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    以下是文章内容：
    longtermintegrationsandstabilityofplanetaryorbitsinoursolarsystem
    abstract
    wepresenttheresultsofverylongtermnumericalintegrationsofplanetaryorbitalmotionsover109yrtimespansincludingallnineplanets.aquickinspectionofournumericaldatashowsthattheplanetarymotion，atleastinoursimpledynamicalmodel，seemstobequitestableevenoverthisverylongtimespan.acloserlookatthelowestfrequencyoscillationsusingalowpassfiltershowsusthepotentiallydiffusivecharacterofterrestrialplanetarymotion，especiallythatofmercury.thebehaviouroftheeccentricityofmercuryinourintegrationsisqualitativelysimilartotheresultsfromjacqueslaskar'ssecularperturbationtheory(e.g.emax～0.35over～±4gyr).however，therearenoapparentsecularincreasesofeccentricityorinclinationinanyorbitalelementsoftheplanets，whichmayberevealedbystilllongertermnumericalintegrations.wehavealsoperformedacoupleoftrialintegrationsincludingmotionsoftheouterfiveplanetsoverthedurationof±5x1010yr.theresultindicatesthatthethreemajorresonancesintheneptune–plutosystemhavebeenmaintainedoverthe1011yrtimespan.
    1introduction
    1.1definitionoftheproblem
    thequestionofthestabilityofoursolarsystemhasbeendebatedoverseveralhundredyears，sincetheeraofnewton.theproblemhasattractedmanyfamousmathematiciansovertheyearsandhasplayedacentralroleinthedevelopmentofnonlineardynamicsandchaostheory.however，wedonotyethaveadefiniteanswertothequestionofwhetheroursolarsystemisstableornot.thisispartlyaresultofthefactthatthedefinitionoftheterm‘stability’isvaguewhenitisusedinrelationtotheproblemofplanetarymotioninthesolarsystem.actuallyitisnoteasytogiveaclear，rigorousandphysicallymeaningfuldefinitionofthestabilityofoursolarsystem.
    amongmanydefinitionsofstability，hereweadoptthehilldefinition(gladman1993):actuallythisisnotadefinitionofstability，butofinstability.wedefineasystemasbecomingunstablewhenacloseencounteroccurssomewhereinthesystem，startingfromacertaininitialconfiguration(chambers，wetherillitotanikawa1999).asystemisdefinedasexperiencingacloseencounterwhentwobodiesapproachoneanotherwithinanareaofthelargerhillradius.otherwisethesystemisdefinedasbeingstable.henceforwardwestatethatourplanetarysystemisdynamicallystableifnocloseencounterhappensduringtheageofoursolarsystem，about±5gyr.incidentally，thisdefinitionmaybereplacedbyoneinwhichanoccurrenceofanyorbitalcrossingbetweeneitherofapairofplanetstakesplace.thisisbecauseweknowfromexperiencethatanorbitalcrossingisverylikelytoleadtoacloseencounterinplanetaryandprotoplanetarysystems(yoshinaga，kokubomakino1999).ofcoursethisstatementcannotbesimplyappliedtosystemswithstableorbitalresonancessuchastheneptune–plutosystem.
    1.2previousstudiesandaimsofthisresearch
    inadditiontothevaguenessoftheconceptofstability，theplanetsinoursolarsystemshowacharactertypicalofdynamicalchaos(sussmanwisdom1988，1992).thecauseofthischaoticbehaviourisnowpartlyunderstoodasbeingaresultofresonanceoverlapping(murraylecar，franklinholman2001).however，itwouldrequireintegratingoveranensembleofplanetarysystemsincludingallnineplanetsforaperiodcoveringseveral10gyrtothoroughlyunderstandthelongtermevolutionofplanetaryorbits，sincechaoticdynamicalsystemsarecharacterizedbytheirstrongdependenceoninitialconditions.
