以下是文章内容：
Long-termintegrationsaaryorbitsinourSolarsystem
Abstrac
Wepresenttheresultsofverylong-termnumericaliaryorbitalmotionsover109-yrtime-spansincs.Aquickinspectionofournumericaldatarymotion，atleastinoursimpledynamicalmodel，seemstobequitestableevenoverthisverylongtime-span.Acloserlookatthelowest-frequencyoscillationsusingalow-passfiltershowsusthepotentiallydiffusivecharacterarymotion，especiallythatofMercury.ThentricityofMercuryinourintegrationsisqualitativelysimilartotheresultsfromJacquesLaskar'ssecularperturbationtheory(e.g.emax∼0.35over∼±4Gyr).However，therearenoapparentsecentricityorinclinationinanyorbitals，whichmayberevealedbystilllonger-termnumericalintegrations.Wehavealsoperformedacoupleoftrialintegrationsincludingmotionsosoverthedurationof±5×1010yr.TheresultindicatesthatthethreemajorresonancesintheNeptune–Plutosystemhavebeenmaintainedoverthe1011-yrtime-span
1Introduction
1.1Definitionoftheproblem
ThequestionofthestabilityofourSolarsystemhasbeendebatedoverseveralhundredyears，sincetheeraofNewton.Theproblemhasattractedmanyfamousmathematiciansovertheyearsandhasplayedacentralroleinthedevelopmentofnon-lineardynamicsandchaostheory.However，wedonotyethaveadefiniteanswertothequestionofwhetherourSolarsystemisstableornot.Thisispartlyaresultofthefactthatthedefinitionoftheterm‘stability’isvaguewhenitisusedinrelationtarymotionintheSolarsystem.Actuallyitisnoteasytogiveaclear，rigorousandphysicallymeaningfuldefinitionofthestabilityofourSolarsyste
Amongmanydefinitionsofstability，hereweadopttheHilldefinition(Gladman1993):actuallythisisnotadefinitionofstability，butofinstability.Wedingunstablewhenurssomewhereinthesystem，startingfromacertaininitialconfiguration(Chambers，Wetherill&Boss1996;Ito&Tanikawa1999).AsystemisdefinedasexperiencingacloseencounterwhentwobodiesapproachoneanotherwithinanareaofthelargerHillradius.Otherwisethesystemisdefinedasbeingstable.HenceforwardwarysystemisdynamicallystableifnocloseencounterhappensduringtheageofourSolarsystem，about±5Gyr.Incidentally，thisdefinitionmaybereplaceurrenceofanyorbitalcrossingbetweeneistakesplace.Thisisbecauseweknowfromexperiencethatanorbitalcrossingisverylikelytoleadtoacloaryandarysystems(Yoshinaga，Kokubo&Makino1999).OfcoursethisstatementcannotbesimplyappliedtosystemswithstableorbitalresonancessuchastheNeptune–Plutosyste
1.2Previousstudiesandaimsofthisresearch
Inadditiontothevaguenessoftheconceptofstability，sinourSolarsystemshowacharactertypicalofdynamicalchaos(Sussman&Wisdom1988，1992).Thecauseofthischaoticbehaviourisnowpartlyunderstoodasbeingaresultofresonanceoverlapping(Murray&Holman1999;Lecar，Franklin&Holman2001).However，itwouldrequireintegratingovearysystemsincsforaperiodcoveringseveral10Gyrtothoroughlyunderstandthelong-tearyorbits，sincechaoticdynamicalsystemsarecharacterizedbytheirstrongdependenceoninitialconditions
Fromthatpointofview，manyofthepreviouslong-termnumericalintegrationsincludedonls(Sussman&Wisdom1988;Kinoshita&Nakai1996).Thisisbecausetheorbitalperisaresomuchlongerthanthoseosthatitismucheasiertofollowthesystemforagivenintegrationperiod.Atpresent，thelongestnumericalintegrationspublishedinjournalsarethoseofDuncan&Lissauer(1998).Althoughtheirmaintargetwastheeffectofpost-main-sequencesolarmasslossontaryorbits，theyperformedmanyintegrationscoveringupto∼1011yroftheorbitalmotionsofs.TheinitialorbitalelemensarethesameasthoseofourSolarsysteminDuncan&Lissauer'spaper，buttheydecreasethemassoftheSungraduallyintheirnumericalexperiments.Thisisbecausetheyconsidertheeffectofpost-main-sequencesolarmasnsequently，theyfoundthatthecrossingtime-aryorbits，whichcanbeatypicalindicatoroftheinstabilitytime-scale，isquitesensitivetotherateofmassdecreaseoftheSun.WhenthemassoftheSunisclosetoitspresentvalue，sremainstableover1010yr，orperhapslonger.Duncan&Lissaueralsoperformedfoursimilarexperimentsontheorbitals(VenustoNeptune)，whichcoveraspanof∼109yr.Theirexperisarenotprehensive，butitseemsthattsalsoremainstableduringtheintegrationperiod，maintainingalmostregularoscillations
Ontheotherhand，uratesemi-analyticalsecularperturbationtheory(Laskar1988)，Laskarfindsthatlargeandirregularvariationentricitiesandinclinationsofts，especiallyofMercuryandMarsonatime-scaleofseveral109yr(Laskar1996).TheresultsofLaskar'ssecularperturbationtheoryshouldbeconfirmedandinvestigatedbyfullynumericalintegrations
Inthispaperwepresentpreliminaryresultsofsixlong-termnumericalintegrataryorbits，coveringaspanofseveral109yr，andoftwootherintegrationscoveringaspanof±5×1010yr.Thetotalelapsedtimeforallintegrationsismorethan5yr，usingseveraldedicatedPCsandworkstations.Oneofthefundamentalconclusionsofourlong-termintegrationsistharymotionseemstobestableintermsoftheHillstabilitymentionedabove，atleastoveratime-spanof±4Gyr.Actually，inournumericalintegrationsthesystemwasfarmorestablethanwhatisdefinedbytheHillstabilitycriterion:notonlydidnocloseencounterhappenduringtheintegrationperiod，aryorbitalelementshavebeenconfinedinanarrowregionbothintimeandfrequencydomain，arymotionsarestochastic.Sincethepurposeofthispaperistoexhibitandoverviewtheresultsofourlong-termnumericalintegrations，weshowtypicalexamplefiguresasevidenceoftheverylong-termstabilityarymotion.Forreaderswhohavemorespecificanddeeperinterestsinournumericalresults，wehavepreparedawebpage(access)，whereweshowraworbitalelements，theirlow-passfilteredresults，variationofDelaunayelementsandangularmomentumdeficit，andresultsofoursimpletime–frequencyanalysisonallofourintegrations
InSection2webrieflyexplainourdynamicalmodel，numericalmethodandinitialconditionsusedinourintegrations.Section3isdevotedtoadescriptionofthequickresultsofthenumericalintegrations.Verylong-termstabilityarymotionisaarypositionsandorbitalelements.Aroughestimationofnumericalerrorsisalsogiven.Section4goesontoadiscussionofthelongest-tearyorbitsusingalow-passfilterandincludesadiscussionofangularmomentumdeficit.InSection5，wepresentasetofnumericalintegrationsfosthatspans±5×1010yr.InSection6wealsodiscussthelong-terarymotionanditspossiblecause
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)
Thestepsizeforthenumericalintegrationsis8dthroughoutallintegrats(N±1，2，3)，whichisabout1/11oftheorbitalperiodo(Mercury).Asforthedeterminationofstepsize，wepartlyfollowthepreviousnumericalintegrasinSussman&Wisdom(1988，7.2d)andSaha&Tremaine(1994，225/32d).Weroundedthedecimalpartofthetheirstepsizesto8tomakethestepsizeamultipleof2inoumulationofround-offerrorinputationprocesses.Inrelationtothis，Wisdom&Holman(1991)performednumericalintegrationsoaryorbitsusingthesymplecticmapwithastepsizeof400d，1/10.83oftheorbitalperiodofJupiter.Theirurateenough，whichpartlyjustifiesourmethodofdeterminingthestepsize.However，entricityofJupiter(∼0.05)ismuchsmallerthanthatofMercury(∼0.2)，weneparetheseintegrationssimplyintermsofstepsizes
