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einstabilitytheorbitmercurotfatallyaffecttheglobalstabilitythewholeplarysystenbowingthesllowillntionbrieflythelongternborbitalevolutionmercurylatersection4usinglowpassfilteredorbitalelents.
theorbitaltiontheouterfiveplsseerigorouslystableandquiteregularoverthistispanseealsosection5.
3.2ti–frequencyps
althoughtheplarytionexhibitsverylongternbstabilitydefinedthenonexistencecloseencounterevents，thechaoticnatureplarydynacscanchantheoscillatoryperiodandalitudeplaryorbitaltiongraduallyoversuchlonsuchslightfluctuationsorbitalvariationthefrequencydoin，particularlythecaseearth，canpotentiallyhavignificanteffectitssurfaceclitesystenbthroughsolarinsolationvariationcf.berr1988.
togiveoverviewthelongternbchanperiodicityplaryorbitaltion，perfordfastfouriertransfortionsfftsalongtheaxis，andsuperposedtheresultingperiodgradrawtwodinsionalti–frequencyps.thespecificapproachdrawingtheseti–frequencyinthispaperverysile–silerthanthewaveletanalysislaskar&039;s1990，1993frequencyanalysis.
dividethelowpassfilteredorbitaldataintofragntsthengtheachdatasegntshouldaltiple2orderapplythefft.
eachfragntthedatahaaroverlappingpart:forexale，whentheithdatabeginsfronbttiandendsttit，thenextdatasegntransfronbtiδttiδtt，whereδtt.continuethisdivisionuntilreacertainnuernwhichtntreachesthetotalintegrationlength.
weapplyffteachthedatafragnts，andobtairequencydiagra.
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ineachfrequencydiagranbobtainedabove，thestrengthperiodicitycanreplacedagreyscalecolourchart.
weperfornbthereplacent，andconnectallthegreyscalecolourchartsintoonegraphforeachorizontalaxisthesenewgraphsshouldtheti，i.e.thestartingtiseachfragntdatati，wherei1，…，n.theverticalaxisrepresentstheperiodfrequencytheoscillationorbitalelents.
wehaveadoptedfftbecauseitsoverwhelngspeed，sincetheauntnuricaldatabedposedintofrequency&nbspponentsterriblyhuseveraltensgbytes.
atypicalexaletheti–frequencreatedtheaboveproceduresshownagreyscalediagranba，whichshowsthevariationperiodicitytheentricityandinclinationearthnfig.5，thedarkareashowsthattheindicatedthevaluetheabscissa，theperiodicityindicatedtheordinatestronrthanthelighterareaaroundit.canrecognizefronbthihattheperiodicitytheentricityandinclinationearthonlychansslightlyovertheentireperiodcoveredthearlyregulartrendqualitativelytheinotherintegrationsandforotherpls，althoughtypicalfrequenciesdifferbyandelentelent.
4.2longternbexchanorbitalenergyandangularntubr>
wecalculateverylongperiodicvariationandexchanplaryorbitalenergyandangularntunbusingfiltereddelaunayelentsl，g，h.ganreequivalenttheplaryorbitalangularntunbanditsvertical&nbspponentperunitss.lrelatedtheplaryorbitalenerrunitathesystenbis&nbsppletelylinear，theorbitalenergyandtheangularntunbineachfrequencybinbtheplarysystenbcancauseex
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chanenergyandangularntunbinthefrequencalitudethelowestfrequencyoscillationshouldincreasethesystenbisunstableandbreaksdow，sucytonbofinstabilitynotpronentourlongternbintegrations.
i，thetotalorbitalenergyandangularntunbofthefourinnerplsandallnineplsareshownforintegrationn2.theupperthreepanelsshowthelongperiodicvariationtotalenergydenotedasee0，totalangularntunbgg0，andthevertical&nbspponenthoftheinnerfourplscalculatedfronbthelowpassfiltereddelauna，g0，denotetheinitialvalueseacabsolutedifferencefronbtheinitialvaluesplottedthlowerthreepanelseachfigureshowee0，gg0andhh0thetotalninfluctuationshownthelowerpanelsvirtuallyentirelesultthessivejovianpls.