    fromthatpointofview，manyofthepreviouslongtermnumericalintegrationsincludedonlytheouterfiveplanets(sussmankinoshitanakai1996).thisisbecausetheorbitalperiodsoftheouterplanetsaresomuchlongerthanthoseoftheinnerfourplanetsthatitismucheasiertofollowthesystemforagivenintegrationperiod.atpresent，thelongestnumericalintegrationspublishedinjournalsarethoseofduncanlissauer(1998).althoughtheirmaintargetwastheeffectofpostmainsequencesolarmasslossonthestabilityofplanetaryorbits，theyperformedmanyintegrationscoveringupto～1011yroftheorbitalmotionsofthefourjovianplanets.theinitialorbitalelementsandmassesofplanetsarethesameasthoseofoursolarsysteminduncanlissauer'spaper，buttheydecreasethemassofthesungraduallyintheirnumericalexperiments.thisisbecausetheyconsidertheeffectofpostmainsequencesolarmasslossinthepaper.consequently，theyfoundthatthecrossingtimescaleofplanetaryorbits，whichcanbeatypicalindicatoroftheinstabilitytimescale，isquitesensitivetotherateofmassdecreaseofthesun.whenthemassofthesunisclosetoitspresentvalue，thejovianplanetsremainstableover1010yr，orperhapslonger.duncanlissaueralsoperformedfoursimilarexperimentsontheorbitalmotionofsevenplanets(venustoneptune)，whichcoveraspanof～109yr.theirexperimentsonthesevenplanetsarenotyetcomprehensive，butitseemsthattheterrestrialplanetsalsoremainstableduringtheintegrationperiod，maintainingalmostregularoscillations.
    ontheotherhand，inhisaccuratesemianalyticalsecularperturbationtheory(laskar1988)，laskarfindsthatlargeandirregularvariationscanappearintheeccentricitiesandinclinationsoftheterrestrialplanets，especiallyofmercuryandmarsonatimescaleofseveral109yr(laskar1996).theresultsoflaskar'ssecularperturbationtheoryshouldbeconfirmedandinvestigatedbyfullynumericalintegrations.
    inthispaperwepresentpreliminaryresultsofsixlongtermnumericalintegrationsonallnineplanetaryorbits，coveringaspanofseveral109yr，andoftwootherintegrationscoveringaspanof±5x1010yr.thetotalelapsedtimeforallintegrationsismorethan5yr，usingseveraldedicatedpcsandworkstations.oneofthefundamentalconclusionsofourlongtermintegrationsisthatsolarsystemplanetarymotionseemstobestableintermsofthehillstabilitymentionedabove，atleastoveratimespanof±4gyr.actually，inournumericalintegrationsthesystemwasfarmorestablethanwhatisdefinedbythehillstabilitycriterion:notonlydidnocloseencounterhappenduringtheintegrationperiod，butalsoalltheplanetaryorbitalelementshavebeenconfinedinanarrowregionbothintimeandfrequencydomain，thoughplanetarymotionsarestochastic.sincethepurposeofthispaperistoexhibitandoverviewtheresultsofourlongtermnumericalintegrations，weshowtypicalexamplefiguresasevidenceoftheverylongtermstabilityofsolarsystemplanetarymotion.forreaderswhohavemorespecificanddeeperinterestsinournumericalresults，wehavepreparedawebpage(access)，whereweshowraworbitalelements，theirlowpassfilteredresults，variationofdelaunayelementsandangularmomentumdeficit，andresultsofoursimpletime–frequencyanalysisonallofourintegrations.
    insection2webrieflyexplainourdynamicalmodel，numericalmethodandinitialconditionsusedinourintegrations.section3isdevotedtoadescriptionofthequickresultsofthenumericalintegrations.verylongtermstabilityofsolarsystemplanetarymotionisapparentbothinplanetarypositionsandorbitalelements.aroughestimationofnumericalerrorsisalsogiven.section4goesontoadiscussionofthelongesttermvariationofplanetaryorbitsusingalowpassfilterandincludesadiscussionofangularmomentumdeficit.insection5，wepresentasetofnumericalintegrationsfortheouterfiveplanetsthatspans±5x1010yr.insection6wealsodiscussthelongtermstabilityoftheplanetarymotionanditspossiblecause.
