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对火星轨道变化问题的最后解释（第1页）

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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。

Verticalviewofthefourinnerplanetaryorbits（fromthez-axisdirection）attheinitialandfinalpartsoftheintegrationsN±1。Theaxesunitsareau。Thexy-planeissettotheinvariantplaneofSolarsystemtotalangularmomentum。（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）。