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-termintegrationsandstabilityofpnetaryorbitsinourSorsystem
Abstract
Wepresenttheresultsofverylong-termnumericalintegrationsofpnetaryorbitalmotionsover109-yrtime-spansincludingallninepnets.Aquickinspectionofournumericaldatashowsthatthepnetarymotion，atleastinoursimpledynamicalmodel，seemstobequitestableevenoverthisverylongtime-span.Acloserlookatthelowest-frequencyosciltionsusingalow-passfiltershowsusthepotentiallydiffusivecharacterofterrestrialpnetarymotion，especiallythatofMercury.ThebehaviouroftheeccentricityofMercuryinourintegrationsisqualitativelysimirtotheresultsfromJacquesskar'ssecurperturbationtheory(e.g.emax0.35over±4Gyr).However，therearenoapparentsecurincreasesofeccentricityorinclinationinanyorbitalelementsofthepnets，whichmayberevealedbystilllonger-termnumericalintegrations.Wehavealsoperformedacoupleoftrialintegrationsincludingmotionsoftheouterfivepnetsoverthedurationof±5×1010yr.TheresultindicatesthatthethreemajorresonancesintheNeptune–Plutosystemhavebeenmaintainedoverthe1011-yrtime-span.
1Introduction
1.1Definitionoftheproblem
ThequestionofthestabilityofourSorsystemhasbeendebatedoverseveralhundredyears，sincetheeraofNewton.Theproblemhasattractedmanyfamousmathematiciansovertheyearsandhaspyedacentralroleinthedevelopmentofnon-lineardynamicsandchaostheory.However，wedonotyethaveadefiniteanswertothequestionofwhetherourSorsystemisstableornot.Thisispartlyaresultofthefactthatthedefinitionoftheterm‘stability’isvaguewhenitisusedinretiontotheproblemofpnetarymotionintheSorsystem.Actuallyitisnoteasytogiveaclear，rigorousandphysicallymeaningfuldefinitionofthestabilityofourSorsystem.
Amongmanydefinitionsofstability，hereweadopttheHilldefinition(Gdman1993):actuallythisisnotadefinitionofstability，butofinstability.Wedefineasystemasbecomingunstablewhenacloseencounteroccurssomewhereinthesystem，startingfromacertaininitialconfiguration(Chambers，Wetherilmp;amp;Boss1996;Itoamp;amp;Tanikawa1999).AsystemisdefinedasexperiencingacloseencounterwhentwobodiesapproachoneanotherwithinanareaofthergerHillradius.Otherwisethesystemisdefinedasbeingstable.HenceforwardwestatethatourpnetarysystemisdynamicallystableifnocloseencounterhappensduringtheageofourSorsystem，about±5Gyr.Incidentally，thisdefinitionmayberepcedbyoneinwhichanoccurrenceofanyorbitalcrossingbetweeneitherofapairofpnetstakespce.Thisisbecauseweknowfromexperiencethatanorbitalcrossingisverylikelytoleadtoacloseencounterinpnetaryandprotopnetarysystems(Yoshinaga，Kokuboamp;amp;Makino1999).OfcoursethisstatementcannotbesimplyappliedtosystemswithstableorbitalresonancessuchastheNeptune–Plutosystem.
1.2Previousstudiesandaimsofthisresearch
Inadditiontothevaguenessoftheconceptofstability，thepnetsinourSorsystemshowacharactertypicalofdynamicalchaos(Sussmanamp;amp;Wisdom1988，1992).Thecauseofthischaoticbehaviourisnowpartlyunderstoodasbeingaresultofresonanceoverpping(Murrayamp;amp;Holman1999;Lecar，Franklinamp;amp;Holman2001).However，itwouldrequireintegratingoveranensembleofpnetarysystemsincludingallninepnetsforaperiodcoveringseveral10Gyrtothoroughlyunderstandthelong-termevolutionofpnetaryorbits，sincechaoticdynamicalsystemsarecharacterizedbytheirstrongdependenceoninitialconditions.
