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Long-termintegrationsandstabilityofplanetaryorbitsinourSolarsystem

Abstract

Wepresenttheresultsofverylong-termnumericalintegrationsofplanetaryorbitalmotionsover109-yrtime-spansincludingallnineplanets.Aquickinspectionofournumericaldatashowsthattheplanetarymotion，atleastinoursimpledynamicalmodel，seemstobequitestableevenoverthisverylongtime-span.Acloserlookatthelowest-frequencyoscillationsusingalow-passfiltershowsusthepotentiallydiffusivecharacterofterrestrialplanetarymotion，especiallythatofMercury.ThebehaviouroftheeccentricityofMercuryinourintegrationsisqualitativelysimilartotheresultsfromJacquesLaskarssecularperturbationtheory（e.g.emax～0.35over～±4Gyr）.However，therearenoapparentsecularincreasesofeccentricityorinclinationinanyorbitalelementsoftheplanets，whichmayberevealedbystilllonger-termnumericalintegrations.Wehavealsoperformedacoupleoftrialintegrationsincludingmotionsoftheouterfiveplanetsoverthedurationof±5×1010yr.TheresultindicatesthatthethreemajorresonancesintheNeptune–Plutosystemhavebeenmaintainedoverthe1011-yrtime-span.

1Introduction

1.1Definitionoftheproblem

ThequestionofthestabilityofourSolarsystemhasbeendebatedoverseveralhundredyears，sincetheeraofNewton.Theproblemhasattractedmanyfamousmathematiciansovertheyearsandhasplayedacentralroleinthedevelopmentofnon-lineardynamicsandchaostheory.However，wedonotyethaveadefiniteanswertothequestionofwhetherourSolarsystemisstableornot.Thisispartlyaresultofthefactthatthedefinitionoftheterm‘stability’isvaguewhenitisusedinrelationtotheproblemofplanetarymotionintheSolarsysteActuallyitisnoteasytogiveaclear，rigorousandphysicallymeaningfuldefinitionofthestabilityofourSolarsyste

Amongmanydefinitionsofstability，hereweadopttheHilldefinition（Gladman1993）:actuallythisisnotadefinitionofstability，butofinstability.Wedefineasystemasbecomingunstablewhenacloseencounteroccurssomewhereinthesystem，startingfromacertaininitialconfiguration（Chambers，Wetherill&Boss1996;Ito&Tanikawa1999）.AsystemisdefinedasexperiencingacloseencounterwhentwobodiesapproachoneanotherwithinanareaofthelargerHillradius.Otherwisethesystemisdefinedasbeingstable.HenceforwardwestatethatourplanetarysystemisdynamicallystableifnocloseencounterhappensduringtheageofourSolarsystem，about±5Gyr.Incidentally，thisdefinitionmaybereplacedbyoneinwhichanoccurrenceofanyorbitalcrossingbetweeneitherofapairofplanetstakesplace.Thisisbecauseweknowfromexperiencethatanorbitalcrossingisverylikelytoleadtoacloseencounterinplanetaryandprotoplanetarysystems（Yoshinaga，Kokubo&Makino1999）.OfcoursethisstatementcannotbesimplyappliedtosystemswithstableorbitalresonancessuchastheNeptune–Plutosyste

1.2Previousstudiesandaimsofthisresearch

Inadditiontothevaguenessoftheconceptofstability，theplanetsinourSolarsystemshowacharactertypicalofdynamicalchaos（Sussman&Wisdom1988，1992）.Thecauseofthischaoticbehaviourisnowpartlyunderstoodasbeingaresultofresonanceoverlapping（Murray&Holman1999;Lecar，Franklin&Holman2001）.However，itwouldrequireintegratingoveranensembleofplanetarysystemsincludingallnineplanetsforaperiodcoveringseveral10Gyrtothoroughlyunderstandthelong-termevolutionofplanetaryorbits，sincechaoticdynamicalsystemsarecharacterizedbytheirstrongdependenceoninitialconditions.

