gu903();thevariationofeentricitiesandorbitalclationsforthenerfourpstheitialandfalpartofthetegrationn+1isshownfig4asexpected，thecharacterofthevariationofparyorbitalelentsdoesnotdiffersignificantlybeeentheitialandfalpartofeachtegration，atleastforven，earthandarstheelentsofrcury，especiallyitseentricity，seetochangetoasignificantextentthisispartlybecaetheorbitalti-scaleofthepistheshortestofalltheps，whichleadstoaorerapidorbitalevotionthanotherps;thenerostpaybenearesttostabilitythisresultappearstobesoagreentwithskar&039;s(1994，1996)expectationsthatrgeandirregurvariationsappeartheeentricitiesandclationsofrcuryonati-scaleofseveral109yrhowever，theeffectofthepossiblestabilityoftheorbitofrcuryaynotfatallyaffecttheglobalstabilityofthewholeparysysteogtothesallassofrcurywewillntionbrieflythelong-terorbitalevotionofrcurytersection4glow-passfilteredorbitalelents

theorbitalotionoftheouterfivepsseesrigorolystableandquitereguroverthisti-span(seealsosection5)

32ti–frequencyaps

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

togiveanoverviewofthelong-terchangeperiodicityparyorbitalotion，weperfordanyfastfouriertransforations(ffts)alongthetiaxis，andsuperposedtheresultgperiodgrastodrawo-dinsionalti–frequencyapsthespecificapproachtodragtheseti–frequencyapsthispaperisverysiple–uchsiplerthanthewaveletanalysisorskar&039;s(1990，1993)frequencyanalysis

dividethelow-passfilteredorbitaldatatoanyfragntsofthesalengththelengthofeachdatasegntshouldbeaultipleof2ordertoapplythefft

eachfragntofthedatahasargeoverppgpart:forexaple，whentheithdatabegsfrot=tiandendsatt=ti+t，thenextdatasegntrangesfroti+δt≤ti+δt+t，whereδt?twentuethisdivisionuntilwereachacertanubernbywhichtn+treachesthetotaltegrationlength

weapplyanffttoeachofthedatafragnts，andobtanfrequencydiagras

eachfrequencydiagraobtaedabove，thestrengthofperiodicitycanberepcedbyagrey-scale(orlour)chart

weperfortherepcent，andnnectallthegrey-scale(orlour)chartstoonegraphforeachtegrationthehorizontalaxisofthesenewgraphsshouldbetheti，iethestartgtisofeachfragntofdata(ti，wherei=1，…，n)theverticalaxisrepresentstheperiod(orfrequency)oftheosciltionoforbitalelents

wehaveadoptedanfftbecaeofitsoverwhelgspeed，scetheaountofnuricaldatatobedeposedtofrequencyponentsisterriblyhuge(severaltensofgbytes)

atypicalexapleoftheti–frequencyapcreatedbytheaboveproceduresisshownagrey-scalediagraasfig5，whichshowsthevariationofperiodicitytheeentricityandclationofearthn+2tegrationfig5，thedarkareashowsthatatthetidicatedbythevaeontheabscissa，theperiodicitydicatedbytheordateisstrongerthanthelighterareaarounditwecanregnizefrothisapthattheperiodicityoftheeentricityandclationofearthonlychangesslightlyovertheentireperiodveredbythen+2tegrationthisnearlyregurtrendisqualitativelythesaothertegrationsandforotherps，althoughtypicalfrequenciesdifferpbypandelentbyelent

42long-terexchangeoforbitalenergyandangurontu

wecalcuteverylong-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，h0denotetheitialvaesofeachquantitytheabsotedifferencefrotheitialvaesisplottedthepanelsthelowerthreepanelseachfigureshowe-e0，g-g0andh-h0ofthetotalofnepsthefctuationshownthelowerpanelsisvirtuallyentirelyaresultoftheassivejovianps

pargthevariationsofenergyandangurontuofthenerfourpsandallneps，itisapparentthattheaplitudesofthoseofthenerpsareuchsallerthanthoseofallneps:theaplitudesoftheouterfivepsareuchrgerthanthoseofthenerpsthisdoesnotanthatthenerterrestrialparysubsysteisorestablethantheouterone:thisissiplyaresultoftheretivesallnessoftheassesofthefourterrestrialpsparedwiththoseoftheouterjovianpsanotherthgwenoticeisthatthenerparysubsysteaybeeunstableorerapidlythantheouteronebecaeofitsshorterorbitalti-scalesthiscanbeseenthepanelsdenotedasner4fig7wherethelonger-periodicandirregurosciltionsareoreapparentthanthepanelsdenotedastotal9actually，thefctuationsthener4panelsaretoargeextentasaresultoftheorbitalvariationofthercuryhowever，wecannotneglectthentributionfrootherterrestrialps，aswewillseesubsequentsections

