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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±5x1010yr.theresultindicatesthatthethreemajorresonancesintheneptune–plutosystemhavebeenmaintainedoverthe1011-yrtime-span.
1introduction
1.1definitionoftheproblem
thequestionofthestabilityofoursolarsystemhasbeendebatedoverseveralhundredyears，sincetheeraofnewton.theproblemhasattractedmanyfamousmathematiciansovertheyearsandhasplayedacentralroleinthedevelopmentofnon-lineardynamicsandchaostheory.however，wedonotyethaveadefiniteanswertothequestionofwhetheroursolarsystemisstableornot.thisispartlyaresultofthefactthatthedefinitionoftheterm‘stability’isvaguewhenitisusedinrelationtotheproblemofplanetarymotioninthesolarsystem.actuallyitisnoteasytogiveaclear，rigorousandphysicallymeaningfuldefinitionofthestabilityofoursolarsystem.
amongmanydefinitionsofstability，hereweadoptthehilldefinition(gladman1993):actuallythisisnotadefinitionofstability，butofinstability.wedefineasystemasbingunstablewhenacloseencounteroccurssomewhereinthesystem，startingfromacertaininitialconfiguration(chambers，wetherill&ito&&tanikawa1999).asystemisdefinedasexperiencingacloseencounterwhentwobodiesapproachoneanotherwithinanareaofthelargerhillradius.otherwisethesystemisdefinedasbeingstable.henceforwardwestatethatourplanetarysystemisdynamicallystableifnocloseencounterhappensduringtheageofoursolarsystem，about±5gyr.incidentally，thisdefinitionmaybereplacedbyoneinwhichanoccurrenceofanyorbitalcrossingbetweeneitherofapairofplanetstakesplace.thisisbecauseweknowfromexperiencethatanorbitalcrossingisverylikelytoleadtoacloseencounterinplanetaryandprotoplanetarysystems(yoshinaga，kokubo&&makino1999).ofcoursethisstatementcannotbesimplyappliedtosystemswithstableorbitalresonancessuchastheneptune–plutosystem.
1.2previousstudiesandaimsofthisresearch
inadditiontothevaguenessoftheconceptofstability，theplanetsinoursolarsystemshowacharactertypicalofdynamicalchaos(sussman&&wisdom1988，1992).thecauseofthischaoticbehaviourisnowpartlyunderstoodasbeingaresultofresonanceoverlapping(murray&lecar，franklin&&holman2001).however，itwouldrequireintegratingoveranensembleofplanetarysystemsincludingallnineplanetsforaperiodcoveringseveral10gyrtothoroughlyunderstandthelong-termevolutionofplanetaryorbits，sincechaoticdynamicalsystemsarecharacterizedbytheirstrongdependenceoninitialconditions.
fromthatpointofview，manyofthepreviouslong-termnumericalintegrationsincludedonlytheouterfiveplanets(sussman&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.theirexperimentsonthesevenplanetsarenotyeprehensive，butitseemsthattheterrestrialplanetsalsoremainstableduringtheintegrationperiod，maintainingalmostregularoscillations.
ontheotherhand，inhisaccuratesemi-analyticalsecularperturbationtheory(laskar1988)，laskarfindsthatlargeandirregularvariationscanappearintheeccentricitiesandinclinationsoftheterrestrialplanets，especiallyofmercuryandmarsonatime-scaleofseveral109yr(laskar1996).theresultsoflaskar'ssecularperturbationtheoryshouldbeconfirmedandinvestigatedbyfullynumericalintegrations.
inthispaperwepresentpreliminaryresultsofsixlong-termnumericalintegrationsonallnineplanetaryorbits，coveringaspanofseveral109yr，andoftwootherintegrationscoveringaspanof±5x1010yr.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±5x1010yr.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&kinoshita，yoshida&&nakai1991)withaspecialstart-upproceduretoreducethetruncationerrorofanglevariables，‘warmstart’(saha&&tremaine1992，1994).
thestepsizeforthenumericalintegrationsis8dthroughoutallintegrationsofthenineplanets(n±1，2，3)，whichisabout111oftheorbitalperiodoftheinnermostplanet(mercury).asforthedeterminationofstepsize，wepartlyfollowthepreviousnumericalintegrationofallnineplanetsinsussman&&wisdom(1988，7.2d)andsaha&&tremaine(1994，22532d).weroundedthedecimalpartofthetheirstepsizesto8tomakethestepsizeamultipleof2inordertoreducetheaccumulationofround-offerrorinthputationprocesses.inrelationtothis，wisdom&&holman(1991)performednumericalintegrationsoftheouterfiveplanetaryorbitsusingthesymplecticmapwithastepsizeof400d，110.83oftheorbitalperiodofjupiter.theirresultseemstobeaccurateenough，whichpartlyjustifiesourmethodofdeterminingthestepsize.however，sincetheeccentricityofjupiter(～0.05)ismuchsmallerthanthatofmercury(～0.2)，weneedsomecarewhenwparetheseintegrationssimplyintermsofstepsizes.
