The Three-Body Problem

353 Pages·1999·5.01 MB·English

THE THREE-BODY PROBLEM
MAURI VALTONEN AND HANNU KARTTUNEN
Va¨isa¨la¨InstituteforSpacePhysicsandAstronomy,
UniversityofTurku
  
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
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Contents
Preface pageix
1 Astrophysicsandthethree-bodyproblem 1
1.1 Aboutthethree-bodyproblem 1
1.2 Thethree-bodyprobleminastrophysics 5
1.3 Shortperiodcomets 8
1.4 Binarystars 12
1.5 Groupsofgalaxies 15
1.6 Binaryblackholes 17
2 Newtonianmechanics 20
2.1 Newton’slaws 20
2.2 Inertialcoordinatesystem 21
2.3 Equationsofmotionfor N bodies 22
2.4 Gravitationalpotential 24
2.5 Constantsofmotion 25
2.6 Thevirialtheorem 27
2.7 TheLagrangeandJacobiformsoftheequationsofmotion 29
2.8 Constantsofmotioninthethree-bodyproblem 31
2.9 Momentofinertia 32
2.10 Scalingofthethree-bodyproblem 34
2.11 Integrationoforbits 34
2.12 Dimensionsandunitsofthethree-bodyproblem 38
2.13 Chaosinthethree-bodyproblem 39
2.14 Rotatingcoordinatesystem 43
Problems 45
3 Thetwo-bodyproblem 47
3.1 Equationsofmotion 47
3.2 Centreofmasscoordinatesystem 48
3.3 Integralsoftheequationofmotion 49
v
vi Contents
3.4 EquationoftheorbitandKepler’sfirstlaw 52
3.5 Kepler’ssecondlaw 53
3.6 Orbitalelements 54
3.7 Orbitalvelocity 57
3.8 Trueandeccentricanomalies 58
3.9 MeananomalyandKepler’sequation 60
3.10 SolutionofKepler’sequation 61
3.11 Kepler’sthirdlaw 63
3.12 Positionandspeedasfunctionsofeccentricanomaly 64
3.13 Hyperbolicorbit 66
3.14 Dynamicalfriction 68
3.15 Seriesexpansions 70
Problems 78
4 Hamiltonianmechanics 80
4.1 Generalisedcoordinates 80
4.2 Hamiltonianprinciple 81
4.3 Variationalcalculus 82
4.4 Lagrangianequationsofmotion 85
4.5 Hamiltonianequationsofmotion 87
4.6 PropertiesoftheHamiltonian 89
4.7 Canonicaltransformations 92
4.8 Examplesofcanonicaltransformations 95
4.9 TheHamilton–Jacobiequation 95
4.10 Two-bodyprobleminHamiltonianmechanics:twodimensions 97
4.11 Two-bodyprobleminHamiltonianmechanics:threedimensions 103
4.12 Delaunay’selements 108
4.13 Hamiltonianformulationofthethree-bodyproblem 109
4.14 Eliminationofnodes 111
4.15 Eliminationofmeananomalies 113
Problems 113
5 Theplanarrestrictedcircularthree-bodyproblemandotherspecialcases115
5.1 Coordinateframes 115
5.2 Equationsofmotion 116
5.3 Jacobianintegral 119
5.4 Lagrangianpoints 123
5.5 StabilityoftheLagrangianpoints 125
5.6 Satelliteorbits 130
5.7 TheLagrangianequilateraltriangle 133
5.8 One-dimensionalthree-bodyproblem 136
Problems 139
Contents vii
6 Three-bodyscattering 141
6.1 Scatteringofsmallfastbodiesfromabinary 141
6.2 Evolutionofthesemi-majoraxisandeccentricity 148
6.3 Captureofsmallbodiesbyacircularbinary 152
6.4 Orbitalchangesinencounterswithplanets 154
6.5 Inclinationandperiheliondistance 157
6.6 Largeanglescattering 162
6.7 Changesintheorbitalelements 165
6.8 Changesintherelativeorbitalenergyofthebinary 169
Problems 170
7 Escapeinthegeneralthree-bodyproblem 171
7.1 Escapesinaboundthree-bodysystem 171
7.2 Aplanarcase 179
7.3 Escapevelocity 180
7.4 Escapermass 183
7.5 Angularmomentum 184
7.6 Escapeangle 188
Problems 195
8 Scatteringandcaptureinthegeneralproblem 197
8.1 Three-bodyscattering 197
8.2 Capture 203
8.3 Ejectionsandlifetime 207
8.4 Exchangeandflyby 211
8.5 Ratesofchangeofthebindingenergy 214
8.6 Collisions 216
Problems 219
9 Perturbationsinhierarchicalsystems 221
9.1 Osculatingelements 221
9.2 Lagrangianplanetaryequations 222
9.3 Three-bodyperturbingfunction 225
9.4 Doublyorbit-averagedperturbingfunction 227
9.5 Motionsinthehierarchicalthree-bodyproblem 231
Problems 239
10 Perturbationsinstrongthree-bodyencounters 240
10.1 Perturbationsoftheintegralskande 240
10.2 Binaryevolutionwithaconstantperturbingforce 243
10.3 Slowencounters 246
10.4 Inclinationdependence 260
10.5 Changeineccentricity 264
10.6 Stabilityoftriplesystems 268
viii Contents
10.7 Fastencounters 274
10.8 Averageenergyexchange 281
Problems 285
11 Someastrophysicalproblems 288
11.1 Binaryblackholesincentresofgalaxies 288
11.2 Theproblemofthreeblackholes 296
11.3 Satelliteblackholesystems 310
11.4 Threegalaxies 310
11.5 BinarystarsintheGalaxy 313
11.6 Evolutionofcometorbits 320
Problems 327
References 329
Authorindex 341
Subjectindex 343
Preface
ClassicalorbitcalculationinNewtonianmechanicshasexperiencedarenaissance
in recent decades. With the beginning of space flights there was suddenly a great
practicalneedtocalculateorbitswithhighaccuracy.Atthesametime,advancesin
computertechnologyhaveimprovedthespeedoforbitcalculationsenormously.
