pected，thecharacterofthevariationofparyorbitalelentsdoesnotdiffersignificantlybeeentheitialandfalpartofeachtegration，atleastforven，earthandarstheelentsofrcury，especiallyitseentricity，seetochangetoasignificantextentthisispartlybecaetheorbitalti-scaleofthepistheshortestofalltheps，&nbshichleadstoaorerapidorbitalevolutionthanotherpsthenerostpaybenearesttostabilitythisresultappearstobesoagreent&nbsithskar's(1994，1996)expectationsthatrgeandirregurvariationsappeartheeentricitiesandclationsofrcuryonati-scaleofseveral109yrhowever，theeffectofthepossiblestabilityoftheorbitofrcuryaynotfatallyaffecttheglobalstabilityofthe&nbsholeparysysteogtothesallassofrcury&nbse&nbsillntionbrieflythelong-terorbitalevolutionofrcurytersection4glow-passfilteredorbitalelents

    theorbitalotionoftheouterfivepsseesrigorolystableandquitereguroverthisti-span(seealsosection5)

    32ti–frequencyaps

    althoughtheparyotionexhibitsverylong-terstabilitydefedasthenon-existenceofcloseenunterevents，thechaoticnatureofparydynaicscanchangetheosciltoryperiodandaplitudeofparyorbitalotiongraduallyoversuchlongti-spansevensuchslightfluctuationsoforbitalvariationthefrequencydoa，particurlythecaseofearth，canpotentiallyhaveasignificanteffectonitssurfacecliatesystethroughsorsotionvariation(cfberger1988)

    togiveanoverviewofthelong-terchangeperiodicityparyorbitalotion，&nbseperfordanyfastfouriertransforations(ffts)alongthetiaxis，andsuperposedtheresultgperiodgrastodrawo-dinsionalti–frequencyapsthespecificapproachtodragtheseti–frequencyapsthispaperisverysiple–uchsiplerthanthe&nbsaveletanalysisorskar's(1990，1993)frequencyanalysis

    dividethelow-passfilteredorbitaldatatoanyfragntsofthesalengththelengthofeachdatasegntshouldbeaultipleof2ordertoapplythefft

    eachfragntofthedatahasargeoverppgpart:forexaple，&nbshentheithdatabegsfrot=tiandendsatt=ti+t，thenextdatasegntrangesfroti+δt≤ti+δt+t，&nbshereδt?t&nbsentuethisdivisionuntil&nbsereachacertanubernby&nbshichtn+treachesthetotaltegrationlength

    &nbseapplyanffttoeachofthedatafragnts，andobtanfrequencydiagras

    eachfrequencydiagraobtaedabove，thestrengthofperiodicitycanberepcedbyagrey-scale(orlour)chart

    &nbseperfortherepcent，andnnectallthegrey-scale(orlour)chartstoonegraphforeachtegrationthehorizontalaxisofthesenewgraphsshouldbetheti，iethestartgtisofeachfragntofdata(ti，&nbsherei=1，…，n)theverticalaxisrepresentstheperiod(orfrequency)oftheosciltionoforbitalelents

    &nbsehaveadoptedanfftbecaeofitsoverwhelgspeed，scetheaountofnuricaldatatobedeposedtofrequencyponentsisterriblyhuge(severaltensofgbytes)

    atypicalexapleoftheti–frequencyapcreatedbytheaboveproceduresisshownagrey-scalediagraasfig5，&nbshichshowsthevariationofperiodicitytheeentricityandclationofearthn+2tegrationfig5，thedarkareashowsthatatthetidicatedbythevalueontheabscissa，theperiodicitydicatedbytheordateisstrongerthanthelighterareaaroundit&nbsecanregnizefrothisapthattheperiodicityoftheeentricityandclationofearthonlychangesslightlyovertheentireperiodveredbythen+2tegrationthisnearlyregurtrendisqualitativelythesaothertegrationsandforotherps，althoughtypicalfrequenciesdifferpbypandelentbyelent

    42long-terexchangeoforbitalenergyandangurontu

    &nbsecalcuteverylong-periodicvariationandexchangeofparyorbitalenergyandangurontugfiltereddeunayelentsl，g，hgandhareequivalenttotheparyorbitalangurontuanditsverticalponentperunitasslisretedtotheparyorbitalenergyeperunitassase=?μ2/2l2ifthesysteispletelylear，theorbitalenergyandtheangurontueachfrequencybtbenstantnon-learitytheparysystecancaeanexchangeofenergyandangurontuthefrequencydoatheaplitudeofthelowest-frequencyosciltionshouldcreaseifthesysteisunstableandbreaksdowngraduallyhowever，suchasyptoofstabilityisnotproentourlong-tertegrations