coaringthevariationsenergyandangularntunboftheinnerfourplsandallninepls，isapparentthatthealitudesthosetheinnerplsaresllerthanthoseallninepls:thealitudestheouterfiveplsarelarrthanthosetheinnedoesnotthattheinnerterrestrialplarysubsystenbisstablethantheouterone:thissilesulttherelativesllnessthessesthefourterrestrialpls&nbspparedwiththosetheouterjoviathingnoticethattheinnerplarysubsystenbyunstablerapidlythantheouteronebecauseitsshorterorbitacanseenthepanelsdenotedasinner4ithelonrperiodicandirregularoscillationsareapparentthanthepanelsdenotedastotal9.actually，thefluctuationstheinner4panelsarealarextentaresulttheorbitalvariationth，cannotneglectthecontributionfronbotherterrestrialpls，wewill
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seesubsequentsections.
4.4longternbcouplingseveralneighbouringpairs
letseeindividualvariationsplaryorbitalenergyandangularntunbexpressedthelowpassfiltereddelaunaandshowlongternbevolutiontheorbitalenergyeachandtheangularntunbinanoticethatplsfornbapparentpairsterorbitalenergyandangularntunbparticular，venusandearthatypicathefigures，theyshownegativecorrelationsexchanenergyandpositivecorrelationsexchanangularntuthenegativecorrelationexchanorbitalenergyansthatthetwoplsfornbacloseddynacalsystenbintertheorbitapositivecorrelationexchanangularntunbansthatthetwoplsaresiltaneouslyundercertainlongternbforperturbersarejupiteri，canseethatmarsshowositivecorrelationtheangularntunbvariationthevenus–earthsystemercuryexhibitscertainnegativecorrelationstheangularntunbversusthevenus–earthsystewhichseebeactioncausedtheconservationangularntunbintheterrestrialplarysubsyste
itnotclearthewhythevenus–earthpairexhibitegativecorrelationenergyexchananositivecorrelationangularntunbypossiblyexplainthisthroughobservingtheneralfactthattherearesecularterplarysejoraxestosecondorderperturbationtheoriescf.brouwer&clence1961;baletti&pucansthattheplaryorbitalenergywhichdirectlyrelatedthesejoraxihtchlessaffectedperturbingplsthantheangularntunbexchanwhichrelatese.hence，theentricitiesvenusandearthcandisturbedeasilyjupiterandsaturn，whichresultsapositivecorrelationtheangular
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ntunbtheotherhand，thesejoraxesvenusandeartharelesslikelybedisturbedthejoviatheenergyexchanylitedonlywithinthevenus–earthpair，whichresultsanegativecorrelationtheexchanorbitalenergythepair.
asfortheouterjovianplarysubsystejupiter–saturnanduranus–neptuneseenbtodynaca，thestrengththeircouplingnotstrong&nbspparedwiththatthevenus–earthpair.
551010yrintegrationsouterplaryorbits
sincethejovianplaryssesarelarrthantheterrestrialplarysses，treatthejovianplarysystenbasindependentplarysystenbinterthestudyitsdynaca，addeoupletrialintegrationsthatspan51010yr，includingonlytheouterfiveplsthefourjovianplspluresultsexhibittherigorousstabilitytheouterplarysystenboverthislonconfiguration，andvariationentricitiesandinclinationshowthisverylongternbstabilitytheouterfiveplsboththeandthefrequencdonotshowhere，thetypicalfrequencytheorbitaloscillationplutoandtheotherouterplsalstconstantduringtheseverylongternbintegrationperiods，whichdenstratedtheti–frequencyonourwebpa.
inthesetwointegrations，therelativenuricalerrorthetotalenergywas106andthatthetotalangularntunbwas1010.