Intheintegrationos(F±)，wefixedthestepsizeat400d
WeadoptGauss'fandgfunctionsinthesymplecticmaptogetherwiththethird-orderHalleymethod(Danby1992)asasolverforKeplerequations.ThenumberofmaximumiterationswesetinHalley'smethodis15，buttheyneverreachedthemaximuminanyofourintegrations
Theintervalofthedataoutputis200000d(∼547yr)forthecalculats(N±1，2，3)，andabout8000000d(∼21903yr)fortheintegrationos(F±)
Althoughnooutputfilteringwasdonewhenthenumericalintegrationswereinprocess，weappliedalow-passfiltertotheraworbitaldataafterwepletedallthecalculations.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.arylongitudes
SincethesymplecticmapspreservetotalenergyandtotalangularmomentumofN-bodydynamicalsystemsinherentlywell，thedegreeoftheirpreservationmaynotbeaguracyofnumericalintegrations，especiallyasameasureoftheposis，i.e.arylongitudes.Toestimatethenumeriarylongitudes，weperformedthefollowingprocedures.Wecomparedtheresultofourmainlong-termintegrationswithsometestintegrations，whichspanmuchshorterperiodsbuuracythanthemainintegrations.Forthispurpose，weperurateintegrationwithastepsizeof0.125d(1/64ofthemainintegrations)spanning3×105yr，startingwiththesameinitialconditionsasintheN−1integration.Weconsiderthatthistestintegrationprovidesuswitha‘pseudo-true’aryorbitalevolution.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-tearyorbitalmotionthroughsomesnapshotsofrawnumericaldata.Theorsindicateslong-termstabilityinallofournumericalintegrations:noorbitalcrossingsnorcloseencountersbetstookplace
3.1Generaldescriptionoftaryorbits
First，webrieflylookatthegeneralcharacterofthelong-tearyorbits.Ourinterestherefocusesparticularlyontheinnerfosforwhichtheorbitaltime-scalesaremuchshorterthanthoseos.AswecanseeclearlyfromtheplanarorbitalconfigurationsshowninFigs2and3，orbitalpositionsoftsdifferlittlebetweentheinitialandfinalpartofeachnumericalintegration，whichspansseveralGyr.Thesolidlinesdenotingthepresesliealmostwithintheswarmofdotseveninthefinalpartofintegrations(b)and(d).Thisindicatesthatthroughouttheentireintegrationperiodthealmostregulaaryorbitalmotionremainnearlythesameastheyareatpresen
Verticalviewoaryorbits(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.Solidlinesineachpaneldenotethepresentorbitsofthefos(takenfromDE245)
entricitiesandorbitalinclinationsfosintheinitialandfinalpartoftheintegrationN+1isshowninFig.4.Asexpected，thecharacteroftaryorbitalelementsdoesnotdiffersignificantlybetweentheinitialandfinalpartofeachintegration，atleastforVenus，EarthandMars.TheelementsofMercury，entricity，seemtochangetoasignificantextent.Thisispartlybecausetheorbitaltime-istheshos，whichleadstoamorerapidorbitalevolus;maybenearesttoinstability.ThisresultappearstobeinsomeagreementwithLaskar's(1994，1996)expectationsthatlargeandirregularvariatentricitiesandinclinationsofMercuryonatime-scaleofseveral109yr.However，theeffectofthepossibleinstabilityoftheorbitofMercurymaynotfatallyaffecttheglobalstabilarysystemowingtothesmallmassofMercury.Wewillmentionbrieflythelong-termorbitalevolutionofMercurylaterinSection4usinglow-passfilteredorbitalelements
Theorbitalmotionosseemsrigorouslystableandquiteregularoverthistime-span(seealsoSection5)
3.2Time–frequencymaps