Theseadvanceshavealsomadeitpossibletostudythegravitationalthree-body
problemwithnewrigour.Thesolutionsofthisproblemgobeyondthepracticalities
ofspaceflightintotheareaofmodernastrophysics.Theyincludeproblemsinthe
SolarSystem,inthestellarsystemsofourGalaxyaswellasinothergalaxies.The
presentbookhasbeenwrittenwiththeastrophysicalapplicationsinmind.
The book is based on two courses which have been taught by us: Celestial
MechanicsandAstrodynamics.TheformercourseincludesapproximatelyChapters
2–5 of the book, with some material from later chapters. It is a rather standard
introductiontothesubjectwhichformsthenecessarybackgroundtomoderntopics.
The celestial mechanics course has been developed in the University of Helsinki
by one of us (H. K.) over about two decades. The remainder of the book is based
ontheastrodynamicscoursewhicharosesubsequentlyintheUniversityofTurku.
Muchofthematerialinthecourseisnewinthesensethatithasnotbeenpresented
atatextbooklevelpreviously.
In our experience there has been a continuous need for specialists in classical
orbitdynamicswhileatthesametimethisareaofstudyhasreceivedlessattention
thanitusedtointhestandardastronomycurriculum.Bywritingthisbookwehope
tohelpthesituationandtoattractnewstudentstotheresearcharea,whichisstill
modernaftermorethan300yearsofstudies.
We have been privileged to receive a great deal of help and encouragement
frommanycolleagues.EspeciallywewouldliketothankDouglasHeggie,Kimmo
InnanenandBillSaslawwhohavebetweenthemreadnearlythewholemanuscript
andsuggestednumerousimprovements.WealsoappreciatethecommentsbyVictor
Orlov,HarryLehtoandTian-YiHuangwhichhavebeenmostuseful.SeppoMikkola
ix
x Preface
has generously provided research tools for the calculations in this book. Other
membersoftheTurkuresearchgrouphavealsohelpeduswithillustrations.Most
ofall,wewouldliketothankSirpaReinikainenfortypingmuchofthefinaltext,
includingthegreatmanymathematicalformulae.
Financial support for this project has been provided by Finland’s Society for
SciencesandLettersandtheAcademyofFinland(project‘CalculationofOrbits’),
whichgavetheopportunityforoneofus(M.V.)toconcentrateonwritingthebook
for a period of two years. The generous support by the Department of Computer
Science, Mathematics and Physics (in Barbados) and the Department of Physics
(in Trinidad) of the University of the West Indies made it possible to carry out
the writing in optimal surroundings. Parts of the text were originally published in
Finnish:H.Karttunen,JohdatusTaivaanmekaniikkaan,Helsinki:Ursa,2001.
Finally,M.V.wouldliketoexpresshisappreciationtoSverreAarsethwhotaught
himhowtocalculateorbits(andmuchelse),andtohiswifeKathleen,theCaribbean
link,whoseencouragementwasvitalfortheaccomplishmentofthebook.
1
Astrophysics and the three-body problem
1.1 Aboutthethree-bodyproblem
Thethree-bodyproblemarisesinmanydifferentcontextsinnature.Thisbookdeals
with the classical three-body problem, the problem of motion of three celestial
bodiesundertheirmutualgravitationalattraction.Itisanoldproblemandlogically
followsfromthetwo-bodyproblemwhichwassolvedbyNewtoninhisPrincipia
in 1687. Newton also considered the three-body problem in connection with the
motionoftheMoonundertheinfluencesoftheSunandtheEarth,theconsequences
ofwhichincludedaheadache.