    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
    weutilizeasecondorderwisdom–holmansymplecticmapasourmainintegrationmethod(wisdomkinoshita，yoshidanakai1991)withaspecialstartupproceduretoreducethetruncationerrorofanglevariables，‘warmstart’(sahatremaine1992，1994).
    thestepsizeforthenumericalintegrationsis8dthroughoutallintegrationsofthenineplanets(n±1，2，3)，whichisabout111oftheorbitalperiodoftheinnermostplanet(mercury).asforthedeterminationofstepsize，wepartlyfollowthepreviousnumericalintegrationofallnineplanetsinsussmanwisdom(1988，7.2d)andsahatremaine(1994，22532d).weroundedthedecimalpartofthetheirstepsizesto8tomakethestepsizeamultipleof2inordertoreducetheaccumulationofroundofferrorinthecomputationprocesses.inrelationtothis，wisdomholman(1991)performednumericalintegrationsoftheouterfiveplanetaryorbitsusingthesymplecticmapwithastepsizeof400d，110.83oftheorbitalperiodofjupiter.theirresultseemstobeaccurateenough，whichpartlyjustifiesourmethodofdeterminingthestepsize.however，sincetheeccentricityofjupiter(～0.05)ismuchsmallerthanthatofmercury(～0.2)，weneedsomecarewhenwecomparetheseintegrationssimplyintermsofstepsizes.
    intheintegrationoftheouterfiveplanets(f±)，wefixedthestepsizeat400d.
    weadoptgauss'fandgfunctionsinthesymplecticmaptogetherwiththethirdorderhalleymethod(danby1992)asasolverforkeplerequations.thenumberofmaximumiterationswesetinhalley'smethodis15，buttheyneverreachedthemaximuminanyofourintegrations.
    theintervalofthedataoutputis200000d(～547yr)forthecalculationsofallnineplanets(n±1，2，3)，andabout8000000d(～21903yr)fortheintegrationoftheouterfiveplanets(f±).
    althoughnooutputfilteringwasdonewhenthenumericalintegrationswereinprocess，weappliedalowpassfiltertotheraworbitaldataafterwehadcompletedallthecalculations.seesection4.1formoredetail.
    2.4errorestimation
    2.4.1relativeerrorsintotalenergyandangularmomentum
    accordingtooneofthebasicpropertiesofsymplecticintegrators，whichconservethephysicallyconservativequantitieswell(totalorbitalenergyandangularmomentum)，ourlongtermnumericalintegrationsseemtohavebeenperformedwithverysmallerrors.theaveragedrelativeerrorsoftotalenergy(～10?9)andoftotalangularmomentum(～10?11)haveremainednearlyconstantthroughouttheintegrationperiod(fig.1).thespecialstartupprocedure，warmstart，wouldhavereducedtheaveragedrelativeerrorintotalenergybyaboutoneorderofmagnitudeormore.
    relativenumericalerrorofthetotalangularmomentumδaa0andthetotalenergyδee0inournumericalintegrationsn±1，2，3，whereδeandδaaretheabsolutechangeofthetotalenergyandtotalangularmomentum，respectively，ande0anda0aretheirinitialvalues.thehorizontalunitisgyr.
    notethatdifferentoperatingsystems，differentmathematicallibraries，anddifferenthardwarearchitecturesresultindifferentnumericalerrors，throughthevariationsinroundofferrorhandlingandnumericalalgorithms.intheupperpaneloffig.1，wecanrecognizethissituationinthesecularnumericalerrorinthetotalangularmomentum，whichshouldberigorouslypreserveduptomachineeprecision.
    2.4.2errorinplanetarylongitudes
    sincethesymplecticmapspreservetotalenergyandtotalangularmomentumofnbodydynamicalsystemsinherentlywell，thedegreeoftheirpreservationmaynotbeagoodmeasureoftheaccuracyofnumericalintegrations，especiallyasameasureofthepositionalerrorofplanets，i.e.theerrorinplanetarylongitudes.toestimatethenumericalerrorintheplanetarylongitudes，weperformedthefollowingprocedures.wecomparedtheresultofourmainlongtermintegrationswithsometestintegrations，whichspanmuchshorterperiodsbutwithmuchhigheraccuracythanthemainintegrations.forthispurpose，weperformedamuchmoreaccurateintegrationwithastepsizeof0.125d(164ofthemainintegrations)spanning3x105yr，startingwiththesameinitialconditionsasinthen?1integration.weconsiderthatthistestintegrationprovidesuswitha‘pseudotrue’solutionofplanetaryorbitalevolution.next，wecomparethetestintegrationwiththemainintegration，n?1.fortheperiodof3x105yr，weseeadifferenceinmeananomaliesoftheearthbetweenthetwointegrationsof～0.52°(inthecaseofthen?1integration).thisdifferencecanbeextrapolatedtothevalue～8700°，about25rotationsofearthafter5gyr，sincetheerroroflongitudesincreaseslinearlywithtimeinthesymplecticmap.similarly，thelongitudeerrorofplutocanbeestimatedas～12°.thisvalueforplutoismuchbetterthantheresultinkinoshitanakai(1996)wherethedifferenceisestimatedas～60°.