Fromthatpointofview，manyofthepreviouslong-termnumericalintegrationsincludedonlytheouterfivepnets(Sussmanamp;amp;Wisdom1988;Kinoshitaamp;amp;Nakai1996).Thisisbecausetheorbitalperiodsoftheouterpnetsaresomuchlongerthanthoseoftheinnerfourpnetsthatitismucheasiertofollowthesystemforagivenintegrationperiod.Atpresent，thelongestnumericalintegrationspublishedinjournalsarethoseofDuncanamp;amp;Lissauer(1998).Althoughtheirmaintargetwastheeffectofpost-main-sequencesormasslossonthestabilityofpnetaryorbits，theyperformedmanyintegrationscoveringupto1011yroftheorbitalmotionsofthefourjovianpnets.TheinitialorbitalelementsandmassesofpnetsarethesameasthoseofourSorsysteminDuncanamp;amp;Lissauer'spaper，buttheydecreasethemassoftheSungraduallyintheirnumericalexperiments.Thisisbecausetheyconsidertheeffectofpost-main-sequencesormasslossinthepaper.Consequently，theyfoundthatthecrossingtime-scaleofpnetaryorbits，whichcanbeatypicalindicatoroftheinstabilitytime-scale，isquitesensitivetotherateofmassdecreaseoftheSun.WhenthemassoftheSunisclosetoitspresentvalue，thejovianpnetsremainstableover1010yr，orperhapslonger.Duncanamp;amp;Lissaueralsoperformedfoursimirexperimentsontheorbitalmotionofsevenpnets(VenustoNeptune)，whichcoveraspanof109yr.Theirexperimentsonthesevenpnetsarenotyetcomprehensive，butitseemsthattheterrestrialpnetsalsoremainstableduringtheintegrationperiod，maintainingalmostregurosciltions.
Ontheotherhand，inhisaccuratesemi-analyticalsecurperturbationtheory(skar1988)，skarfindsthatrgeandirregurvariationscanappearintheeccentricitiesandinclinationsoftheterrestrialpnets，especiallyofMercuryandMarsonatime-scaleofseveral109yr(skar1996).Theresultsofskar'ssecurperturbationtheoryshouldbeconfirmedandinvestigatedbyfullynumericalintegrations.
Inthispaperwepresentpreliminaryresultsofsixlong-termnumericalintegrationsonallninepnetaryorbits，coveringaspanofseveral109yr，andoftwootherintegrationscoveringaspanof±5×1010yr.Thetotalepsedtimeforallintegrationsismorethan5yr，usingseveraldedicatedPCsandworkstations.Oneofthefundamentalconclusionsofourlong-termintegrationsisthatSorsystempnetarymotionseemstobestableintermsoftheHillstabilitymentionedabove，atleastoveratime-spanof±4Gyr.Actually，inournumericalintegrationsthesystemwasfarmorestablethanwhatisdefinedbytheHillstabilitycriterion:notonlydidnocloseencounterhappenduringtheintegrationperiod，butalsoallthepnetaryorbitalelementshavebeenconfinedinanarrowregionbothintimeandfrequencydomain，thoughpnetarymotionsarestochastic.Sincethepurposeofthispaperistoexhibitandoverviewtheresultsofourlong-termnumericalintegrations，weshowtypicalexamplefiguresasevidenceoftheverylong-termstabilityofSorsystempnetarymotion.Forreaderswhohavemorespecificanddeeperinterestsinournumericalresults，wehavepreparedawebpage(access)，whereweshowraworbitalelements，theirlow-passfilteredresults，variationofDeunayelementsandangurmomentumdeficit，andresultsofoursimpletime–frequencyanalysisonallofourintegrations.
InSection2webrieflyexpinourdynamicalmodel，numericalmethodandinitialconditionsusedinourintegrations.Section3isdevotedtoadescriptionofthequickresultsofthenumericalintegrations.Verylong-termstabilityofSorsystempnetarymotionisapparentbothinpnetarypositionsandorbitalelements.Aroughestimationofnumericalerrorsisalsogiven.Section4goesontoadiscussionofthelongest-termvariationofpnetaryorbitsusingalow-passfilterandincludesadiscussionofangurmomentumdeficit.InSection5，wepresentasetofnumericalintegrationsfortheouterfivepnetsthatspans±5×1010yr.InSection6wealsodiscussthelong-termstabilityofthepnetarymotionanditspossiblecause.
