死在火星上

　　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&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&Lissauer'spaper，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).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.

　　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)，whichisabout1/11oftheorbitalperiodoftheinnermostplanet(Mercury).Asforthedeterminationofstepsize，wepartlyfollowthepreviousnumericalintegrationofallnineplanetsinSussman&Wisdom(1988，7.2d)andSaha&Tremaine(1994，225/32d).Weroundedthedecimalpartofthetheirstepsizesto8tomakethestepsizeamultipleof2inordertoreducetheaccumulationofround-offerrorinthecomputationprocesses.Inrelationtothis，Wisdom&Holman(1991)performednumericalintegrationsoftheouterfiveplanetaryorbitsusingthesymplecticmapwithastepsizeof400d，1/10.83oftheorbitalperiodofJupiter.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.

　　2.4Errorestimation

　　2.4.1Relativeerrorsintotal