intheintegrationoftheouterfiveplanets(f±)，wefixedthestepsizeat400d.
weadoptgauss'fandgfunctionsinthesymplecticmaptogetherwiththethird-orderhalleymethod(danby1992)asasolverforkeplerequations.thenumberofmaximumiterationswesetinhalley'smethodis15，buttheyneverreachedthemaximuminanyofourintegrations.
theintervalofthedataoutputis200000d(～547yr)forthecalculationsofallnineplanets(n±1，2，3)，andabout8000000d(～21903yr)fortheintegrationoftheouterfiveplanets(f±).
althoughnooutputfilteringwasdonewhenthenumericalintegrationswereinprocess，weappliedalow-passfiltertotheraworbitaldataafterwehapletedallthecalculations.seesection4.1formoredetail.
2.4errorestimation
2.4.1relativeerrorsintotalenergyandangularmomentum
accordingtooneofthebasicpropertiesofsymplecticintegrators，whichconservethephysicallyconservativequantitieswell(totalorbitalenergyandangularmomentum)，ourlong-termnumericalintegrationsseemtohavebeenperformedwithverysmallerrors.theaveragedrelativeerrorsoftotalenergy(～10?9)andoftotalangularmomentum(～10?11)haveremainednearlyconstantthroughouttheintegrationperiod(fig.1).thespecialstartupprocedure，warmstart，wouldhavereducedtheaveragedrelativeerrorintotalenergybyaboutoneorderofmagnitudeormore.
relativenumericalerrorofthetotalangularmomentumδaa0andthetotalenergyδee0inournumericalintegrationsn±1，2，3，whereδeandδaaretheabsolutechangeofthetotalenergyandtotalangularmomentum，respectively，ande0anda0aretheirinitialvalues.thehorizontalunitisgyr.
notethatdifferentoperatingsystems，differentmathematicallibraries，anddifferenthardwarearchitecturesresultindifferentnumericalerrors，throughthevariationsinround-offerrorhandlingandnumericalalgorithms.intheupperpaneloffig.1，wecanrecognizethissituationinthesecularnumericalerrorinthetotalangularmomentum，whichshouldberigorouslypreserveduptomachine-eprecision.
2.4.2errorinplanetarylongitudes
sincethesymplecticmapspreservetotalenergyandtotalangularmomentumofn-bodydynamicalsystemsinherentlywell，thedegreeoftheirpreservationmaynotbeagoodmeasureoftheaccuracyofnumericalintegrations，especiallyasameasureofthepositionalerrorofplanets，i.e.theerrorinplanetarylongitudes.toestimatethenumericalerrorintheplanetarylongitudes，weperformedthefollowingprocedures.wparedtheresultofourmainlong-termintegrationswithsometestintegrations，whichspanmuchshorterperiodsbutwithmuchhigheraccuracythanthemainintegrations.forthispurpose，weperformedamuchmoreaccurateintegrationwithastepsizeof0.125d(164ofthemainintegrations)spanning3x105yr，startingwiththesameinitialconditionsasinthen?1integration.weconsiderthatthistestintegrationprovidesuswitha‘pseudo-true’solutionofplanetaryorbitalevolution.next，wparethetestintegrationwiththemainintegration，n?1.fortheperiodof3x105yr，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=0to0.0547x109yr).(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.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-passfilteredorbitaldataintomanyfragmentsofthesamelenh.thelenhofeachdatasegmentshouldbeamultipleof2inordertoapplythefft.
eachfragmentofthedatahasalargeoverlappingpart:forexample，whentheithdatabeginsfromt=tiandendsatt=ti+t，thenextdatasegmentrangesfromti+δt≤ti+δt+t，whereδt?t.wecontinuethisdivisionuntilwereachacertainnumbernbywhichtn+treachesthetotalintegrationlenh.
weapplyanffttoeachofthedatafragments，andobtainnfrequencydiagrams.
ineachfrequencydiagramobtainedabove，thestrenhofperiodicitycanbereplacedbyagrey-scale(orcolour)chart.
weperformthereplacement，andconnectallthegrey-scale(orcolour)chartsintoonegraphforeachintegration.thehorizontalaxisofthesenewgraphsshouldbethetime，i.e.thestartingtimesofeachfragmentofdata(ti，wherei=1，…，n).theverticalaxisrepresentstheperiod(orfrequency)oftheoscillationoforbitalelements.
wehaveadoptedanfftbecauseofitsoverwhelmingspeed，sincetheamountofnumericaldatatobedposedintofrequencponentsisterriblyhuge(severaltensofgbytes).
atypicalexampleofthetime–frequencymapcreatedbytheaboveproceduresisshowninagrey-scalediagramasfig.5，whichshowsthevariationofperiodicityintheeccentricityandinclinationofearthinn+2integration.infig.5，thedarkareashowsthatatthetimeindicatedbythevalueontheabscissa，theperiodicityindicatedbytheordinateisstrongerthaninthelighterareaaroundit.wecanrecognizefromthismapthattheperiodicityoftheeccentricityandinclinationofearthonlychangesslightlyovertheentireperiodcoveredbythen+2integration.thisnearlyregulartrendisqualitativelythesameinotherintegrationsandforotherplanets，althoughtypicalfrequenciesdifferplanetbyplanetandelementbyelement.