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(Wisdomamp;amp;Holman1991;Kinoshita，Yoshidaamp;amp;Nakai1991)withaspecialstart-upproceduretoreducethetruncationerrorofanglevariables，‘warmstart’(Sahaamp;amp;Tremaine1992，1994).
Thestepsizeforthenumericalintegrationsis8dthroughoutallintegrationsoftheninepnets(N±1，2，3)，whichisabout111oftheorbitalperiodoftheinnermostpnet(Mercury).Asforthedeterminationofstepsize，wepartlyfollowthepreviousnumericalintegrationofallninepnetsinSussmanamp;amp;Wisdom(1988，7.2d)andSahaamp;amp;Tremaine(1994，22532d).Weroundedthedecimalpartofthetheirstepsizesto8tomakethestepsizeamultipleof2inordertoreducetheaccumutionofround-offerrorinthecomputationprocesses.Inretiontothis，Wisdomamp;amp;Holman(1991)performednumericalintegrationsoftheouterfivepnetaryorbitsusingthesymplecticmapwithastepsizeof400d，110.83oftheorbitalperiodofJupiter.Theirresultseemstobeaccurateenough，whichpartlyjustifiesourmethodofdeterminingthestepsize.However，sincetheeccentricityofJupiter(0.05)ismuchsmallerthanthatofMercury(0.2)，weneedsomecarewhenwecomparetheseintegrationssimplyintermsofstepsizes.
Intheintegrationoftheouterfivepnets(F±)，wefixedthestepsizeat400d.
WeadoptGauss'fandgfunctionsinthesymplecticmaptogetherwiththethird-orderHalleymethod(Danby1992)asasolverforKeplerequations.ThenumberofmaximumiterationswesetinHalley'smethodis15，buttheyneverreachedthemaximuminanyofourintegrations.
Theintervalofthedataoutputis200000d(547yr)forthecalcutionsofallninepnets(N±1，2，3)，andabout8000000d(21903yr)fortheintegrationoftheouterfivepnets(F±).
Althoughnooutputfilteringwasdonewhenthenumericalintegrationswereinprocess，weappliedalow-passfiltertotheraworbitaldataafterwehadcompletedallthecalcutions.SeeSection4.1formoredetail.
2.4Errorestimation
2.4.1Retiveerrorsintotalenergyandangurmomentum
Accordingtooneofthebasicpropertiesofsymplecticintegrators，whichconservethephysicallyconservativequantitieswell(totalorbitalenergyandangurmomentum)，ourlong-termnumericalintegrationsseemtohavebeenperformedwithverysmallerrors.Theaveragedretiveerrorsoftotalenergy(109)andoftotangurmomentum(1011)haveremainednearlyconstantthroughouttheintegrationperiod(Fig.1).Thespecialstartupprocedure，warmstart，wouldhavereducedtheaveragedretiveerrorintotalenergybyaboutoneorderofmagnitudeormore.
RetivenumericalerrorofthetotangurmomentumδAA0andthetotalenergyδEE0inournumericalintegrationsN±1，2，3，whereδEandδAaretheabsolutechangeofthetotalenergyandtotangurmomentum，respectively，andE0andA0aretheirinitialvalues.ThehorizontalunitisGyr.
Notethatdifferentoperatingsystems，differentmathematicallibraries，anddifferenthardwarearchitecturesresultindifferentnumericalerrors，throughthevariationsinround-offerrorhandlingandnumericalgorithms.IntheupperpanelofFig.1，wecanrecognizethissituationinthesecurnumericalerrorinthetotangurmomentum，whichshouldberigorouslypreserveduptomachine-εprecision.