Fromthatpointofview，manyofthepreviouslong-termnumericalintegrationsincludedonlytheouterfiveplanets（Sussman&Wisdom1988;Kinoshita&Nakai1996）.Thisisbecausetheorbitalperiodsoftheouterplanetsaresomuchlongerthanthoseoftheinnerfourplanetsthatitismucheasiertofollowthesystemforagivenintegrationperiod.Atpresent，thelongestnumericalintegrationspublishedinjournalsarethoseofDuncan&Lissauer（1998）.Althoughtheirmaintargetwastheeffectofpost-main-sequencesolarmasslossonthestabilityofplanetaryorbits，theyperformedmanyintegrationscoveringupto～1011yroftheorbitalmotionsofthefourjovianplanets.TheinitialorbitalelementsandmassesofplanetsarethesameasthoseofourSolarsysteminDuncan&Lissauerspaper，buttheydecreasethemassoftheSungraduallyintheirnumericalexperiments.Thisisbecausetheyconsidertheeffectofpost-main-sequencesolarmasslossinthepaper.Consequently，theyfoundthatthecrossingtime-scaleofplanetaryorbits，whichcanbeatypicalindicatoroftheinstabilitytime-scale，isquitesensitivetotherateofmassdecreaseoftheSun.WhenthemassoftheSunisclosetoitspresentvalue，thejovianplanetsremainstableover1010yr，orperhapslonger.Duncan&Lissaueralsoperformedfoursimilarexperimentsontheorbitalmotionofsevenplanets（VenustoNeptune），whichcoveraspanof～109yr.Theirexperimentsonthesevenplanetsarenotyetcomprehensive，butitseemsthattheterrestrialplanetsalsoremainstableduringtheintegrationperiod，maintainingalmostregularoscillations.

Ontheotherhand，inhisaccuratesemi-analyticalsecularperturbationtheory（Laskar1988），Laskarfindsthatlargeandirregularvariationscanappearintheeccentricitiesandinclinationsoftheterrestrialplanets，especiallyofMercuryandMarsonatime-scaleofseveral109yr（Laskar1996）.TheresultsofLaskarssecularperturbationtheoryshouldbeconfirmedandinvestigatedbyfullynumericalintegrations.

Inthispaperwepresentpreliminaryresultsofsixlong-termnumericalintegrationsonallnineplanetaryorbits，coveringaspanofseveral109yr，andoftwootherintegrationscoveringaspanof±5×1010yr.Thetotalelapsedtimeforallintegrationsismorethan5yr，usingseveraldedicatedPCsandworkstations.Oneofthefundamentalconclusionsofourlong-termintegrationsisthatSolarsystemplanetarymotionseemstobestableintermsoftheHillstabilitymentionedabove，atleastoveratime-spanof±4Gyr.Actually，inournumericalintegrationsthesystemwasfarmorestablethanwhatisdefinedbytheHillstabilitycriterion:notonlydidnocloseencounterhappenduringtheintegrationperiod，butalsoalltheplanetaryorbitalelementshavebeenconfinedinanarrowregionbothintimeandfrequencydomain，thoughplanetarymotionsarestochastic.Sincethepurposeofthispaperistoexhibitandoverviewtheresultsofourlong-termnumericalintegrations，weshowtypicalexamplefiguresasevidenceoftheverylong-termstabilityofSolarsystemplanetarymotion.Forreaderswhohavemorespecificanddeeperinterestsinournumericalresults，wehavepreparedawebpage（access），whereweshowraworbitalelements，theirlow-passfilteredresults，variationofDelaunayelementsandangularmomentumdeficit，andresultsofoursimpletime–frequencyanalysisonallofourintegrations.