    3numericalresults–i.glanceattherawdata
    inthissectionwebrieflyreviewthelongtermstabilityofplanetaryorbitalmotionthroughsomesnapshotsofrawnumericaldata.theorbitalmotionofplanetsindicateslongtermstabilityinallofournumericalintegrations:noorbitalcrossingsnorcloseencountersbetweenanypairofplanetstookplace.
    3.1generaldescriptionofthestabilityofplanetaryorbits
    first，webrieflylookatthegeneralcharacterofthelongtermstabilityofplanetaryorbits.ourinterestherefocusesparticularlyontheinnerfourterrestrialplanetsforwhichtheorbitaltimescalesaremuchshorterthanthoseoftheouterfiveplanets.aswecanseeclearlyfromtheplanarorbitalconfigurationsshowninfigs2and3，orbitalpositionsoftheterrestrialplanetsdifferlittlebetweentheinitialandfinalpartofeachnumericalintegration，whichspansseveralgyr.thesolidlinesdenotingthepresentorbitsoftheplanetsliealmostwithintheswarmofdotseveninthefinalpartofintegrations(b)and(d).thisindicatesthatthroughouttheentireintegrationperiodthealmostregularvariationsofplanetaryorbitalmotionremainnearlythesameastheyareatpresent.
    verticalviewofthefourinnerplanetaryorbits(fromthezaxisdirection)attheinitialandfinalpartsoftheintegrationsn±1.theaxesunitsareau.thexyplaneissettotheinvariantplaneofsolarsystemtotalangularmomentum.(a)theinitialpartofn+1(t=0to0.0547x109yr).(b)thefinalpartofn+1(t=4.9339x108to4.9886x109yr).(c)theinitialpartofn?1(t=0to?0.0547x109yr).(d)thefinalpartofn?1(t=?3.9180x109to?3.9727x109yr).ineachpanel，atotalof23684pointsareplottedwithanintervalofabout2190yrover5.47x107yr.solidlinesineachpaneldenotethepresentorbitsofthefourterrestrialplanets(takenfromde245).
    thevariationofeccentricitiesandorbitalinclinationsfortheinnerfourplanetsintheinitialandfinalpartoftheintegrationn+1isshowninfig.4.asexpected，thecharacterofthevariationofplanetaryorbitalelementsdoesnotdiffersignificantlybetweentheinitialandfinalpartofeachintegration，atleastforvenus，earthandmars.theelementsofmercury，especiallyitseccentricity，seemtochangetoasignificantextent.thisispartlybecausetheorbitaltimescaleoftheplanetistheshortestofalltheplanets，whichleadstoamorerapidorbitalevolutionthanotherplanets;theinnermostplanetmaybenearesttoinstability.thisresultappearstobeinsomeagreementwithlaskar's(1994，1996)expectationsthatlargeandirregularvariationsappearintheeccentricitiesandinclinationsofmercuryonatimescaleofseveral109yr.however，theeffectofthepossibleinstabilityoftheorbitofmercurymaynotfatallyaffecttheglobalstabilityofthewholeplanetarysystemowingtothesmallmassofmercury.wewillmentionbrieflythelongtermorbitalevolutionofmercurylaterinsection4usinglowpassfilteredorbitalelements.
    theorbitalmotionoftheouterfiveplanetsseemsrigorouslystableandquiteregularoverthistimespan(seealsosection5).
    3.2time–frequencymaps
    althoughtheplanetarymotionexhibitsverylongtermstabilitydefinedasthenonexistenceofcloseencounterevents，thechaoticnatureofplanetarydynamicscanchangetheoscillatoryperiodandamplitudeofplanetaryorbitalmotiongraduallyoversuchlongtimespans.evensuchslightfluctuationsoforbitalvariationinthefrequencydomain，particularlyinthecaseofearth，canpotentiallyhaveasignificanteffectonitssurfaceclimatesystemthroughsolarinsolationvariation(cf.berger1988).