arymotionexhibitsverylong-termstabilitydefinedasthenon-existenceofcloseencounterevents，arydynamicscanchangetheoscillatoryperiodaaryorbitalmotiongraduallyoversuchlongtime-spans.Evensuchslightfluctuationsoforbitalvariationinthefrequencydomain，particularlyinthecaseofEarth，canpotentiallyhaveasignificanteffectonitssurfaceclimatesystemthroughsolarinsolationvariation(cf.Berger198
Togiveanoverviewofthelong-termchangeinaryorbitalmotion，weperformedmanyfastFouriertransformations(FFTs)alongthetimeaxis，andsuperposedtheresultingperiodgramstodrawtwo-dimensionaltime–frequencymaps.Thespecificapproachtodrawingthesetime–frequencymapsinthispaperisverysimple–muchsimplerthanthewaveletanalysisorLaskar's(1990，1993)frequencyanalysis
Dividethelow-passfilteredorbitaldataintomanyfragmentsofthesamelength.Thelengthofeachdatasegmentshouldbeamultipleof2inordertoapplytheFF
Eachfragmentofthedatahasalargeoverlappingpart:forexample，whentheithdatabeginsfromt=tiandendsatt=ti+T，thenextdatasegmentrangesfromti+δT≤ti+δT+T，whereδT?T.WecontinuethisdivisionuntilwereachacertainnumberNbywhichtn+Treachesthetotalintegrationlength
WeapplyanFFTtoeachofthedatafragments，andobtainnfrequencydiagrams
Ineachfrequencydiagramobtainedabove，thestrengthofperiodicitycanbereplacedbyagrey-scale(orcolour)char
Weperformthereplacement，andconnectallthegrey-scale(orcolour)chartsintoonegraphforeachintegration.Thehorizontalaxisofthesenewgraphsshouldbethetime，i.e.thestartingtimesofeachfragmentofdata(ti，wherei=1，…，n).Theverticalaxisrepresentstheperiod(orfrequency)oftheoscillationoforbitalelements
WehaveadoptedanFFTbecauseofitsoverwhelmingspeed，sincetheamountofnuposedintoponentsisterriblyhuge(severaltensofGbytes)
Atypicalexampleofthetime–frequencymapcreatedbytheaboveproceduresisshowninagrey-scalediagramasFig.5，whichshowsthevariationofpentricityandinclinationofEarthinN+2integration.InFig.5，thedarkareashowsthatatthetimeindicatedbythevalueontheabscissa，theperiodicityindicatedbytheordinateisstrongerthaninthelighterareaaroundit.WecanrecognizefromthismapthatthepentricityandinclinationofEarthonlychangesslightlyovertheentireperiodcoveredbytheN+2integration.Thisnearlyregulartrendisqualitativelythesameinotherintegratios，althoughtypicalfreqandelementbyelemen
4.2Long-termexchangeoforbitalenergyandangularmomentum
Wecalculateverylong-periodicvariationaryorbitalenergyandangularmomentumusingfilteredDelaunayelementsL，G，H.GandHareeqaryorbitalangularmomentumanditsponentperunitmass.LiaryorbitalenergyEperunitmassasE=−μ2/2L2.pletelylinear，theorbitalenergyandtheangularmomentumineachfrequencybinmustbeconstant.Non-larysystemcancauseanexchangeofenergyandangularmomentuminthefrequencydomain.Theamplitudeofthelowest-frequencyoscillationshouldincreaseifthesystemisunstableandbreaksdowngradually.However，suchasymptomofinstabilityisnotprominentinourlong-termintegrations
InFig.7，thetotalorbitalenergyandangularmomentusareshownforintegrationN+2.Theupperthreepanelsshowthelong-periodicvariationoftotalenergy(denotedasE-E0)，totalangularmomentum(G-G0)，andtheponent(H-H0)oscalculatedfromthelow-passfilteredDelaunayelements.E0，G0，H0denotetheinitialvaluesofeachquantity.Theabsolutedifferencefromtheinitialvaluesisplottedinthepanels.ThelowerthreepanelsineachfigureshowE-E0，G-G0andH-H0ofts.Thefluctuationshowninthelowerpanelsisvirtuallyentirelyaresultofthes