InSection2webrieflyexplainourdynamicalmodel，numericalmethodandinitialconditionsusedinourintegrations.Section3isdevotedtoadescriptionofthequickresultsofthenumericalintegrations.Verylong-termstabilityofSolarsystemplanetarymotionisapparentbothinplanetarypositionsandorbitalelements.Aroughestimationofnumericalerrorsisalsogiven.Section4goesontoadiscussionofthelongest-termvariationofplanetaryorbitsusingalow-passfilterandincludesadiscussionofangularmomentumdeficit.InSection5，wepresentasetofnumericalintegrationsfortheouterfiveplanetsthatspans±5×1010yr.InSection6wealsodiscussthelong-termstabilityoftheplanetarymotionanditspossiblecause.

2Descriptionofthenumericalintegrations

（本部分涉及比较复杂的积分计算，作者君就不贴上来了，贴上来了起点也不一定能成功显示。）

2.3Numericalmethod

Weutilizeasecond-orderWisdom–Holmansymplecticmapasourmainintegrationmethod（Wisdom&Holman1991;Kinoshita，Yoshida&Nakai1991）withaspecialstart-upproceduretoreducethetruncationerrorofanglevariables，‘warmstart’（Saha&Tremaine1992，1994）.

Thestepsizeforthenumericalintegrationsis8dthroughoutallintegrationsofthenineplanets（N±1，2，3），whichisabout111oftheorbitalperiodoftheinnermostplanet（Mercury）.Asforthedeterminationofstepsize，wepartlyfollowthepreviousnumericalintegrationofallnineplanetsinSussman&Wisdom（1988，7.2d）andSaha&Tremaine（1994，22532d）.Weroundedthedecimalpartofthetheirstepsizesto8tomakethestepsizeamultipleof2inordertoreducetheaccumulationofround-offerrorinthecomputationprocesses.Inrelationtothis，Wisdom&Holman（1991）performednumericalintegrationsoftheouterfiveplanetaryorbitsusingthesymplecticmapwithastepsizeof400d，110.83oftheorbitalperiodofJupiter.Theirresultseemstobeaccurateenough，whichpartlyjustifiesourmethodofdeterminingthestepsize.However，sincetheeccentricityofJupiter（～0.05）ismuchsmallerthanthatofMercury（～0.2），weneedsomecarewhenwecomparetheseintegrationssimplyintermsofstepsizes.

Intheintegrationoftheouterfiveplanets（F±），wefixedthestepsizeat400d.

WeadoptGaussfandgfunctionsinthesymplecticmaptogetherwiththethird-orderHalleymethod（Danby1992）asasolverforKeplerequations.ThenumberofmaximumiterationswesetinHalleysmethodis15，buttheyneverreachedthemaximuminanyofourintegrations.

Theintervalofthedataoutputis200000d（～547yr）forthecalculationsofallnineplanets（N±1，2，3），andabout8000000d（～21903yr）fortheintegrationoftheouterfiveplanets（F±）.

Althoughnooutputfilteringwasdonewhenthenumericalintegrationswereinprocess，weappliedalow-passfiltertotheraworbitaldataafterwehadcompletedallthecalculations.SeeSection4.1formoredetail.

2.4Errorestimation

2.4.1Relativeerrorsintotalenergyandangularmomentum

Accordingtooneofthebasicpropertiesofsymplecticintegrators，whichconservethephysicallyconservativequantitieswell（totalorbitalenergyandangularmomentum），ourlong-termnumericalintegrationsseemtohavebeenperformedwithverysmallerrors.Theaveragedrelativeerrorsoftotalenergy（～10?9）andoftotalangularmomentum（～10?11）haveremainednearlyconstantthroughouttheintegrationperiod（Fig.1）.Thespecialstartupprocedure，warmstart，wouldhavereducedtheaveragedrelativeerrorintotalenergybyaboutoneorderofmagnitudeormore.

RelativenumericalerrorofthetotalangularmomentumδAA0andthetotalenergyδEE0inournumericalintegrationsN±1，2，3，whereδEandδAaretheabsolutechangeofthetotalenergyandtotalangularmomentum，respectively，andE0andA0aretheirinitialvalues.ThehorizontalunitisGyr.