    fig7，thetotalorbitalenergyandangurontuofthefournerpsandallnepsareshownfortegrationn+2theupperthreepanelsshowthelong-periodicvariationoftotalenergy(denotedase-e0)，totalangurontu(g-g0)，andtheverticalponent(h-h0)ofthenerfourpscalcutedfrothelow-passfiltereddeunayelentse0，g0，h0denotetheitialvaluesofeachquantitytheabsolutedifferencefrotheitialvaluesisplottedthepanelsthelowerthreepanelseachfigureshowe-e0，g-g0andh-h0ofthetotalofnepsthefluctuationshownthelowerpanelsisvirtuallyentirelyaresultoftheassivejovianps

    pargthevariationsofenergyandangurontuofthenerfourpsandallneps，itisapparentthattheaplitudesofthoseofthenerpsareuchsallerthanthoseofallneps:theaplitudesoftheouterfivepsareuchrgerthanthoseofthenerpsthisdoesnotanthatthenerterrestrialparysubsysteisorestablethantheouterone:thisissiplyaresultoftheretivesallnessoftheassesofthefourterrestrialpspared&nbsiththoseoftheouterjovianpsanotherthg&nbsenoticeisthatthenerparysubsysteaybeeunstableorerapidlythantheouteronebecaeofitsshorterorbitalti-scalesthiscanbeseenthepanelsdenotedasner4fig7&nbsherethelonger-periodicandirregurosciltionsareoreapparentthanthepanelsdenotedastotal9actually，thefluctuationsthener4panelsaretoargeextentasaresultoftheorbitalvariationofthercuryhowever，&nbsecannotneglectthentributionfrootherterrestrialps，as&nbse&nbsillseesubsequentsections

    44long-teruplgofseveralneighbourgppairs

    letseesodividualvariationsofparyorbitalenergyandangurontuexpressedbythelow-passfiltereddeunayelentsfigs10and11showlong-terevolutionoftheorbitalenergyofeachpandtheangurontun+1andn?2tegrations&nbsenoticethatsopsforapparentpairstersoforbitalenergyandangurontuexchangeparticur，venandearthakeatypicalpairthefigures，theyshownegativerretionsexchangeofenergyandpositiverretionsexchangeofangurontuthenegativerretionexchangeoforbitalenergyansthattheopsforacloseddynaicalsystetersoftheorbitalenergythepositiverretionexchangeofangurontuansthattheopsaresiultaneolyundercertalong-terperturbationscandidatesforperturbersarejupiterandsaturnalsofig11，&nbsecanseethatarsshowsapositiverretiontheangurontuvariationtotheven–earthsystercuryexhibitscertanegativerretionstheangurontuverstheven–earthsyste，&nbshichseestobeareactioncaedbythenservationofangurontutheterrestrialparysubsyste

    itisnotclearattheont&nbshytheven–earthpairexhibitsanegativerretionenergyexchangeandapositiverretionangurontuexchange&nbseaypossiblyexpthisthroughobservgthegeneralfactthattherearenosecurtersparyseiajoraxesuptosend-orderperturbationtheories(cfbrouwer&apapclence1961boaletti&apappucao1998)thisansthattheparyorbitalenergy(whichisdirectlyretedtotheseiajoraxisa)ightbeuchlessaffectedbyperturbgpsthanistheangurontuexchange(whichretestoe)hence，theeentricitiesofvenandearthcanbedisturbedeasilybyjupiterandsaturn，&nbshichresultsapositiverretiontheangurontuexchangeontheotherhand，theseiajoraxesofvenandeartharelesslikelytobedisturbedbythejovianpsththeenergyexchangeaybeliitedonly&nbsiththeven–earthpair，&nbshichresultsanegativerretiontheexchangeoforbitalenergythepair

    asfortheouterjovianparysubsyste，jupiter–saturnanduran–neptuneseetoakedynaicalpairshowever，thestrengthoftheiruplgisnotasstrongpared&nbsiththatoftheven–earthpair

    5±5x1010-yrtegrationsofouterparyorbits

    scethejovianparyassesareuchrgerthantheterrestrialparyasses，&nbsetreatthejovianparysysteasandependentparysystetersofthestudyofitsdynaicalstabilityhence，&nbseaddedaupleoftrialtegrationsthatspan±5x1010yr，cludgonlytheouterfiveps(thefourjovianpsplpluto)theresultsexhibittherigorostabilityoftheouterparysysteoverthislongti-spanorbitalnfigurations(fig12)，andvariationofeentricitiesandclations(fig13)showthisverylong-terstabilityoftheouterfivepsboththetiandthefrequencydoasalthough&nbsedonotshowapshere，thetypicalfrequencyoftheorbitalosciltionofplutoandtheotherouterpsisalostnstantdurgtheseverylong-tertegrationperiods，&nbshichisdeonstratedtheti–frequencyapsonour&nbsebpage

    theseotegrations，theretivenuricalerrorthetotalenergy&nbsas～10?6andthatofthetotalangurontu&nbsas～10?10