44long-teruplgofseveralneighbourgppairs

letseesodividualvariationsofparyorbitalenergyandangurontuexpressedbythelow-passfiltereddeunayelentsfigs10and11showlong-terevotionoftheorbitalenergyofeachpandtheangurontun+1andn?2tegrationswenoticethatsopsforapparentpairstersoforbitalenergyandangurontuexchangeparticur，venandearthakeatypicalpairthefigures，theyshownegativerretionsexchangeofenergyandpositiverretionsexchangeofangurontuthenegativerretionexchangeoforbitalenergyansthattheopsforacloseddynaicalsystetersoftheorbitalenergythepositiverretionexchangeofangurontuansthattheopsaresiultaneolyundercertalong-terperturbationscandidatesforperturbersarejupiterandsaturnalsofig11，wecanseethatarsshowsapositiverretiontheangurontuvariationtotheven–earthsystercuryexhibitscertanegativerretionstheangurontuverstheven–earthsyste，whichseestobeareactioncaedbythenservationofangurontutheterrestrialparysubsyste

itisnotclearattheontwhytheven–earthpairexhibitsanegativerretionenergyexchangeandapositiverretionangurontuexchangeweaypossiblyexpthisthroughobservgthegeneralfactthattherearenosecurtersparyseiajoraxesuptosend-orderperturbationtheories(cfbrouwer≈ap;ap;ap;clence1961;boaletti≈ap;ap;ap;pucao1998)thisansthattheparyorbitalenergy(whichisdirectlyretedtotheseiajoraxisa)ightbeuchlessaffectedbyperturbgpsthanistheangurontuexchange(whichretestoe)hence，theeentricitiesofvenandearthcanbedisturbedeasilybyjupiterandsaturn，whichresultsapositiverretiontheangurontuexchangeontheotherhand，theseiajoraxesofvenandeartharelesslikelytobedisturbedbythejovianpsththeenergyexchangeaybeliitedonlywiththeven–earthpair，whichresultsanegativerretiontheexchangeoforbitalenergythepair

asfortheouterjovianparysubsyste，jupiter–saturnanduran–neptuneseetoakedynaicalpairshowever，thestrengthoftheiruplgisnotasstrongparedwiththatoftheven–earthpair

5±5x1010-yrtegrationsofouterparyorbits

scethejovianparyassesareuchrgerthantheterrestrialparyasses，wetreatthejovianparysysteasandependentparysystetersofthestudyofitsdynaicalstabilityhence，weaddedaupleoftrialtegrationsthatspan±5x1010yr，cdgonlytheouterfiveps(thefourjovianpspspto)theresultsexhibittherigorostabilityoftheouterparysysteoverthislongti-spanorbitalnfigurations(fig12)，andvariationofeentricitiesandclations(fig13)showthisverylong-terstabilityoftheouterfivepsboththetiandthefrequencydoasalthoughwedonotshowapshere，thetypicalfrequencyoftheorbitalosciltionofptoandtheotherouterpsisalostnstantdurgtheseverylong-tertegrationperiods，whichisdeonstratedtheti–frequencyapsonourwebpage

theseotegrations，theretivenuricalerrorthetotalenergywas～10?6andthatofthetotalangurontuwas～10?10

