对火星轨道变化问题的最后解释 (第1/1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。

    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

    2.4.1Relativeerrorsintotalenergyandangularmomentum

    Accordingtooneofthebasicpropertiesofsymplecticintegrators，whichconservethephysicallyconservativequantitieswell(totalorbitalenergyandangularmomentum)，ourlong-termnumericalintegrationsseemtohavebeenperformedwithverysmallerrors.Theaveragedrelativeerrorsoftotalenergy(～10?9)andoftotalangularmomentum(～10?11)haveremainednearlyconstantthroughouttheintegrationperiod(Fig.1).Thespecialstartupprocedure，warmstart，wouldhavereducedtheaveragedrelativeerrorintotalenergybyaboutoneorderofmagnitudeormore.

    RelativenumericalerrorofthetotalangularmomentumδA/A0andthetotalenergyδE/E0inournumericalintegrationsN±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(1/64ofthemainintegrations)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).

    ThevariationofeccentricitiesandorbitalinclinationsfortheinnerfourplanetsintheinitialandfinalpartoftheintegrationN 1isshowninFig.4.Asexpected，thecharacterofthevariationofplanetaryorbitalelementsdoesnotdiffersignificantlybetweentheinitialandfinalpartofeachintegration，atleastforVenus，EarthandMars.TheelementsofMercury，especiallyitseccentricity，seemtochangetoasignificantextent.Thisispartlybecausetheorbitaltime-scaleoftheplanetistheshortestofalltheplanets，whichleadstoamorerapidorbitalevolutionthanotherplanets;theinnermostplanetmaybenearesttoinstability.ThisresultappearstobeinsomeagreementwithLaskar's(1994，1996)expectationsthatlargeandirregularvariationsappearintheeccentricitiesandinclinationsofMercuryonatime-scaleofseveral109yr.However，theeffectofthepossibleinstabilityoftheorbitofMercurymaynotfatallyaffecttheglobalstabilityofthewholeplanetarysystemowingtothesmallmassofMercury.Wewillmentionbrieflythelong-termorbitalevolutionofMercurylaterinSection4usinglow-passfilteredorbitalelements.

    Theorbitalmotionoftheouterfiveplanetsseemsrigorouslystableandquiteregularoverthistime-span(seealsoSection5).

    3.2Time–frequencymaps

    Althoughtheplanetarymotionexhibitsverylong-termstabilitydefinedasthenon-existenceofcloseencounterevents，thechaoticnatureofplanetarydynamicscanchangetheoscillatoryperiodandamplitudeofplanetaryorbitalmotiongraduallyoversuchlongtime-spans.Evensuchslightfluctuationsoforbitalvariationinthefrequencydomain，particularlyinthecaseofEarth，canpotentiallyhaveasignificanteffectonitssurfaceclimatesystemthroughsolarinsolationvariation(cf.Berger1988).

    Togiveanoverviewofthelong-termchangeinperiodicityinplanetaryorbitalmotion，weperformedmanyfastFouriertransformations(FFTs)alongthetimeaxis，andsuperposedtheresultingperiodgramstodrawtwo-dimensionaltime–frequencymaps.Thespecificapproachtodrawingthesetime–frequencymapsinthispaperisverysimple–muchsimplerthanthewaveletanalysisorLaskar's(1990，1993)frequencyanalysis.

    Dividethelow-passfilteredorbitaldataintomanyfragmentsofthesamelength.Thelengthofeachdatasegmentshouldbeamultipleof2inordertoapplytheFFT.

    Eachfragmentofthedatahasalargeoverlappingpart:forexample，whentheithdatabeginsfromt=tiandendsatt=ti T，thenextdatasegmentrangesfromti δT≤ti δT T，whereδT?T.WecontinuethisdivisionuntilwereachacertainnumberNbywhichtn Treachesthetotalintegrationlength.

    WeapplyanFFTtoeachofthedatafragments，andobtainnfrequencydiagrams.

    Ineachfrequencydiagramobtainedabove，thestrengthofperiodicitycanbereplacedbyagrey-scale(orcolour)chart.

    Weperformthereplacement，andconnectallthegrey-scale(orcolour)chartsintoonegraphforeachintegration.Thehorizontalaxisofthesenewgraphsshouldbethetime，i.e.thestartingtimesofeachfragmentofdata(ti，wherei=1，…，n).Theverticalaxisrepresentstheperiod(orfrequency)oftheoscillationoforbitalelements.

