Weutilizeasecond-orderWisdom–Holmansymplecticmapasourmainintegrationmethod(Wisdom&HolmanKinoshita，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-elationtothis，Wisdom&Holman(1991)performednumericalintegrationsoftheouterfiveplanetaryorbitsusingthesymplecticmapwithastepsizeof400d，1/10.irresultseemstobeaccurateenough，ever，sincetheeccentricityofJupiter(?0.05)ismuchsmallerthanthatofMercury(?0.2)，weneedsomecarewhenweparetheseintegrationssimplyintermsofstepsizes.

Intheintegrationoftheouterfiveplanets(F±)，wefixedthestepsizeat400d.

以下是文章内容：

Long-termintegrationsandstabilityofplanetaryorbitsinourSolarsystem

Abstract

Ontheotherhand，inhisaccuratesemi-analyticalsecularperturbationtheory(Laskar198，Laskarfindsthatlargeandirregularvariationscanappearintheeccentricitiesandinclinationsoftheterrestrialplanets，especiallyofMercuryandMarsonatime-scaleofseveral109yr(Laskar1996).TheresultsofLaskar‘ssecularperturbationtheoryshouldbeconfirmedandinvestigatedbyfullynumericalintegrations.

Inthispaperwepresentpreliminaryresultsofsixlong-termnumericalintegrationsonallnineplanetaryorbits，coveringaspanofseveral109yr，andoftwootherintegrationscoveringaspanof±5xtotalelapsedtimeforallintegrationsismorethan5yr，ofthefundamentalconclusionsofourlong-termintegrationsisthatSolarsystemplanetarymotionseemstobestableintermsoftheHillstabilitymentionedabove，atleastoveratime-spanof±ually，inournumericalintegrationsthesystemwasfarmorestablethanwhatisdefinedbytheHillstabilitycriterion:notonlydidnocloseencounterhappenduringtheintegrationperiod，butalsoalltheplanetaryorbitalelementshavebeenconfinedinanarrowregionbothintimeandfrequencydomain，cethepurposeofthispaperistoexhibitandoverviewtheresultsofourlong-termnumericalintegrations，weshowtypicalexamplefiguresasevidenceoftheverylong-readerswhohavemorespecificanddeeperinterestsinournumericalresults，wehavepreparedawebpage(access)，whereweshowraworbitalelements，theirlow-passfilteredresults，variationofDelaunayelementsandangularmomentumdeficit，andresultsofoursimpletime–frequencyanalysisonallofourintegrations.

InSection2webrieflyexplainourdynamicalmodel，tion3isdylong-termstabilityofSolarsystemplantion4goesontoadiscussionofthelongest-termvariationofplanetaryorbitsusingalow-ection5，wepresentasetofnumericalintegrationsfortheouterfiveplanetsthatspans±5xection6wealsodiscussthelong-termstabilityoftheplanetarymotionanditspossiblecause.

Wepresenttheresultsofverylong-termnumericalintegrationsofplanetaryorbitalmotionsover109-yrtime-ickinspectionofournumericaldatashowsthattheplanetarymotion，atleastinoursimpledynamicalmodel，seemstobequitestableevenoverthisverylongtime-oserlookatthelowest-frequencyoscillationsusingalow-passfiltershowsusthepotentiallydiffusivecharacterofterrestrialplanetarymotion，behaviouroftheeccentricityofMercuryinourintegrationsisqualitativelysimilartotheresultsfromJacquesLaskar‘ssecularperturbationtheory(e.x?0.35over?±4Gyr).However，therearenoapparentsecularincreasesofeccentricityorinclinationinanyorbitalelementsoftheplanets，whichmayberevealedbystilllonger-avealsoperformedacoupleoftrialintegrationsincludingmotionsoftheouterfiveplanetsoverthedurationof±5xresultindicatesthatthethreemajorresonancesintheNeptune–Plutosystemhavebeenmaintainedoverthe1011-yrtime-span.

1Introduction

1.1Definitionoftheproblem

ThequestionofthestabilityofourSolarsystemhasbeendebatedoverseveralhundredyears，problemhasattractedmanyfamousmathematiciansovertheyearsandhasplayedacentralroleinthedevelopmentofnon-ever，wedonotyethaveasispartlyaresultofthefactthatthedefinitionoftheterm‘stability’isvaguewhenuallyitisnoteasytogiveaclear，rigorousandphysicallymeaningfuldefinitionofthestabilityofourSolarsystem.

Amongmanydefinitionsofstability，hereweadopttheHilldefinition(Gladman1993):actuallythisisnotadefinitionofstability，efineasystemasbeingunstablewhenacloseencounteroccurssomewhereinthesystem，startingfromacertaininitialconfiguration(Chambers，Wetherill&BossIto&Tanikawa1999).AsystemisdefinedasexperiencingacloseencounterwceforwardwestatethatourplanetarysystemisdynamicallystableifnocloseencounterhappensduringtheageofourSolarsystem，about±identally，thisdefinitionmaybereplacedbyoneinwhichanoccurrsisbecauseweknowfromexperiencethatanorbitalcrossingisverylikelytoleadtoacloseencounterinplanetaryandprotoplanetarysystems(Yoshinaga，Kokubo&Makino1999).OfcoursethisstatementcannotbesimplyappliedtosystemswithstableorbitalresonancessuchastheNeptune–Plutosystem.

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

WeadoptGauss‘fandgfunctionsinthesy

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对火星轨道变化问题的最后解释 (第1/3页)

作者君在作品相关中其实已经解释过这个问题。

不过仍然有人质疑——“你说得太含糊了”，“火星轨道的变化比你想象要大得多！”

那好吧，既然作者君的简单解释不够有力，那咱们就看看严肃的东西，反正这本书写到现在，嚷嚷着本书BUG一大堆，用初高中物理在书中挑刺的人也不少。

1.2Previousstudiesandaimsofthisresearch

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

Fromthatpointofview，manyofthepreviouslong-termnumericalintegrationsincludedonlytheouterfiveplanets(Sussman&WisdomKinoshita&Nakai1996).Thisisbecausetheorbitalperiodsoftheouterplanetsaresomuchlongerthanthoseoftheinnerfourplaneresent，thelongestnumericalintegrationspublishedinjournalsarethoseofDuncan&Lissauer(houghtheirmaintargetwastheeffectofpost-main-sequencesolarmasslossonthestabilityofplanetaryorbits，theyperformedmanyintegrationscoveringupto?initialorbitalelementsandmassesofplanetsarethesameasthoseofourSolarsysteminDuncan&Lissauer‘spaper，butsisbecausetheyconsidertheeffectofpost-main-sequently，theyfoundthatthecrossingtime-scaleofplanetaryorbits，whichcanbeatypicalindicatoroftheinstabilitytime-scale，nthemassoftheSunisclosetoitspresentvalue，thejovianplanetsremainstableover1010yr，can&Lissaueralsoperformedfoursimilarexperimentsontheorbitalmotionofsevenplanets(VenustoNeptune)，whichcoveraspanof?irexperimentsonthesevenplanetsarenotyetprehensive，butitseemsthattheterrestrialplanetsalsoremainstableduringtheintegrationperiod，maintainingalmostregularoscillations.

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