51resonancestheneptune–ptosyste

koshita≈ap;ap;ap;nakai(1996)tegratedtheouterfiveparyorbitsover±55x109yrtheyfoundthatfourajorresonancesbeeenneptuneandptoareataeddurgthewholetegrationperiod，andthattheresonancesaybetheacaesofthestabilityoftheorbitofptotheajorfourresonancesfoundprevioresearchareasfollowsthefollogdescription，λdenotestheanlongitude，Ωisthelongitudeoftheascendgnodeand?isthelongitudeofperihelionsubscriptspandndenoteptoandneptune

anotionresonancebeeenneptuneandpto(3:2)thecriticalarguntθ1=3λp?2λn??plibratesaround180°withanaplitudeofabout80°andalibrationperiodofabout2x104yr

thearguntofperihelionofptowp=θ2=?p?Ωplibratesaround90°withaperiodofabout38x106yrthedoantperiodicvariationsoftheeentricityandclationofptoaresynchronizedwiththelibrationofitsarguntofperihelionthisisanticipatedthesecurperturbationtheorynstructedbykozai(1962)

thelongitudeofthenodeofptoreferredtothelongitudeofthenodeofneptune，θ3=Ωp?Ωn，circutesandtheperiodofthiscircutionisequaltotheperiodofθ2librationwhenθ3beeszero，iethelongitudesofascendgnodesofneptuneandptooverp，theclationofptobeesaxiu，theeentricitybeesiuandthearguntofperihelionbees90°whenθ3bees180°，theclationofptobeesiu，theeentricitybeesaxiuandthearguntofperihelionbees90°agawillias≈ap;ap;ap;benson(1971)anticipatedthistypeofresonance，ternfirdbyini，nobili≈ap;ap;ap;carpo(1989)

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

ournuricaltegrations，theresonances(i)–(iii)arewellataed，andvariationofthecriticalarguntsθ1，θ2，θ3reasiirdurgthewholetegrationperiod(figs14–16)however，thefourthresonance(iv)appearstobedifferent:thecriticalarguntθ4alternateslibrationandcircutionovera1010-yrti-scale(fig17)thisisanterestgfactthatkoshita≈ap;ap;ap;nakai&039;s(1995，1996)shortertegrationswerenotabletodisclose

6discsion

whatkdofdynaicalchanisatasthislong-terstabilityoftheparysyste?wecanidiatelythkofoajorfeaturesthatayberesponsibleforthelong-terstabilityfirst，thereseetobenosignificantlower-orderresonances(anotionandsecur)beeenanypairaongthenepsjupiterandsaturnareclosetoa5:2anotionresonance(thefao‘greatequality’)，butnotjttheresonancezonehigher-orderresonancesaycaethechaoticnatureoftheparydynaicalotion，buttheyarenotsostrongastodestroythestableparyotionwiththelifetioftherealsorsystethesendfeature，whichwethkisoreiportantforthelong-terstabilityofourparysyste，isthedifferencedynaicaldistancebeeenterrestrialandjovianparysubsystes(ito≈ap;ap;ap;tanikawa1999，2001)whenweasureparyseparationsbytheutualhillradii(r_)，separationsaongterrestrialpsaregreaterthan26rh，whereasthoseaongjovianpsarelessthan14rhthisdifferenceisdirectlyretedtothedifferencebeeendynaicalfeaturesofterrestrialandjovianpsterrestrialpshavesallerasses，shorterorbitalperiodsandwiderdynaicalseparationtheyarestronglyperturbedbyjovianpsthathavergerasses，longerorbitalperiodsandnarrowerdynaicalseparationjovianpsarenotperturbedbyanyotherassivebodies

thepresentterrestrialparysysteisstillbegdisturbedbytheassivejovianpshowever，thewideseparationandutualteractionaongtheterrestrialpsrendersthedisturbanceeffective;thedegreeofdisturbancebyjovianpsiso(ej)(orderofagnitudeoftheeentricityofjupiter)，scethedisturbancecaedbyjovianpsisaforcedosciltionhavganaplitudeofo(ej)heightengofeentricity，forexapleo(ej)～005，isfarfrosufficienttoprovokestabilitytheterrestrialpshavgsuchawideseparationas26rhthweassuthatthepresentwidedynaicalseparationaongterrestrialps(≈ap;ap;gt;26rh)isprobablyoneoftheostsignificantnditionsforatagthestabilityoftheparysysteovera109-yrti-spanourdetailedanalysisoftheretionshipbeeendynaicaldistancebeeenpsandthestabilityti-scaleofsorsysteparyotionisnowon-gog

althoughournuricaltegrationsspanthelifetiofthesorsyste，thenuberoftegrationsisfarfrosufficienttofilltheitialphasespaceitisnecessarytoperfororeandorenuricaltegrationstonfirandexaedetailthelong-terstabilityofourparydynaics

——以上文段引自ito，t≈ap;ap;tanikawa，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。

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