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