2.4.2Errorinpnetarylongitudes
SincethesymplecticmapspreservetotalenergyandtotangurmomentumofN-bodydynamicalsystemsinherentlywell，thedegreeoftheirpreservationmaynotbeagoodmeasureoftheaccuracyofnumericalintegrations，especiallyasameasureofthepositionalerrorofpnets，i.e.theerrorinpnetarylongitudes.Toestimatethenumericalerrorinthepnetarylongitudes，weperformedthefollowingprocedures.Wecomparedtheresultofourmainlong-termintegrationswithsometestintegrations，whichspanmuchshorterperiodsbutwithmuchhigheraccuracythanthemainintegrations.Forthispurpose，weperformedamuchmoreaccurateintegrationwithastepsizeof0.125d(164ofthemainintegrations)spanning3×105yr，startingwiththesameinitialconditionsasintheN1integration.Weconsiderthatthistestintegrationprovidesuswitha‘pseudo-true’solutionofpnetaryorbitalevolution.Next，wecomparethetestintegrationwiththemainintegration，N1.Fortheperiodof3×105yr，weseeadifferenceinmeananomaliesoftheEarthbetweenthetwointegrationsof0.52°(inthecaseoftheN1integration).Thisdifferencecanbeextrapotedtothevalue8700°，about25rotationsofEarthafter5Gyr，sincetheerroroflongitudesincreaseslinearlywithtimeinthesymplecticmap.Simirly，thelongitudeerrorofPlutocanbeestimatedas12°.ThisvalueforPlutoismuchbetterthantheresultinKinoshitaamp;amp;Nakai(1996)wherethedifferenceisestimatedas60°.
3Numericalresults–I.Gnceattherawdata
Inthissectionwebrieflyreviewthelong-termstabilityofpnetaryorbitalmotionthroughsomesnapshotsofrawnumericaldata.Theorbitalmotionofpnetsindicateslong-termstabilityinallofournumericalintegrations:noorbitalcrossingsnorcloseencountersbetweenanypairofpnetstookpce.
3.1Generaldescriptionofthestabilityofpnetaryorbits
First，webrieflylookatthegeneralcharacterofthelong-termstabilityofpnetaryorbits.Ourinterestherefocusesparticurlyontheinnerfourterrestrialpnetsforwhichtheorbitaltime-scalesaremuchshorterthanthoseoftheouterfivepnets.AswecanseeclearlyfromthepnarorbitalconfigurationsshowninFigs2and3，orbitalpositionsoftheterrestrialpnetsdifferlittlebetweentheinitiandfinalpartofeachnumericalintegration，whichspansseveralGyr.Thesolidlinesdenotingthepresentorbitsofthepnetsliealmostwithintheswarmofdotseveninthefinalpartofintegrations(b)and(d).Thisindicatesthatthroughouttheentireintegrationperiodthealmostregurvariationsofpnetaryorbitalmotionremainnearlythesameastheyareatpresent.
Verticalviewofthefourinnerpnetaryorbits(fromthez-axisdirection)attheinitiandfinalpartsoftheintegrationsN±1.Theaxesunitsareau.Thexy-pneissettotheinvariantpneofSorsystemtotangurmomentum.(a)TheinitialpartofN+1(t=0to0.0547×109yr).(b)ThefinalpartofN+1(t=4.9339×108to4.9886×109yr).(c)TheinitialpartofN1(t=0to0.0547×109yr).(d)ThefinalpartofN1(t=3.9180×109to3.9727×109yr).Ineachpanel，atotalof23684pointsareplottedwithanintervalofabout2190yrover5.47×107yr.Solidlinesineachpaneldenotethepresentorbitsofthefourterrestrialpnets(takenfromDE245).