Therearegoodreasonstostudythethree-bodygravitationalproblem.Themotion
of the Earth and other planets around the Sun is not strictly a two-body problem.
The gravitational pull by another planet constitutes an extra force which tries to
steertheplanetoffitsellipticalpath.Onemayevenworry,asscientistsdidinthe
eighteenthcentury,whethertheextraforcemightchangetheorbitalcourseofthe
EarthentirelyandmakeitfallintotheSunorescapetocoldouterspace.Thiswasa
legitimateworryatthetimewhentheEarthwasthoughttobeonlyafewthousand
yearsold,andallpossiblecombinationsofplanetaryinfluencesontheorbitofthe
Earthhadnotyethadtimetooccur.
Another serious question was the influence of the Moon on the motion of the
Earth.Wouldithavelongtermmajoreffects?IstheMooninastableorbitaboutthe
Earthormightitonedaycrashonus?ThemotionoftheMoonwasalsoaquestion
ofmajorpracticalsignificance,sincetheMoonwasusedasauniversaltimekeeping
deviceintheabsenceofclockswhichwereaccurateoverlongperiodsoftime.After
Newton,thelunartheorywasstudiedintheeighteenthcenturyusingtherestricted
problemofthreebodies(Euler1772).Intherestrictedproblem,oneofthebodies
is regarded as massless in comparison with the other two which are in a circular
orbitrelativetoeachother.Ataboutthesametime,thefirstspecialsolutionofthe
general three-body problem was discovered, the Lagrangian equilateral triangle
1
2 Astrophysicsandthethree-bodyproblem
solution (Lagrange 1778). The theory of the restricted three-body problem was
furtherdevelopedbyJacobi(1836),anditwasusedforthepurposeofidentifying
comets by Tisserand (1889, 1896) and reached its peak in the later nineteenth
centurywiththeworkofHill(1878)andDelaunay(1860).The‘classical’period
reacheditsfinalphasewithPoincare´ (1892–1899).
Inspiteofthesesuccessesinspecialcases,thesolutionofthegeneralthree-body
problem remained elusive even after two hundred years following the publication
ofPrincipia.Inthegeneralthree-bodyproblemallthreemassesarenon-zeroand
their initial positions and velocities are not arranged in any particular way. The
difficultyofthegeneralthree-bodyproblemderivesfromthefactthatthereareno
coordinate transformations which would simplify the problem greatly. This is in
contrasttothetwo-bodyproblemwherethesolutionsarefoundmosteasilyinthe
centreofmasscoordinatesystem.Themutualforcebetweenthetwobodiespoints
towards the centre of mass, a stationary point in this coordinate system. Thus the
solution is derived from the motion in the inverse square force field. Similarly, in
the restricted three-body problem one may transfer to a coordinate system which
rotatesatthesameratearoundthecentreofmassasthetwoprimarybodies.Then
the problem is reduced to the study of motions in two stationary inverse square
forcefields.Inthegeneralproblem,thelinesofmutualforcesdonotpassthrough
the centre of mass of the system. The motion of each body has to be considered
inconjunctionwiththemotionsoftheothertwobodies,whichmadetheproblem
ratherintractableanalyticallybeforetheageofpowerfulcomputers.
Atthesuggestionofleadingscientists,theKingofSwedenOscarIIestablished
a prize for the solution of the general three-body problem. The solution was to
be in the form of a series expansion which describes the positions of the three
bodiesatallfuturemomentsoftimefollowinganarbitrarystartingconfiguration.
Nobody was able to claim the prize for many years and finally it was awarded in
1889 to Poincare´ who was thought to have made the most progress in the subject
eventhoughhehadnotsolvedthespecificproblem.Ittookmorethantwentyyears
before Sundman completed the given task (Sundman 1912). Unfortunately, the
extremelypoorconvergenceoftheseriesexpansiondiscoveredbySundmanmakes
this method useless for the purpose of calculating the orbits of the three bodies.
Nowthattheorbitscanbecalculatedquicklybycomputer,itisquiteobviouswhy
thislineofresearchcouldnotleadtoarealsolutionofthethree-bodyproblem:the
orbits are good examples of chaos in nature, and deterministic series expansions
are utterly unsuitable for their description. Poincare´ was on the right track in this
regardandwiththecurrentknowledgewasthusamostreasonablerecipientofthe
prize.
Ataboutthesametime,anewapproachbeganwhichhasbeensosuccessfulin
recentyears:theintegrationoforbitsstepbystep.Inorbitintegration,eachbody,