    togiveanoverviewofthelongtermchangeinperiodicityinplanetaryorbitalmotion，weperformedmanyfastfouriertransformations(ffts)alongthetimeaxis，andsuperposedtheresultingperiodgramstodrawtwodimensionaltime–frequencymaps.thespecificapproachtodrawingthesetime–frequencymapsinthispaperisverysimple–muchsimplerthanthewaveletanalysisorlaskar's(1990，1993)frequencyanalysis.
    dividethelowpassfilteredorbitaldataintomanyfragmentsofthesamelenh.thelenhofeachdatasegmentshouldbeamultipleof2inordertoapplythefft.
    eachfragmentofthedatahasalargeoverlappingpart:forexample，whentheithdatabeginsfromt=tiandendsatt=ti+t，thenextdatasegmentrangesfromti+δt≤ti+δt+t，whereδt?t.wecontinuethisdivisionuntilwereachacertainnumbernbywhichtn+treachesthetotalintegrationlenh.
    weapplyanffttoeachofthedatafragments，andobtainnfrequencydiagrams.
    ineachfrequencydiagramobtainedabove，thestrenhofperiodicitycanbereplacedbyagreyscale(orcolour)chart.
    weperformthereplacement，andconnectallthegreyscale(orcolour)chartsintoonegraphforeachintegration.thehorizontalaxisofthesenewgraphsshouldbethetime，i.e.thestartingtimesofeachfragmentofdata(ti，wherei=1，…，n).theverticalaxisrepresentstheperiod(orfrequency)oftheoscillationoforbitalelements.
    wehaveadoptedanfftbecauseofitsoverwhelmingspeed，sincetheamountofnumericaldatatobedecomposedintofrequencycomponentsisterriblyhuge(severaltensofgbytes).
    atypicalexampleofthetime–frequencymapcreatedbytheaboveproceduresisshowninagreyscalediagramasfig.5，whichshowsthevariationofperiodicityintheeccentricityandinclinationofearthinn+2integration.infig.5，thedarkareashowsthatatthetimeindicatedbythevalueontheabscissa，theperiodicityindicatedbytheordinateisstrongerthaninthelighterareaaroundit.wecanrecognizefromthismapthattheperiodicityoftheeccentricityandinclinationofearthonlychangesslightlyovertheentireperiodcoveredbythen+2integration.thisnearlyregulartrendisqualitativelythesameinotherintegrationsandforotherplanets，althoughtypicalfrequenciesdifferplanetbyplanetandelementbyelement.
    4.2longtermexchangeoforbitalenergyandangularmomentum
    wecalculateverylongperiodicvariationandexchangeofplanetaryorbitalenergyandangularmomentumusingfiltereddelaunayelementsl，g，h.gandhareequivalenttotheplanetaryorbitalangularmomentumanditsverticalcomponentperunitmass.lisrelatedtotheplanetaryorbitalenergyeperunitmassase=?μ22l2.ifthesystemiscompletelylinear，theorbitalenergyandtheangularmomentumineachfrequencybinmustbeconstant.nonlinearityintheplanetarysystemcancauseanexchangeofenergyandangularmomentuminthefrequencydomain.theamplitudeofthelowestfrequencyoscillationshouldincreaseifthesystemisunstableandbreaksdowngradually.however，suchasymptomofinstabilityisnotprominentinourlongtermintegrations.
    infig.7，thetotalorbitalenergyandangularmomentumofthefourinnerplanetsandallnineplanetsareshownforintegrationn+2.theupperthreepanelsshowthelongperiodicvariationoftotalenergy(denotedasee0)，totalangularmomentum(gg0)，andtheverticalcomponent(hh0)oftheinnerfourplanetscalculatedfromthelowpassfiltereddelaunayelements.e0，g0，h0denotetheinitialvaluesofeachquantity.theabsolutedifferencefromtheinitialvaluesisplottedinthepanels.thelowerthreepanelsineachfigureshowee0，gg0andhh0ofthetotalofnineplanets.thefluctuationshowninthelowerpanelsisvirtuallyentirelyaresultofthemassivejovianplanets.