ThevariationofeccentricitiesandorbitalinclinationsfortheinnerfourpnetsintheinitiandfinalpartoftheintegrationN+1isshowninFig.4.Asexpected，thecharacterofthevariationofpnetaryorbitalelementsdoesnotdiffersignificantlybetweentheinitiandfinalpartofeachintegration，atleastforVenus，EarthandMars.TheelementsofMercury，especiallyitseccentricity，seemtochangetoasignificantextent.Thisispartlybecausetheorbitaltime-scaleofthepnetistheshortestofallthepnets，whichleadstoamorerapidorbitalevolutionthanotherpnets;theinnermostpnetmaybenearesttoinstability.Thisresultappearstobeinsomeagreementwithskar's(1994，1996)expectationsthatrgeandirregurvariationsappearintheeccentricitiesandinclinationsofMercuryonatime-scaleofseveral109yr.However，theeffectofthepossibleinstabilityoftheorbitofMercurymaynotfatallyaffecttheglobalstabilityofthewholepnetarysystemowingtothesmallmassofMercury.Wewillmentionbrieflythelong-termorbitalevolutionofMercuryterinSection4usinglow-passfilteredorbitalelements.
Theorbitalmotionoftheouterfivepnetsseemsrigorouslystableandquitereguroverthistime-span(seealsoSection5).
3.2Time–frequencymaps
Althoughthepnetarymotionexhibitsverylong-termstabilitydefinedasthenon-existenceofcloseencounterevents，thechaoticnatureofpnetarydynamicscanchangetheosciltoryperiodandamplitudeofpnetaryorbitalmotiongraduallyoversuchlongtime-spans.Evensuchslightfluctuationsoforbitalvariationinthefrequencydomain，particurlyinthecaseofEarth，canpotentiallyhaveasignificanteffectonitssurfaceclimatesystemthroughsorinsotionvariation(cf.Berger1988).
Togiveanoverviewofthelong-termchangeinperiodicityinpnetaryorbitalmotion，weperformedmanyfastFouriertransformations(FFTs)alongthetimeaxis，andsuperposedtheresultingperiodgramstodrawtwo-dimensionaltime–frequencymaps.Thespecificapproachtodrawingthesetime–frequencymapsinthispaperisverysimple–muchsimplerthanthewaveletanalysisorskar's(1990，1993)frequencyanalysis.
Dividethelow-passfilteredorbitaldataintomanyfragmentsofthesamelenh.Thelenhofeachdatasegmentshouldbeamultipleof2inordertoapplytheFFT.
Eachfragmentofthedatahasargeoverppingpart:forexample，whentheithdatabeginsfromt=tiandendsatt=ti+T，thenextdatasegmentrangesfromti+δT≤ti+δT+T，whereδT?T.WecontinuethisdivisionuntilwereachacertainnumberNbywhichtn+Treachesthetotalintegrationlenh.
WeapplyanFFTtoeachofthedatafragments，andobtainnfrequencydiagrams.
Ineachfrequencydiagramobtainedabove，thestrenhofperiodicitycanberepcedbyagrey-scale(orcolour)chart.
Weperformtherepcement，andconnectallthegrey-scale(orcolour)chartsintoonegraphforeachintegration.Thehorizontaxisofthesenewgraphsshouldbethetime，i.e.thestartingtimesofeachfragmentofdata(ti，wherei=1，…，n).Theverticaxisrepresentstheperiod(orfrequency)oftheosciltionoforbitalelements.
WehaveadoptedanFFTbecauseofitsoverwhelmingspeed，sincetheamountofnumericaldatatobedecomposedintofrequencycomponentsisterriblyhuge(severaltensofGbytes).
Atypicalexampleofthetime–frequencymapcreatedbytheaboveproceduresisshowninagrey-scalediagramasFig.5，whichshowsthevariationofperiodicityintheeccentricityandinclinationofEarthinN+2integration.InFig.5，thedarkareashowsthatatthetimeindicatedbythevalueontheabscissa，theperiodicityindicatedbytheordinateisstrongerthaninthelighterareaaroundit.WecanrecognizefromthismapthattheperiodicityoftheeccentricityandinclinationofEarthonlychangesslightlyovertheentireperiodcoveredbytheN+2integration.Thisnearlyregurtrendisqualitativelythesameinotherintegrationsandforotherpnets，althoughtypicalfrequenciesdifferpnetbypnetandelementbyelement.