    WehaveadoptedanFFTbecauseofitsoverwhelmingspeed，sincetheamountofnumericaldatatobedecomposedintofrequencycomponentsisterriblyhuge(severaltensofGbytes).

    Atypicalexampleofthetime–frequencymapcreatedbytheaboveproceduresisshowninagrey-scalediagramasFig.5，whichshowsthevariationofperiodicityintheeccentricityandinclinationofEarthinN 2integration.InFig.5，thedarkareashowsthatatthetimeindicatedbythevalueontheabscissa，theperiodicityindicatedbytheordinateisstrongerthaninthelighterareaaroundit.WecanrecognizefromthismapthattheperiodicityoftheeccentricityandinclinationofEarthonlychangesslightlyovertheentireperiodcoveredbytheN 2integration.Thisnearlyregulartrendisqualitativelythesameinotherintegrationsandforotherplanets，althoughtypicalfrequenciesdifferplanetbyplanetandelementbyelement.

    4.2Long-termexchangeoforbitalenergyandangularmomentum

    Wecalculateverylong-periodicvariationandexchangeofplanetaryorbitalenergyandangularmomentumusingfilteredDelaunayelementsL，G，H.GandHareequivalenttotheplanetaryorbitalangularmomentumanditsverticalcomponentperunitmass.LisrelatedtotheplanetaryorbitalenergyEperunitmassasE=?μ2/2L2.Ifthesystemiscompletelylinear，theorbitalenergyandtheangularmomentumineachfrequencybinmustbeconstant.Non-linearityintheplanetarysystemcancauseanexchangeofenergyandangularmomentuminthefrequencydomain.Theamplitudeofthelowest-frequencyoscillationshouldincreaseifthesystemisunstableandbreaksdowngradually.However，suchasymptomofinstabilityisnotprominentinourlong-termintegrations.

    InFig.7，thetotalorbitalenergyandangularmomentumofthefourinnerplanetsandallnineplanetsareshownforintegrationN 2.Theupperthreepanelsshowthelong-periodicvariationoftotalenergy(denotedasE-E0)，totalangularmomentum(G-G0)，andtheverticalcomponent(H-H0)oftheinnerfourplanetscalculatedfromthelow-passfilteredDelaunayelements.E0，G0，H0denotetheinitialvaluesofeachquantity.Theabsolutedifferencefromtheinitialvaluesisplottedinthepanels.ThelowerthreepanelsineachfigureshowE-E0，G-G0andH-H0ofthetotalofnineplanets.Thefluctuationshowninthelowerpanelsisvirtuallyentirelyaresultofthemassivejovianplanets.

    Comparingthevariationsofenergyandangularmomentumoftheinnerfourplanetsandallnineplanets，itisapparentthattheamplitudesofthoseoftheinnerplanetsaremuchsmallerthanthoseofallnineplanets:theamplitudesoftheouterfiveplanetsaremuchlargerthanthoseoftheinnerplanets.Thisdoesnotmeanthattheinnerterrestrialplanetarysubsystemismorestablethantheouterone:thisissimplyaresultoftherelativesmallnessofthemassesofthefourterrestrialplanetscomparedwiththoseoftheouterjovianplanets.Anotherthingwenoticeisthattheinnerplanetarysubsystemmaybecomeunstablemorerapidlythantheouteronebecauseofitsshorterorbitaltime-scales.Thiscanbeseeninthepanelsdenotedasinner4inFig.7wherethelonger-periodicandirregularoscillationsaremoreapparentthaninthepanelsdenotedastotal9.Actually，thefluctuationsintheinner4panelsaretoalargeextentasaresultoftheorbitalvariationoftheMercury.However，wecannotneglectthecontributionfromotherterrestrialplanets，aswewillseeinsubsequentsections.

    4.4Long-termcouplingofseveralneighbouringplanetpairs

    Letusseesomeindividualvariationsofplanetaryorbitalenergyandangularmomentumexpressedbythelow-passfilteredDelaunayelements.Figs10and11showlong-termevolutionoftheorbitalenergyofeachplanetandtheangularmomentuminN 1andN?2integrations.Wenoticethatsomeplanetsformapparentpairsintermsoforbitalenergyandangularmomentumexchange.Inparticular，VenusandEarthmakeatypicalpair.Inthefigures，theyshownegativecorrelationsinexchangeofenergyandpositivecorrelationsinexchangeofangularmomentum.Thenegativecorrelationinexchangeoforbitalenergymeansthatthetwoplanetsformacloseddynamicalsystemintermsoftheorbitalenergy.Thepositivecorrelationinexchangeofangularmomentummeansthatthetwoplanetsaresimultaneouslyundercertainlong-termperturbations.CandidatesforperturbersareJupiterandSaturn.AlsoinFig.11，wecanseethatMarsshowsapositivecorrelationintheangularmomentumvariationtotheVenus–Earthsystem.MercuryexhibitscertainnegativecorrelationsintheangularmomentumversustheVenus–Earthsystem，whichseemstobeareactioncausedbytheconservationofangularmomentumintheterrestrialplanetarysubsystem.