    51resonancestheneptune–plutosyste

    koshita&apapnakai(1996)tegratedtheouterfiveparyorbitsover±55x109yrtheyfoundthatfourajorresonancesbeeenneptuneandplutoareataeddurgthe&nbsholetegrationperiod，andthattheresonancesaybetheacaesofthestabilityoftheorbitofplutotheajorfourresonancesfoundprevioresearchareasfollowsthefollogdescription，λdenotestheanlongitude，Ωisthelongitudeoftheascendgnodeand?isthelongitudeofperihelionsubscriptspandndenoteplutoandneptune

    anotionresonancebeeenneptuneandpluto(3:2)thecriticalarguntθ1=3λp?2λn??plibratesaround180°&nbsithanaplitudeofabout80°andalibrationperiodofabout2x104yr

    thearguntofperihelionofpluto=θ2=?p?Ωplibratesaround90°&nbsithaperiodofabout38x106yrthedoantperiodicvariationsoftheeentricityandclationofplutoaresynchronized&nbsiththelibrationofitsarguntofperihelionthisisanticipatedthesecurperturbationtheorynstructedbykozai(1962)

    thelongitudeofthenodeofplutoreferredtothelongitudeofthenodeofneptune，θ3=Ωp?Ωn，circutesandtheperiodofthiscircutionisequaltotheperiodofθ2libration&nbshenθ3beeszero，iethelongitudesofascendgnodesofneptuneandplutooverp，theclationofplutobeesaxiu，theeentricitybeesiuandthearguntofperihelionbees90°&nbshenθ3bees180°，theclationofplutobeesiu，theeentricitybeesaxiuandthearguntofperihelionbees90°aga&nbsillias&apapbenson(1971)anticipatedthistypeofresonance，ternfirdbyini，nobili&apapcarpo(1989)

    anarguntθ4=?p??n+3(Ωp?Ωn)libratesaround180°&nbsithalongperiod，～57x108yr

    ournuricaltegrations，theresonances(i)–(iii)are&nbsellataed，andvariationofthecriticalarguntsθ1，θ2，θ3reasiirdurgthe&nbsholetegrationperiod(figs14–16)however，thefourthresonance(iv)appearstobedifferent:thecriticalarguntθ4alternateslibrationandcircutionovera1010-yrti-scale(fig17)thisisanterestgfactthatkoshita&apapnakai's(1995，1996)shortertegrations&nbserenotabletodisclose

    6discsion

    &nbshatkdofdynaicalchanisatasthislong-terstabilityoftheparysyste?&nbsecanidiatelythkofoajorfeaturesthatayberesponsibleforthelong-terstabilityfirst，thereseetobenosignificantlower-orderresonances(anotionandsecur)beeenanypairaongthenepsjupiterandsaturnareclosetoa5:2anotionresonance(thefao‘greatequality’)，butnotjttheresonancezonehigher-orderresonancesaycaethechaoticnatureoftheparydynaicalotion，buttheyarenotsostrongastodestroythestableparyotion&nbsiththelifetioftherealsorsystethesendfeature，&nbshich&nbsethkisoreiportantforthelong-terstabilityofourparysyste，isthedifferencedynaicaldistancebeeenterrestrialandjovianparysubsystes(ito&apaptanikawa1999，2001)&nbshen&nbseasureparyseparationsbytheutualhillradii(r_)，separationsaongterrestrialpsaregreaterthan26rh，&nbshereasthoseaongjovianpsarelessthan14rhthisdifferenceisdirectlyretedtothedifferencebeeendynaicalfeaturesofterrestrialandjovianpsterrestrialpshavesallerasses，shorterorbitalperiodsand&nbsiderdynaicalseparationtheyarestronglyperturbedbyjovianpsthathavergerasses，longerorbitalperiodsandnarrowerdynaicalseparationjovianpsarenotperturbedbyanyotherassivebodies

    thepresentterrestrialparysysteisstillbegdisturbedbytheassivejovianpshowever，the&nbsideseparationandutualteractionaongtheterrestrialpsrendersthedisturbanceeffectivethedegreeofdisturbancebyjovianpsiso(ej)(orderofagnitudeoftheeentricityofjupiter)，scethedisturbancecaedbyjovianpsisaforcedosciltionhavganaplitudeofo(ej)heightengofeentricity，forexapleo(ej)～005，isfarfrosufficienttoprovokestabilitytheterrestrialpshavgsucha&nbsideseparationas26rhth&nbseassuthatthepresent&nbsidedynaicalseparationaongterrestrialps(&apgt26rh)isprobablyoneoftheostsignificantnditionsforatagthestabilityoftheparysysteovera109-yrti-spanourdetailedanalysisoftheretionshipbeeendynaicaldistancebeeenpsandthestabilityti-scaleofsorsysteparyotionisnowon-gog

    althoughournuricaltegrationsspanthelifetiofthesorsyste，thenuberoftegrationsisfarfrosufficienttofilltheitialphasespaceitisnecessarytoperfororeandorenuricaltegrationstonfirandexaedetailthelong-terstabilityofourparydynaics

    ——以上文段引自ito，t&aptanikawa，klong-tertegrationsandstabilityofparyorbitsoursorsysteonnotrastronsoc336，483–500(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。