4.2Long-termexchangeoforbitalenergyandangurmomentum
Wecalcuteverylong-periodicvariationandexchangeofpnetaryorbitalenergyandangurmomentumusingfilteredDeunayelementsL，G，H.GandHareequivalenttothepnetaryorbitangurmomentumanditsverticalcomponentperunitmass.LisretedtothepnetaryorbitalenergyEperunitmassasE=μ22L2.Ifthesystemiscompletelylinear，theorbitalenergyandtheangurmomentumineachfrequencybinmustbeconstant.Non-linearityinthepnetarysystemcancauseanexchangeofenergyandangurmomentuminthefrequencydomain.Theamplitudeofthelowest-frequencyosciltionshouldincreaseifthesystemisunstableandbreaksdowngradually.However，suchasymptomofinstabilityisnotprominentinourlong-termintegrations.
InFig.7，thetotalorbitalenergyandangurmomentumofthefourinnerpnetsandallninepnetsareshownforintegrationN+2.Theupperthreepanelsshowthelong-periodicvariationoftotalenergy(denotedasE-E0)，totangurmomentum(G-G0)，andtheverticalcomponent(H-H0)oftheinnerfourpnetscalcutedfromthelow-passfilteredDeunayelements.E0，G0，H0denotetheinitialvaluesofeachquantity.Theabsolutedifferencefromtheinitialvaluesisplottedinthepanels.ThelowerthreepanelsineachfigureshowE-E0，G-G0andH-H0ofthetotalofninepnets.Thefluctuationshowninthelowerpanelsisvirtuallyentirelyaresultofthemassivejovianpnets.
Comparingthevariationsofenergyandangurmomentumoftheinnerfourpnetsandallninepnets，itisapparentthattheamplitudesofthoseoftheinnerpnetsaremuchsmallerthanthoseofallninepnets:theamplitudesoftheouterfivepnetsaremuchrgerthanthoseoftheinnerpnets.Thisdoesnotmeanthattheinnerterrestrialpnetarysubsystemismorestablethantheouterone:thisissimplyaresultoftheretivesmallnessofthemassesofthefourterrestrialpnetscomparedwiththoseoftheouterjovianpnets.Anotherthingwenoticeisthattheinnerpnetarysubsystemmaybecomeunstablemorerapidlythantheouteronebecauseofitsshorterorbitaltime-scales.Thiscanbeseeninthepanelsdenotedasinner4inFig.7wherethelonger-periodicandirregurosciltionsaremoreapparentthaninthepanelsdenotedastotal9.Actually，thefluctuationsintheinner4panelsaretoargeextentasaresultoftheorbitalvariationoftheMercury.However，wecannotneglectthecontributionfromotherterrestrialpnets，aswewillseeinsubsequentsections.
4.4Long-termcouplingofseveralneighbouringpnetpairs
Letusseesomeindividualvariationsofpnetaryorbitalenergyandangurmomentumexpressedbythelow-passfilteredDeunayelements.Figs10and11showlong-termevolutionoftheorbitalenergyofeachpnetandtheangurmomentuminN+1andN2integrations.Wenoticethatsomepnetsformapparentpairsintermsoforbitalenergyandangurmomentumexchange.Inparticur，VenusandEarthmakeatypicalpair.Inthefigures，theyshownegativecorretionsinexchangeofenergyandpositivecorretionsinexchangeofangurmomentum.Thenegativecorretioninexchangeoforbitalenergymeansthatthetwopnetsformacloseddynamicalsystemintermsoftheorbitalenergy.Thepositivecorretioninexchangeofangurmomentummeansthatthetwopnetsaresimultaneouslyundercertainlong-termperturbations.CandidatesforperturbersareJupiterandSaturn.AlsoinFig.11，wecanseethatMarsshowsapositivecorretionintheangurmomentumvariationtotheVenus–EarthsystemrcuryexhibitscertainnegativecorretionsintheangurmomentumversustheVenus–Earthsystem，whichseemstobeareactioncausedbytheconservationofangurmomentumintheterrestrialpnetarysubsystem.