    comparingthevariationsofenergyandangularmomentumoftheinnerfourplanetsandallnineplanets，itisapparentthattheamplitudesofthoseoftheinnerplanetsaremuchsmallerthanthoseofallnineplanets:theamplitudesoftheouterfiveplanetsaremuchlargerthanthoseoftheinnerplanets.thisdoesnotmeanthattheinnerterrestrialplanetarysubsystemismorestablethantheouterone:thisissimplyaresultoftherelativesmallnessofthemassesofthefourterrestrialplanetscomparedwiththoseoftheouterjovianplanets.anotherthingwenoticeisthattheinnerplanetarysubsystemmaybecomeunstablemorerapidlythantheouteronebecauseofitsshorterorbitaltimescales.thiscanbeseeninthepanelsdenotedasinner4infig.7wherethelongerperiodicandirregularoscillationsaremoreapparentthaninthepanelsdenotedastotal9.actually，thefluctuationsintheinner4panelsaretoalargeextentasaresultoftheorbitalvariationofthemercury.however，wecannotneglectthecontributionfromotherterrestrialplanets，aswewillseeinsubsequentsections.
    4.4longtermcouplingofseveralneighbouringplanetpairs
    letusseesomeindividualvariationsofplanetaryorbitalenergyandangularmomentumexpressedbythelowpassfiltereddelaunayelements.figs10and11showlongtermevolutionoftheorbitalenergyofeachplanetandtheangularmomentuminn+1andn?2integrations.wenoticethatsomeplanetsformapparentpairsintermsoforbitalenergyandangularmomentumexchange.inparticular，venusandearthmakeatypicalpair.inthefigures，theyshownegativecorrelationsinexchangeofenergyandpositivecorrelationsinexchangeofangularmomentum.thenegativecorrelationinexchangeoforbitalenergymeansthatthetwoplanetsformacloseddynamicalsystemintermsoftheorbitalenergy.thepositivecorrelationinexchangeofangularmomentummeansthatthetwoplanetsaresimultaneouslyundercertainlongtermperturbations.candidatesforperturbersarejupiterandsaturn.alsoinfig.11，wecanseethatmarsshows'itivecorrelationintheangularmomentumvariationtothevenus–earthsystem.mercuryexhibitscertainnegativecorrelationsintheangularmomentumversusthevenus–earthsystem，whichseemstobeareactioncausedbytheconservationofangularmomentumintheterrestrialplanetarysubsystem.
    itisnotclearatthemomentwhythevenus–earthpairexhibitsanegativecorrelationinenergyexchangeand'itivecorrelationinangularmomentumexchange.wemaypossiblyexplainthisthroughobservingthegeneralfactthattherearenoseculartermsinplanetarysemimajoraxesuptosecondorderperturbationtheories(cf.brouwerboccalettipucacco1998).thismeansthattheplanetaryorbitalenergy(whichisdirectlyrelatedtothesemimajoraxisa)mightbemuchlessaffectedbyperturbingplanetsthanistheangularmomentumexchange(whichrelatestoe).hence，theeccentricitiesofvenusandearthcanbedisturbedeasilybyjupiterandsaturn，whichresultsin'itivecorrelationintheangularmomentumexchange.ontheotherhand，thesemimajoraxesofvenusandeartharelesslikelytobedisturbedbythejovianplanets.thustheenergyexchangemaybelimitedonlywithinthevenus–earthpair，whichresultsinanegativecorrelationintheexchangeoforbitalenergyinthepair.
    asfortheouterjovianplanetarysubsystem，jupiter–saturnanduranus–neptuneseemtomakedynamicalpairs.however，thestrenhoftheircouplingisnotasstrongcomparedwiththatofthevenus–earthpair.
    5±5x1010yrintegrationsofouterplanetaryorbits
    sincethejovianplanetarymassesaremuchlargerthantheterrestrialplanetarymasses，wetreatthejovianplanetarysystemasanindependentplanetarysystemintermsofthestudyofitsdynamicalstability.hence，weaddedacoupleoftrialintegrationsthatspan±5x1010yr，includingonlytheouterfiveplanets(thefourjovianplanetspluspluto).theresultsexhibittherigorousstabilityoftheouterplanetarysystemoverthislongtimespan.orbitalconfigurations(fig.12)，andvariationofeccentricitiesandinclinations(fig.13)showthisverylongtermstabilityoftheouterfiveplanetsinboththetimeandthefrequencydomains.althoughwedonotshowmapshere，thetypicalfrequencyoftheorbitaloscillationofplutoandtheotherouterplanetsisalmostconstantduringtheseverylongtermintegrationperiods，whichisdemonstratedinthetime–frequencymapsonourwebpage.
    inthesetwointegrations，therelativenumericalerrorinthetotalenergywas～10?6andthatofthetotalangularmomentumwas～10?10.