Notethatdifferentoperatingsystems，differentmathematicallibraries，anddifferenthardwarearchitecturesresultindifferentnumericalerrors，throughthevariationsinround-offerrorhandlingandnumericalalgorithms.IntheupperpanelofFig.1，wecanrecognizethissituationinthesecularnumericalerrorinthetotalangularmomentum，whichshouldberigorouslypreserveduptomachine-εprecision.

2.4.2Errorinplanetarylongitudes

SincethesymplecticmapspreservetotalenergyandtotalangularmomentumofN-bodydynamicalsystemsinherentlywell，thedegreeoftheirpreservationmaynotbeagoodmeasureoftheaccuracyofnumericalintegrations，especiallyasameasureofthepositionalerrorofplanets，i.e.theerrorinplanetarylongitudes.Toestimatethenumericalerrorintheplanetarylongitudes，weperformedthefollowingprocedures.Wecomparedtheresultofourmainlong-termintegrationswithsometestintegrations，whichspanmuchshorterperiodsbutwithmuchhigheraccuracythanthemainintegrations.Forthispurpose，weperformedamuchmoreaccurateintegrationwithastepsizeof0.125d（164ofthemainintegrations）spanning3×105yr，startingwiththesameinitialconditionsasintheN?1integration.Weconsiderthatthistestintegrationprovidesuswitha‘pseudo-true’solutionofplanetaryorbitalevolution.Next，wecomparethetestintegrationwiththemainintegration，N?1.Fortheperiodof3×105yr，weseeadifferenceinmeananomaliesoftheEarthbetweenthetwointegrationsof～0.52°（inthecaseoftheN?1integration）.Thisdifferencecanbeextrapolatedtothevalue～8700°，about25rotationsofEarthafter5Gyr，sincetheerroroflongitudesincreaseslinearlywithtimeinthesymplecticmap.Similarly，thelongitudeerrorofPlutocanbeestimatedas～12°.ThisvalueforPlutoismuchbetterthantheresultinKinoshita&Nakai（1996）wherethedifferenceisestimatedas～60°.

3Numericalresults–I.Glanceattherawdata

Inthissectionwebrieflyreviewthelong-termstabilityofplanetaryorbitalmotionthroughsomesnapshotsofrawnumericaldata.Theorbitalmotionofplanetsindicateslong-termstabilityinallofournumericalintegrations:noorbitalcrossingsnorcloseencountersbetweenanypairofplanetstookplace.

3.1Generaldescriptionofthestabilityofplanetaryorbits

First，webrieflylookatthegeneralcharacterofthelong-termstabilityofplanetaryorbits.Ourinterestherefocusesparticularlyontheinnerfourterrestrialplanetsforwhichtheorbitaltime-scalesaremuchshorterthanthoseoftheouterfiveplanets.AswecanseeclearlyfromtheplanarorbitalconfigurationsshowninFigs2and3，orbitalpositionsoftheterrestrialplanetsdifferlittlebetweentheinitialandfinalpartofeachnumericalintegration，whichspansseveralGyr.Thesolidlinesdenotingthepresentorbitsoftheplanetsliealmostwithintheswarmofdotseveninthefinalpartofintegrations（b）and（d）.Thisindicatesthatthroughouttheentireintegrationperiodthealmostregularvariationsofplanetaryorbitalmotionremainnearlythesameastheyareatpresent.

Verticalviewofthefourinnerplanetaryorbits（fromthez-axisdirection）attheinitialandfinalpartsoftheintegrationsN±1.Theaxesunitsareau.Thexy-planeissettotheinvariantplaneofSolarsystemtotalangularmomentu（a）TheinitialpartofN+1（t=0to0.0547×109yr）.（b）ThefinalpartofN+1（t=4.9339×108to4.9886×109yr）.（c）TheinitialpartofN?1（t=0to?0.0547×109yr）.（d）ThefinalpartofN?1（t=?3.9180×109to?3.9727×109yr）.Ineachpanel，atotalof23684pointsareplottedwithanintervalofabout2190yrover5.47×107yr.Solidlinesineachpaneldenotethepresentorbitsofthefourterrestrialplanets（takenfromDE245）.

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