4.2long-termexchangeoforbitalenergyandangularmomentum
wecalculateverylong-periodicvariationandexchangeofplanetaryorbitalenergyandangularmomentumusingfiltereddelaunayelementsl，g，h.gandhareequivalenttotheplanetaryorbitalangularmomentumanditsverticaponentperunitmass.lisrelatedtotheplanetaryorbitalenergyeperunitmassase=?μ22l2.ifthesystemipletelylinear，theorbitalenergyandtheangularmomentumineachfrequencybinmustbeconstant.non-linearityintheplanetarysystemcancauseanexchangeofenergyandangularmomentuminthefrequencydomain.theamplitudeofthelowest-frequencyoscillationshouldincreaseifthesystemisunstableandbreaksdowngradually.however，suchasymptomofinstabilityisnotprominentinourlong-termintegrations.
infig.7，thetotalorbitalenergyandangularmomentumofthefourinnerplanetsandallnineplanetsareshownforintegrationn+2.theupperthreepanelsshowthelong-periodicvariationoftotalenergy(denotedase-e0)，totalangularmomentum(g-g0)，andtheverticaponent(h-h0)oftheinnerfourplanetscalculatedfromthelow-passfiltereddelaunayelements.e0，g0，h0denotetheinitialvaluesofeachquantity.theabsolutedifferencefromtheinitialvaluesisplottedinthepanels.thelowerthreepanelsineachfigureshowe-e0，g-g0andh-h0ofthetotalofnineplanets.thefluctuationshowninthelowerpanelsisvirtuallyentirelyaresultofthemassivejovianplanets.
comparingthevariationsofenergyandangularmomentumoftheinnerfourplanetsandallnineplanets，itisapparentthattheamplitudesofthoseoftheinnerplanetsaremuchsmallerthanthoseofallnineplanets:theamplitudesoftheouterfiveplanetsaremuchlargerthanthoseoftheinnerplanets.thisdoesnotmeanthattheinnerterrestrialplanetarysubsystemismorestablethantheouterone:thisissimplyaresultoftherelativesmallnessofthemassesofthefourterrestrialplanetparedwiththoseoftheouterjovianplanets.anotherthingwenoticeisthattheinnerplanetarysubsystemmaybeunstablemorerapidlythantheouteronebecauseofitsshorterorbitaltime-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，wecanseethatmarsshows'itivecorrelationintheangularmomentumvariationtothevenus–earthsystem.mercuryexhibitscertainnegativecorrelationsintheangularmomentumversusthevenus–earthsystem，whichseemstobeareactioncausedbytheconservationofangularmomentumintheterrestrialplanetarysubsystem.
itisnotclearatthemomentwhythevenus–earthpairexhibitsanegativecorrelationinenergyexchangeand'itivecorrelationinangularmomentumexchange.wemaypossiblyexplainthisthroughobservingthegeneralfactthattherearenoseculartermsinplanetarysemimajoraxesuptosecond-orderperturbationtheories(cf.brouwer&boccaletti&&pucacco1998).thismeansthattheplanetaryorbitalenergy(whichisdirectlyrelatedtothesemimajoraxisa)mightbemuchlessaffectedbyperturbingplanetsthanistheangularmomentumexchange(whichrelatestoe).hence，theeccentricitiesofvenusandearthcanbedisturbedeasilybyjupiterandsaturn，whichresultsin'itivecorrelationintheangularmomentumexchange.ontheotherhand，thesemimajoraxesofvenusandeartharelesslikelytobedisturbedbythejovianplanets.thustheenergyexchangemaybelimitedonlywithinthevenus–earthpair，whichresultsinanegativecorrelationintheexchangeoforbitalenergyinthepair.
asfortheouterjovianplanetarysubsystem，jupiter–saturnanduranus–neptuneseemtomakedynamicalpairs.however，thestrenhoftheircouplingisnotasstronparedwiththatofthevenus–earthpair.
5±5x1010-yrintegrationsofouterplanetaryorbits
sincethejovianplanetarymassesaremuchlargerthantheterrestrialplanetarymasses，wetreatthejovianplanetarysystemasanindependentplanetarysystemintermsofthestudyofitsdynamicalstability.hence，weaddedacoupleoftrialintegrationsthatspan±5x1010yr，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±5.5x109yr.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.whenθ3beszero，i.e.thelongitudesofascendingnodesofneptuneandplutooverlap，theinclinationofplutobesmaximum，theeccentricitybesminimumandtheargumentofperihelionbes90°.whenθ3bes180°，theinclinationofplutobesminimum，theeccentricitybesmaximumandtheargumentofperihelionbes90°again.williams&&benson(1971)anticipatedthistypeofresonance，laterconfirmedbymilani，nobili&&carpino(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: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。