    5.1resonancesintheneptune–plutosystem
    kinoshitanakai(1996)integratedtheouterfiveplanetaryorbitsover±5.5x109yr.theyfoundthatfourmajorresonancesbetweenneptuneandplutoaremaintainedduringthewholeintegrationperiod，andthattheresonancesmaybethemaincausesofthestabilityoftheorbitofpluto.themajorfourresonancesfoundinpreviousresearchareasfollows.inthefollowingdescription，λdenotesthemeanlongitude，Ωisthelongitudeoftheascendingnodeand?isthelongitudeofperihelion.subscriptspandndenoteplutoandneptune.
    meanmotionresonancebetweenneptuneandpluto(3:2).thecriticalargumentθ1=3λp?2λn??plibratesaround180°withanamplitudeofabout80°andalibrationperiodofabout2x104yr.
    theargumentofperihelionofplutowp=θ2=?p?Ωplibratesaround90°withaperiodofabout3.8x106yr.thedominantperiodicvariationsoftheeccentricityandinclinationofplutoaresynchronizedwiththelibrationofitsargumentofperihelion.thisisanticipatedinthesecularperturbationtheoryconstructedbykozai(1962).
    thelongitudeofthenodeofplutoreferredtothelongitudeofthenodeofneptune，θ3=Ωp?Ωn，circulatesandtheperiodofthiscirculationisequaltotheperiodofθ2libration.whenθ3becomeszero，i.e.thelongitudesofascendingnodesofneptuneandplutooverlap，theinclinationofplutobecomesmaximum，theeccentricitybecomesminimumandtheargumentofperihelionbecomes90°.whenθ3becomes180°，theinclinationofplutobecomesminimum，theeccentricitybecomesmaximumandtheargumentofperihelionbecomes90°again.williamsbenson(1971)anticipatedthistypeofresonance，laterconfirmedbymilani，nobilicarpino(1989).
    anargumentθ4=?p??n+3(Ωp?Ωn)libratesaround180°withalongperiod，～5.7x108yr.
    inournumericalintegrations，theresonances(i)–(iii)arewellmaintained，andvariationofthecriticalargumentsθ1，θ2，θ3remainsimilarduringthewholeintegrationperiod(figs14–16).however，thefourthresonance(iv)appearstobedifferent:thecriticalargumentθ4alternateslibrationandcirculationovera1010yrtimescale(fig.17).thisisaninterestingfactthatkinoshitanakai's(1995，1996)shorterintegrationswerenotabletodisclose.
    6discussion
    whatkindofdynamicalmechanismmaintainsthislongtermstabilityoftheplanetarysystem?wecanimmediatelythinkoftwomajorfeaturesthatmayberesponsibleforthelongtermstability.first，thereseemtobenosignificantlowerorderresonances(meanmotionandsecular)betweenanypairamongthenineplanets.jupiterandsaturnareclosetoa5:2meanmotionresonance(thefamous‘greatinequality’)，butnotjustintheresonancezone.higherorderresonancesmaycausethechaoticnatureoftheplanetarydynamicalmotion，buttheyarenotsostrongastodestroythestableplanetarymotionwithinthelifetimeoftherealsolarsystem.thesecondfeature，whichwethinkismoreimportantforthelongtermstabilityofourplanetarysystem，isthedifferenceindynamicaldistancebetweenterrestrialandjovianplanetarysubsystems(itotanikawa1999，2001).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)isprobablyoneofthemostsignificantconditionsformaintainingthestabilityoftheplanetarysystemovera109yrtimespan.ourdetailedanalysisoftherelationshipbetweendynamicaldistancebetweenplanetsandtheinstabilitytimescaleofsolarsystemplanetarymotionisnowongoing.
    althoughournumericalintegrationsspanthelifetimeofthesolarsystem，thenumberofintegrationsisfarfromsufficienttofilltheinitialphasespace.itisnecessarytoperformmoreandmorenumericalintegrationstoconfirmandexamineindetailthelongtermstabilityofourplanetarydynamics.
    ——以上文段引自ito，t.tanikawa，k.longtermintegrationsandstabilityofplanetaryorbitsinoursolarsystem.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。