作者君在作品相关中已经解释过这个问题，并在此列出相关参考文献中的一篇开源论文。

以下是文章内容：

long-termintegrationsandstabilityofplanetaryorbitsinoursolarsystem

abstract

wepresenttheresultsofverylong-termnumericalintegrationsofplanetaryorbitalmotionsover109-yrtime-spansincludingallnineplanets.aquickinspectionofournumericaldatashowsthattheplanetarymotion，atleastinoursimpledynamicalmodel，seemstobequitestableevenoverthisverylongtime-span.acloserlookatthelowest-frequencyoscillationsusingalow-passfiltershowsusthepotentiallydiffusivecharacterofterrestrialplanetarymotion，especiallythatofmercury.thebehaviouroftheeccentricityofmercuryinourintegrationsisqualitativelysimilartotheresultsfromjacqueslaskar'ssecularperturbationtheory(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&ito&&tanikawa1999).asystemisdefinedasexperiencingacloseencounterwhentwobodiesapproachoneanotherwithinanareaofthelargerhillradius.otherwisethesystemisdefinedasbeingstable.henceforwardwestatethatourplanetarysystemisdynamicallystableifnocloseencounterhappensduringtheageofoursolarsystem，about±5gyr.incidentally，thisdefinitionmaybereplacedbyoneinwhichanoccurrenceofanyorbitalcrossingbetweeneitherofapairofplanetstakesplace.thisisbecauseweknowfromexperiencethatanorbitalcrossingisverylikelytoleadtoacloseencounterinplanetaryandprotoplanetarysystems(yoshinaga，kokubo&&makino1999).ofcoursethisstatementcannotbesimplyappliedtosystemswithstableorbitalresonancessuchastheneptune–plutosystem.

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1.2previousstudiesandaimsofthisresearch

inadditiontothevaguenessoftheconceptofstability，theplanetsinoursolarsystemshowacharactertypicalofdynamicalchaos(sussman&&wisdom1988，1992).thecauseofthischaoticbehaviourisnowpartlyunderstoodasbeingaresultofresonanceoverlapping(murray&lecar，franklin&&holman2001).however，itwouldrequireintegratingoveranensembleofplanetarysystemsincludingallnineplanetsforaperiodcoveringseveral10gyrtothoroughlyunderstandthelong-termevolutionofplanetaryorbits，sincechaoticdynamicalsystemsarecharacterizedbytheirstrongdependenceoninitialconditions.

fromthatpointofview，manyofthepreviouslong-termnumericalintegrationsincludedonlytheouterfiveplanets(sussman&kinoshita&&nakai1996).thisisbecausetheorbitalperiodsoftheouterplanetsaresomuchlongerthanthoseoftheinnerfourplanetsthatitismucheasiertofollowthesystemforagivenintegrationperiod.atpresent，thelongestnumericalintegrationspublishedinjournalsarethoseofduncan&&lissauer(1998).althoughtheirmaintargetwastheeffectofpost-main-sequencesolarmasslossonthestabilityofplanetaryorbits，theyperformedmanyintegrationscoveringupto～1011yroftheorbitalmotionsofthefourjovianplanets.theinitialorbitalelementsandmassesofplanetsarethesameasthoseofoursolarsysteminduncan&&lissauer'spaper，buttheydecreasethemassofthesungraduallyintheirnumericalexperiments.thisisbecausetheyconsidertheeffectofpost-main-sequencesolarmasslossinthepaper.consequently，theyfoundthatthecrossingtime-scaleofplanetaryorbits，whichcanbeatypicalindicatoroftheinstabilitytime-scale，isquitesensitivetotherateofmassdecreaseofthesun.whenthemassofthesunisclosetoitspresentvalue，thejovianplanetsremainstableover1010yr，orperhapslonger.duncan&&lissaueralsoperformedfoursimilarexperimentsontheorbitalmotionofsevenplanets(venustoneptune)，whichcoveraspanof～109yr.theirexperimentsonthesevenplanetsarenotyetcomprehensive，butitseemsthattheterrestrialplanetsalsoremainstableduringtheintegrationperiod，maintainingalmostregularoscillations.

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ontheotherhand，inhisaccuratesemi-analyticalsecularperturbationtheory(laskar1988)，laskarfindsthatlargeandirregularvariationscanappearintheeccentricitiesandinclinationsoftheterrestrialplanets，especiallyofmercuryandmarsonatime-scaleofseveral109yr(laskar1996).theresultsoflaskar'ssecularperturbationtheoryshouldbeconfirmedandinvestigatedbyfullynumericalintegrations.

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.

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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&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.

weadoptgauss'fandgfunctionsinthesymplecticmaptogetherwiththethird-orderhalleymethod(danby1992)asasolverforkeplerequations.thenumberofmaximumiterationswesetinhalley'smethodis15，buttheyneverreachedthemaximuminanyofourintegrations.

theintervalofthedataoutputis200000d(～547yr)forthecalculationsofallnineplanets(n±1，2，3)，andabout8000000d(～21903yr)fortheintegrationoftheouterfiveplanets(f±).

althoughnooutputfilteringwasdonewhenthenumericalintegrationswereinprocess，weappliedalow-passfiltertotheraworbitaldataafterwehadcompletedallthecalculations.seesection4.1formoredetail.

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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.