    ItisnotclearatthemomentwhytheVenus–Earthpairexhibitsanegativecorrelationinenergyexchangeandapositivecorrelationinangularmomentumexchange.Wemaypossiblyexplainthisthroughobservingthegeneralfactthattherearenoseculartermsinplanetarysemimajoraxesuptosecond-orderperturbationtheories(cf.Brouwer&Clemence1961;Boccaletti&Pucacco1998).Thismeansthattheplanetaryorbitalenergy(whichisdirectlyrelatedtothesemimajoraxisa)mightbemuchlessaffectedbyperturbingplanetsthanistheangularmomentumexchange(whichrelatestoe).Hence，theeccentricitiesofVenusandEarthcanbedisturbedeasilybyJupiterandSaturn，whichresultsinapositivecorrelationintheangularmomentumexchange.Ontheotherhand，thesemimajoraxesofVenusandEartharelesslikelytobedisturbedbythejovianplanets.ThustheenergyexchangemaybelimitedonlywithintheVenus–Earthpair，whichresultsinanegativecorrelationintheexchangeoforbitalenergyinthepair.

    Asfortheouterjovianplanetarysubsystem，Jupiter–SaturnandUranus–Neptuneseemtomakedynamicalpairs.However，thestrengthoftheircouplingisnotasstrongcomparedwiththatoftheVenus–Earthpair.

    5±5×1010-yrintegrationsofouterplanetaryorbits

    Sincethejovianplanetarymassesaremuchlargerthantheterrestrialplanetarymasses，wetreatthejovianplanetarysystemasanindependentplanetarysystemintermsofthestudyofitsdynamicalstability.Hence，weaddedacoupleoftrialintegrationsthatspan±5×1010yr，includingonlytheouterfiveplanets(thefourjovianplanetsplusPluto).Theresultsexhibittherigorousstabilityoftheouterplanetarysystemoverthislongtime-span.Orbitalconfigurations(Fig.12)，andvariationofeccentricitiesandinclinations(Fig.13)showthisverylong-termstabilityoftheouterfiveplanetsinboththetimeandthefrequencydomains.Althoughwedonotshowmapshere，thetypicalfrequencyoftheorbitaloscillationofPlutoandtheotherouterplanetsisalmostconstantduringtheseverylong-termintegrationperiods，whichisdemonstratedinthetime–frequencymapsonourwebpage.

    Inthesetwointegrations，therelativenumericalerrorinthetotalenergywas～10?6andthatofthetotalangularmomentumwas～10?10.

    5.1ResonancesintheNeptune–Plutosystem

    Kinoshita&Nakai(1996)integratedtheouterfiveplanetaryorbitsover±5.5×109yr.TheyfoundthatfourmajorresonancesbetweenNeptuneandPlutoaremaintainedduringthewholeintegrationperiod，andthattheresonancesmaybethemaincausesofthestabilityoftheorbitofPluto.Themajorfourresonancesfoundinpreviousresearchareasfollows.Inthefollowingdescription，λdenotesthemeanlongitude，Ωisthelongitudeoftheascendingnodeand?isthelongitudeofperihelion.SubscriptsPandNdenotePlutoandNeptune.

    MeanmotionresonancebetweenNeptuneandPluto(3:2).Thecriticalargumentθ1=3λP?2λN??Plibratesaround180°withanamplitudeofabout80°andalibrationperiodofabout2×104yr.

    TheargumentofperihelionofPlutoωP=θ2=?P?ΩPlibratesaround90°withaperiodofabout3.8×106yr.ThedominantperiodicvariationsoftheeccentricityandinclinationofPlutoaresynchronizedwiththelibrationofitsargumentofperihelion.ThisisanticipatedinthesecularperturbationtheoryconstructedbyKozai(1962).