&paringthevariationsofenergyandangularmomentus，itisapparentthattheamplitudesofthsaremuchsmallerthants:theamplitudesosaremuchlargerthanths.Thisdoesnotmeanthattheinnarysubsystemismorestablethantheouterone:thisissimplyaresultoftherelativesmallnessofthemassesofthefourterrestrialparedwiththoseofts.Anotherthingwenoticeiaeunstablemorerapidlythantheouteronebecauseofitsshorterorbitaltime-scales.Thiscanbeseeninthepanelsdenotedasinner4inFig.7wherethelonger-periodicandirregularoscillationsaremoreapparentthaninthepanelsdenotedastotal9.Actually，thefluctuationsintheinner4panelsaretoalargeextentasaresultoftheorbitalvariationoftheMercury.However，wecannotneglectthecontributionfromoths，aswewillseeinsubsequentsections
4.4Long-termcouplingofseverapai
Letusseesomeindividuaaryorbitalenergyandangularmomentumexpressedbythelow-passfilteredDelaunayelements.Figs10and11showlong-termevolutionoftheorbitaandtheangularmomentuminN+1andN−2integrations.Wensformapparentpairsintermsoforbitalenergyandangularmomentumexchange.Inparticular，VenusandEarthmakeatypicalpair.Inthefigures，theyshownegativecorrelationsinexchangeofenergyandpositivecorrelationsinexchangeofangularmomentum.Thenegativecorrelationinexchangeoforbitalenergysformacloseddynamicalsystemintermsoftheorbitalenergy.Thepositivecorrelationinexchangeofangularmomentumsaresimultaneouslyundercertainlong-termperturbations.CandidatesforperturbersareJupiterandSaturn.AlsoinFig.11，wecanseethatMarsshowsapositivecorrelationintheangularmomentumvariationtotheVenus–Earthsystem.MercuryexhibitscertainnegativecorrelationsintheangularmomentumversustheVenus–Earthsystem，whichseemstobeareactioncausedbytheconservationofangularmomentumintarysubsyste
ItisnotclearatthemomentwhytheVenus–Earthpairexhibitsanegativecorrelationinenergyexchangeandapositivecorrelationinangularmomentumexchange.Wemaypossiblyexplainthisthroughobservingthegeneralfactthattherearenosarysemimajoraxesuptosecond-orderperturbationtheories(cf.Brouwer&Clemence1961;Boccaletti&o1998).Thiaryorbitalenergy(whichisdirectlyrelatedtothesemimajoraxisa)mightbemuchlessaffectesthanistheangularmomentumexchange(whichrelatestoe).Hence，entricitiesofVenusandEarthcanbedisturbedeasilybyJupiterandSaturn，whichresultsinapositivecorrelationintheangularmomentumexchange.Ontheotherhand，thesemimajoraxesofVenusandEartharelesslikelytobedisturbs.ThustheenergyexchangemaybelimitedonlywithintheVenus–Earthpair，whichresultsinanegativecorrelationintheexchangeoforbitalenergyinthepair
Asfortarysubsystem，Jupiter–SaturnandUranus–Neptuneseemtomakedynamicalpairs.However，thestrengthoftheircouplingisnotasparedwiththatoftheVenus–Earthpair
5±5×1010-yrintegraryorbits
Sarymassesaremuchlargerthantarymasses，wetarysystemasarysystemintermsofthestudyofitsdynamicalstability.Hence，weaddedacoupleoftrialintegrationsthatspan±5×1010yr，includingonls(splusPluto).Theresultsexhibittherigorousstabilarysystemoverthislongtime-span.Orbitalconfigurations(Fig.12)，entricitiesandinclinations(Fig.13)showthisverylong-termstabilityosinboththetimeandthefrequencydomains.Althoughwedonotshowmapshere，thetypicalfrequencyoftheorbitaloscillationofPlutoandsisalmostconstantduringtheseverylong-termintegrationperiods，whichisdemonstratedinthetime–frequencymapsonourwebpage
Inthesetwointegrations，therelativenumericalerrorinthetotalenergywas∼10−6andthatofthetotalangularmomentumwas∼10−10
5.1ResonancesintheNeptune–Plutosystem