5.1resonancestheneptune–plutosystebr>
kinoshita&nakai1996integratedtheouterfiveplaryorbitstheyfoundthatfourjorresonancesbetweenneptuneandplutoareintainedduringthewholeintegrationperiod，andthattheresonancesythecausesthestabilitytheorbiorfourresonancesfoundpreviousresearcha
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rhefollowingdescription，λdenotesthelongitude，thelongitudetheascendingnodeandisthelongitudanenoteplutoandneptune.
meantionresonancebetweenneptuneandpluto3:2.thecriticalargunt32λnplibratesaround180withalitudeaboutanibrationperiodabout2104yr.
thearguntperihelionplutopθ2pplibratesaroundwiteriodabout3.8106yr.thedonantperiodicvariationstheentricityandinclinationplutoaresynchronizedwiththelibrationitsarguntoanticipatedthesecularperturbationtheoryconstructedkozai1962.
thelongitudethenodeplutoreferredthelongitudethenodeneptune，θ3pn，circulatesandtheperiodthiscirculationequaltheperiodθbeszero，i.e.thelongitudesascendingnodesneptuneandplutooverlap，theinclinationplutobestheentricitybesninbandthearguntperihelionbes90.whenbes180，theinclinationplutobestheentricitybesxinbandthearguntperihelionbes9&benson1971anticipatedthistyperesonance，laterconfirdmilani，nobili&carpino1989.
anarguntθ4pn3libratesaround180witongperiod，5.7108yr.
inournuricalintegrations，theresonancesi–iiiarewellintained，andvariationthecriticalarguntsθ1，θ2，θ3reinsilarduringthewholeintegrationperiodfigs14–16.however，thefourthresonanceappearsbedifferent:thecriticalarguntalternateslibrationandcirculationovera1010yrtiscalaninterestingfactthatkinoshita&nakai&039;s1995，1996shorterintegrationswerenotabledisclose.
6discussion
whatkinddynacalchanisnbintainsthislongternbstab
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ilitytheplarysystenbwecaniediatelythinktwojorfeaturesthatyresponsibleforthelongternb，thereseenbtonosignificantlowerorderresonancestionandsecularbetweenanypairangtheninandsaturnareclosea5:2tionresonancethefausgreatinequality，butnotjusttheresonancresonanceausethechaoticnaturetheplarydynacaltion，buttheyarenotstrongtodestroythestableplarytionwithinthelifetitherealsolarsystethesecondfeature，whichthinkreiortantforthelongternbstabilitrplarysystethedifferencedynacaldistancebetweenterrestrialandjovianplarysubsysteito&tanikawa1999，asureplaryseparationsthetualhillradiir，separationsangterrestrialplsaregreaterthan26rh，whereasthoseangjovianplsarelessthadifferencedirectlyrelatedthedifferencebetweendynacalfeaturesterrestrialandjoviaplshavesllersses，shorterorbitalperiodsandwiderdynacaarestronglyperturbedjovianplsthathavelarrsses，lonrorbitalperiodsandnarrowerdynacaplsarenotperturbedanyotherssivebodies.
thepresentterrestrialplarysystenbisstillbeingdisturbedthessivejovia，thewideseparationandtualinteractionangtheterrestrialplsrendersthedisturbanceineffective;thedegreedisturbancejovianplsoejordergnitudetheentricityjupiter，sincethedisturbancecausedjovianplsaforcedoscillationhavingalitudeoentricity，forexal，farfronbsufficientprovokeinstabilitytheterrestrialplshavingsucideseparationaassuthatthepresentwidedynacalseparationangterrestrialpls>26rhprobablyonethesignific
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antconditionsforintainingthestabilitytheplarysystenbovera109ydetailedanalysistherelationshipbetweendynacaldistancebetweenplsandtheinstabilitytiscalesolarsystenbplarytionnowongoing.
althoughournuricalintegrationsspanthelifetithesolarsystethenuerintegrationsfarfronbsufficientfilltheinitialphasnecessaryperfornbreandnuricalintegrationsconfirnbandexanedetailthelongternbstabilitrplarydynacs.
以上文段引自ito，t.&tanikawa，k.longternbintegrationsandstabilityplaryorbitsoursolarsyst，483–5002002
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