ItisnotclearatthemomentwhytheVenus–Earthpairexhibitsanegativecorretioninenergyexchangeandapositivecorretioninangurmomentumexchange.Wemaypossiblyexpinthisthroughobservingthegeneralfactthattherearenosecurtermsinpnetarysemimajoraxesuptosecond-orderperturbationtheories(cf.Brouweramp;amp;Clemence1961;Boccalettiamp;amp;Pucacco1998).Thismeansthatthepnetaryorbitalenergy(whichisdirectlyretedtothesemimajoraxisa)mightbemuchlessaffectedbyperturbingpnetsthanistheangurmomentumexchange(whichretestoe).Hence，theeccentricitiesofVenusandEarthcanbedisturbedeasilybyJupiterandSaturn，whichresultsinapositivecorretionintheangurmomentumexchange.Ontheotherhand，thesemimajoraxesofVenusandEartharelesslikelytobedisturbedbythejovianpnets.ThustheenergyexchangemaybelimitedonlywithintheVenus–Earthpair，whichresultsinanegativecorretionintheexchangeoforbitalenergyinthepair.
Asfortheouterjovianpnetarysubsystem，Jupiter–SaturnandUranus–Neptuneseemtomakedynamicalpairs.However，thestrenhoftheircouplingisnotasstrongcomparedwiththatoftheVenus–Earthpair.
5±5×1010-yrintegrationsofouterpnetaryorbits
Sincethejovianpnetarymassesaremuchrgerthantheterrestrialpnetarymasses，wetreatthejovianpnetarysystemasanindependentpnetarysystemintermsofthestudyofitsdynamicalstability.Hence，weaddedacoupleoftrialintegrationsthatspan±5×1010yr，includingonlytheouterfivepnets(thefourjovianpnetsplusPluto).Theresultsexhibittherigorousstabilityoftheouterpnetarysystemoverthislongtime-span.Orbitalconfigurations(Fig.12)，andvariationofeccentricitiesandinclinations(Fig.13)showthisverylong-termstabilityoftheouterfivepnetsinboththetimeandthefrequencydomains.Althoughwedonotshowmapshere，thetypicalfrequencyoftheorbitalosciltionofPlutoandtheotherouterpnetsisalmostconstantduringtheseverylong-termintegrationperiods，whichisdemonstratedinthetime–frequencymapsonourwebpage.
Inthesetwointegrations，theretivenumericalerrorinthetotalenergywas106andthatofthetotangurmomentumwas1010.
5.1ResonancesintheNeptune–Plutosystem
Kinoshitaamp;amp;Nakai(1996)integratedtheouterfivepnetaryorbitsover±5.5×109yr.TheyfoundthatfourmajorresonancesbetweenNeptuneandPlutoaremaintainedduringthewholeintegrationperiod，andthattheresonancesmaybethemaincausesofthestabilityoftheorbitofPluto.Themajorfourresonancesfoundinpreviousresearchareasfollows.Inthefollowingdescription，λdenotesthemeanlongitude，Ωisthelongitudeoftheascendingnodeandisthelongitudeofperihelion.SubscriptsPandNdenotePlutoandNeptune.
MeanmotionresonancebetweenNeptuneandPluto(3:2).Thecriticargumentθ1=3λP2λNPlibratesaround180°withanamplitudeofabout80°andalibrationperiodofabout2×104yr.
TheargumentofperihelionofPlutoωP=θ2=PΩPlibratesaround90°withaperiodofabout3.8×106yr.ThedominantperiodicvariationsoftheeccentricityandinclinationofPlutoaresynchronizedwiththelibrationofitsargumentofperihelion.ThisisanticipatedinthesecurperturbationtheoryconstructedbyKozai(1962).
ThelongitudeofthenodeofPlutoreferredtothelongitudeofthenodeofNeptune，θ3=ΩPΩN，circutesandtheperiodofthiscircutionisequaltotheperiodofθ2libration.Whenθ3becomeszero，i.e.thelongitudesofascendingnodesofNeptuneandPlutooverp，theinclinationofPlutobecomesmaximum，theeccentricitybecomesminimumandtheargumentofperihelionbecomes90°.Whenθ3becomes180°，theinclinationofPlutobecomesminimum，theeccentricitybecomesmaximumandtheargumentofperihelionbecomes90°again.Williamsamp;amp;Benson(1971)anticipatedthistypeofresonance，terconfirmedbyMini，Nobiliamp;amp;Carpino(1989).