Kinoshita&Nakai(1996)integratearyorbitsover±5.5×109yr.TheyfoundthatfourmajorresonancesbetweenNeptuneandPlutoaremaintainedduringthewholeintegrationperiod，andthattheresonancesmaybethemaincausesofthestabilityoftheorbitofPluto.Themajorfourresonancesfoundinpreviousresearchareasfollows.Inthefollowingdescription，λdenotesthemeanlongitude，Ωisthelongitudeoftheascendingnodeandϖisthelongitudeofperihelion.SubscriptsPandNdenotePlutoandNeptune
MeanmotionresonancebetweenNeptuneandPluto(3:2).Thecriticalargumentθ1=3λP−2λN−ϖPlibratesaround180°withanamplitudeofabout80°andalibrationperiodofabout2×104yr
TheargumentofperihelionofPlutoωP=θ2=ϖP−ΩPlibratesaround90°withaperiodofabout3.8×106yr.ThedominantperiodicentricityandinclinationofPlutoaresynchronizedwiththelibrationofitsargumentofperihelion.ThisisanticipatedinthesecularperturbationtheoryconstructedbyKozai(1962)
ThelongitudeofthenodeofPlutoreferredtothelongitudeofthenodeofNeptune，θ3=ΩP−ΩN，circulatesandtheperiodofthiscirculationisequaltotheperiodofθ2libration.Whenθ3becomeszero，i.e.thelongitudesofascendingnodesofNeptuneandPlutooverlap，theinclesmaximum，esminimumandtheargues90°.Whenθ3becomes180°，theinclesminimum，esmaximumandtheargues90°again.Williams&Benson(1971)anticipatedthistypeofresonance，laterconfirmedbyMilani，Nobili&Carpino(1989)
Anargumentθ4=ϖP−ϖN+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-terarysystem?Wecanimmediatelythinkoftwomajorfeaturesthatmayberesponsibleforthelong-termstability.First，thereseemtobenosignificantlower-orderresonances(meanmotionandsecular)betweenanypais.JupiterandSaturnareclosetoa5:2meanmotionresonance(thefamous‘greatinequality’)，butnotjustintheresonancezone.Higher-orderresonancesarydynamicalmotion，buttheyarenotsostrongastodesarymotionwithinthelifetimeoftherealSolarsystem.Thesecondfeature，whichwethinkismoreimportantforthelong-terarysystem，isthedifferenceindynamicaldistancebetweenterrestarysubsystems(Ito&Tanikawa1999，2001).aryseparationsbythemutualHillradii(R_)，separationsasaregreaterthan26RH，whereasthosarelessthan14RH.Thisdifferenceisdirectlyrelatedtothedifferencebetweendynamicalfeaturesofterrests.shavesmallermasses，shorterorbitalperiodsandwiderdynamicalseparation.Theyarestronglypertsthathavelargermasses，longerorbitalperiodsandnarrowerdynamicalseparation.sarenotperturbedbyanyothermassivebodies
Thepresearysystemisstillbeingdisturbedbythes.However，thewideseparationandmutualinteractionamongtsrendersthedisturbanceineffective;thedegreeofdistursisO(eJ)(orderoentricityofJupiter)，sincethedisturbancecsisaforcedoscillationhavinganamplitudeofO(eJ).entricity，forexampleO(eJ)∼0.05，isfarfromsufficienttoprovokeinstabilityintshavingsuchawideseparationas26RH.Thusweassumethatthepresentwidedynamicalseparationas(>26RH)isprobablyoneofthemostsignificantconditionsformaintainingthesarysystemovera109-yrtime-span.Ourdetailedanalysisoftherelationshipbetweendynamicaldisandtheinstabilitytime-scalearymotionisnowon-going
AlthoughournumericalintegrationsspanthelifetimeoftheSolarsystem，thenumberofintegrationsisfarfromsufficienttofilltheinitialphasespace.Itisnecessarytoperformmoreandmorenumericalintegrationstoconfirmandexamineindetailthelong-terarydynamics
——以上文段引自Ito，T.&Tanikawa，K.Long-termintegrationsaaryorbitsinourSolarSystem.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。
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