    ThelongitudeofthenodeofPlutoreferredtothelongitudeofthenodeofNeptune，θ3=ΩP?ΩN，circulatesandtheperiodofthiscirculationisequaltotheperiodofθ2libration.Whenθ3becomeszero，i.e.thelongitudesofascendingnodesofNeptuneandPlutooverlap，theinclinationofPlutobecomesmaximum，theeccentricitybecomesminimumandtheargumentofperihelionbecomes90°.Whenθ3becomes180°，theinclinationofPlutobecomesminimum，theeccentricitybecomesmaximumandtheargumentofperihelionbecomes90°again.Williams&Benson(1971)anticipatedthistypeofresonance，laterconfirmedbyMilani，Nobili&Carpino(1989).

    Anargumentθ4=?P??N 3(ΩP?ΩN)libratesaround180°withalongperiod，～5.7×108yr.

    Inournumericalintegrations，theresonances(i)–(iii)arewellmaintained，andvariationofthecriticalargumentsθ1，θ2，θ3remainsimilarduringthewholeintegrationperiod(Figs14–16).However，thefourthresonance(iv)appearstobedifferent:thecriticalargumentθ4alternateslibrationandcirculationovera1010-yrtime-scale(Fig.17).ThisisaninterestingfactthatKinoshita&Nakai's(1995，1996)shorterintegrationswerenotabletodisclose.

    6Discussion

    Whatkindofdynamicalmechanismmaintainsthislong-termstabilityoftheplanetarysystem?Wecanimmediatelythinkoftwomajorfeaturesthatmayberesponsibleforthelong-termstability.First，thereseemtobenosignificantlower-orderresonances(meanmotionandsecular)betweenanypairamongthenineplanets.JupiterandSaturnareclosetoa5:2meanmotionresonance(thefamous‘greatinequality’)，butnotjustintheresonancezone.Higher-orderresonancesmaycausethechaoticnatureoftheplanetarydynamicalmotion，buttheyarenotsostrongastodestroythestableplanetarymotionwithinthelifetimeoftherealSolarsystem.Thesecondfeature，whichwethinkismoreimportantforthelong-termstabilityofourplanetarysystem，isthedifferenceindynamicaldistancebetweenterrestrialandjovianplanetarysubsystems(Ito&Tanikawa1999，2001).WhenwemeasureplanetaryseparationsbythemutualHillradii(R_)，separationsamongterrestrialplanetsaregreaterthan26RH，whereasthoseamongjovianplanetsarelessthan14RH.Thisdifferenceisdirectlyrelatedtothedifferencebetweendynamicalfeaturesofterrestrialandjovianplanets.Terrestrialplanetshavesmallermasses，shorterorbitalperiodsandwiderdynamicalseparation.Theyarestronglyperturbedbyjovianplanetsthathavelargermasses，longerorbitalperiodsandnarrowerdynamicalseparation.Jovianplanetsarenotperturbedbyanyothermassivebodies.

    Thepresentterrestrialplanetarysystemisstillbeingdisturbedbythemassivejovianplanets.However，thewideseparationandmutualinteractionamongtheterrestrialplanetsrendersthedisturbanceineffective;thedegreeofdisturbancebyjovianplanetsisO(eJ)(orderofmagnitudeoftheeccentricityofJupiter)，sincethedisturbancecausedbyjovianplanetsisaforcedoscillationhavinganamplitudeofO(eJ).Heighteningofeccentricity，forexampleO(eJ)～0.05，isfarfromsufficienttoprovokeinstabilityintheterrestrialplanetshavingsuchawideseparationas26RH.Thusweassumethatthepresentwidedynamicalseparationamongterrestrialplanets(>26RH)isprobablyoneofthemostsignificantconditionsformaintainingthestabilityoftheplanetarysystemovera109-yrtime-span.Ourdetailedanalysisoftherelationshipbetweendynamicaldistancebetweenplanetsandtheinstabilitytime-scaleofSolarsystemplanetarymotionisnowon-going.

    AlthoughournumericalintegrationsspanthelifetimeoftheSolarsystem，thenumberofintegrationsisfarfromsufficienttofilltheinitialphasespace.Itisnecessarytoperformmoreandmorenumericalintegrationstoconfirmandexamineindetailthelong-termstabilityofourplanetarydynamics.

    ——以上文段引自Ito，T.&Tanikawa，K.Long-termintegrationsandstabilityofplanetaryorbitsinourSolarSystem.Mon.Not.R.Astron.Soc.336，483–500(2002)

    这只是作者君参考的一篇文章，关于太阳系的稳定性。

    还有其他论文，不过也都是英文的，相关课题的中文文献很少，那些论文下载一篇要九美元（《Nature》真是暴利），作者君写这篇文章的时候已经回家，不在检测中心，所以没有数据库的使用权，下不起，就不贴上来了。