Anargumentθ4=PN+3(ΩPΩN)libratesaround180°withalongperiod，5.7×108yr.
Inournumericalintegrations，theresonances(i)–(iii)arewellmaintained，andvariationofthecriticargumentsθ1，θ2，θ3remainsimirduringthewholeintegrationperiod(Figs14–16).However，thefourthresonance(iv)appearstobedifferent:thecriticargumentθ4alternateslibrationandcircutionovera1010-yrtime-scale(Fig.17).ThisisaninterestingfactthatKinoshitaamp;amp;Nakai's(1995，1996)shorterintegrationswerenotabletodisclose.
6Discussion
Whatkindofdynamicalmechanismmaintainsthislong-termstabilityofthepnetarysystem?Wecanimmediatelythinkoftwomajorfeaturesthatmayberesponsibleforthelong-termstability.First，thereseemtobenosignificantlower-orderresonances(meanmotionandsecur)betweenanypairamongtheninepnets.JupiterandSaturnareclosetoa5:2meanmotionresonance(thefamous‘greatinequality’)，butnotjustintheresonancezone.Higher-orderresonancesmaycausethechaoticnatureofthepnetarydynamicalmotion，buttheyarenotsostrongastodestroythestablepnetarymotionwithinthelifetimeoftherealSorsystem.Thesecondfeature，whichwethinkismoreimportantforthelong-termstabilityofourpnetarysystem，isthedifferenceindynamicaldistancebetweenterrestriandjovianpnetarysubsystems(Itoamp;amp;Tanikawa1999，2001).WhenwemeasurepnetaryseparationsbythemutualHillradii(R_)，separationsamongterrestrialpnetsaregreaterthan26RH，whereasthoseamongjovianpnetsarelessthan14RH.Thisdifferenceisdirectlyretedtothedifferencebetweendynamicalfeaturesofterrestriandjovianpnets.Terrestrialpnetshavesmallermasses，shorterorbitalperiodsandwiderdynamicalseparation.Theyarestronglyperturbedbyjovianpnetsthathavergermasses，longerorbitalperiodsandnarrowerdynamicalseparation.Jovianpnetsarenotperturbedbyanyothermassivebodies.
Thepresentterrestrialpnetarysystemisstillbeingdisturbedbythemassivejovianpnets.However，thewideseparationandmutualinteractionamongtheterrestrialpnetsrendersthedisturbanceineffective;thedegreeofdisturbancebyjovianpnetsisO(eJ)(orderofmagnitudeoftheeccentricityofJupiter)，sincethedisturbancecausedbyjovianpnetsisaforcedosciltionhavinganamplitudeofO(eJ).Heighteningofeccentricity，forexampleO(eJ)0.05，isfarfromsufficienttoprovokeinstabilityintheterrestrialpnetshavingsuchawideseparationas26RH.Thusweassumethatthepresentwidedynamicalseparationamongterrestrialpnets(amp;;26RH)isprobablyoneofthemostsignificantconditionsformaintainingthestabilityofthepnetarysystemovera109-yrtime-span.Ourdetailedanalysisoftheretionshipbetweendynamicaldistancebetweenpnetsandtheinstabilitytime-scaleofSorsystempnetarymotionisnowon-going.
AlthoughournumericalintegrationsspanthelifetimeoftheSorsystem，thenumberofintegrationsisfarfromsufficienttofilltheinitialphasespace.Itisnecessarytoperformmoreandmorenumericalintegrationstoconfirmandexamineindetailthelong-termstabilityofourpnetarydynamics.
——以上文段引自Ito，T.amp;Tanikawa，K.Long-termintegrationsandstabilityofpnetaryorbitsinourSorSystem.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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