconnectthemtothemoderntoolsofchaostheory. Yearswentbywithoutsuchabookbeingwritten!Butthen,sometenyearsago,I wasfortunateenoughtohaveabrilliantyoungstudent(CL)workingespeciallyon moderndevelopmentinnonlineardynamicswhoextensivelymadeuseofthetools of computer algebra. After his PhD he then moved as a postdoc to two famous Mathematics Departments (Rome and Namur under the direction of two world knowncolleagues,Profs.AlessandraCelletti,respectivelyAnneLemaitre).Imme- diatelyafterhisPhD,stillinVienna,I(RD)gotanofferbyWileytowriteabookon thesubject Ihavebeen workingonfor40years.Beingawareofthisopportunity I invited my former student to write a book with me which, on one hand, deals withthemoderntoolsofnonlineardynamicsand,ontheotherhand,introducesto theclassicalmethodsusedsincemorethantwocenturieswithgreatsuccess.We realizethatthisisadifficulttaskbecauserecentlytwoexcellentbooksconcerning this subject appeared written by Prof. S. Ferraz-Mello2) and by Prof. A. Celletti3). NeverthelesswehopetocombinewithourbookdifferentaspectsofCelestialMe- chanicsinanunderstandablewayforinterestedstudentsandcolleaguesinPhysics andAstronomy,andtosucceedininfectingthereaderwithourenthusiasmforthe subject. VienneandNamur,December2012 RudolfDvorakandChristophLhotka 1) StableandRandomMotioninDynamicalSystems. 2) CanonicalPerturbationTheories. 3) StabilityandChaosinCelestialMechanics.