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pargthevariationsofenergyandangularontuofthenerfourplasandallneplas,itisapparentthattheaplitudesofthoseofthenerplasareuchsallerthanthoseofallneplas:theaplitudesoftheouterfiveplasareuchlarrthanthoseofthenerplasthisdoesnotanthatthenerterrestrialplaarysubsysteisorestablethantheouterone:thisissiplyaresultoftherelativesallnessoftheassesofthefourterrestrialplasparedwiththoseoftheouterjovianplasanotherthgwenoticeisthatthenerplaarysubsysteaybeeunstableorerapidlythantheouteronebecaeofitsshorterorbitalti-scalesthiscanbeseenthepanelsdenotedasner4fig7wherethelonr-periodicandirregularoscillationsareoreapparentthanthepanelsdenotedastotal9actually,thefctuationsthener4panelsaretoalarextentasaresultoftheorbitalvariationofthercuryhowever,wecannotneglectthentributionfrootherterrestrialplas,aswewillseesubseentsections
44long-teruplgofseveralneighbourgplapairs
letseedividualvariationsofplaaryorbitalenergyandangularontuexpressedbythelow-passfiltereddelaunayelentsfigs10and11showlong-terevotionoftheorbitalenergyofeachplaandtheangularontun+1andn&;2tegrationswenoticethatplasforapparentpairstersoforbitalenergyandangularontuex插nparticular,venandearthakeatypicalpairthefigures,theyshownegativerrelationsex插nofenergyandpositiverrelationsex插nofangularontuthenegativerrelationex插noforbitalenergyansthattheoplasforacloseddynaicalsystetersoftheorbitalenergythepositiverrelationex插nofangularontuansthattheoplasaresiultaneolyundercertalong-terperturbationscandidatesforperturbersarejupiterandsaturnalfig11,wecanseethatarsshowsapositiverrelationtheangularontuvariationtotheven–earthsystercuryexhibitscertanegativerrelationstheangularontuverstheven–earthsyste,whichseestobeareactioncaedbythenservationofangularontutheterrestrialplaarysubsyste
itisnotclearattheontwhytheven–earthpairexhibitsanegativerrelationenergyex插nandapositiverrelationangularontuex插nweaypossiblyexpnthisthroughobservgtheneralfactthattherearenoseculartersplaaryseiajoraxesuptosend-orderperturbationtheories(cfbrouwer≈ap;clence1961;boaletti≈ap;pucao1998)thisansthattheplaaryorbitalenergy(whichisdirectlyrelatedtotheseiajoraxisa)ightbeuchlessaffectedbyperturbgplasthanistheangularontuex插n(whichrelatestoe)hence,theeentricitiesofvenandearthcanbedisturbedeasilybyjupiterandsaturn,whichresultsapositiverrelationtheangularontuex插nontheotherhand,theseiajoraxesofvenandeartharelesslikelytobedisturbedbythejovianplasththeenergyex插naybeliitedonlywiththeven–earthpair,whichresultsanegativerrelationtheex插noforbitalenergythepair
asfortheouterjovianplaarysubsyste,jupiter–saturnanduran–neptuneseetoakedynaicalpairshowever,thestrengthoftheiruplgisnotasstrongparedwiththatoftheven–earthpair
5&psn;5&tis;1010-yrtegrationsofouterplaaryorbits
scethejovianplaaryassesareuchlarrthantheterrestrialplaaryasses,wetreatthejovianplaarysysteasandependentplaarysystetersofthestudyofitsdynaicalstabilityhence,weaddedaupleoftrialtegrationsthatspan&psn;5&tis;1010yr,cdgonlytheouterfiveplas(thefourjovianplaspspto)theresultsexhibittherigorostabilityoftheouterplaarysysteoverthislongti-spanorbitalnfigurations(fig12),andvariationofeentricitiesandclations(fig13)showthisverylong-terstabilityoftheouterfiveplasboththetiandthefreencydoasalthoughwedonotshowapshere,thetypicalfreencyoftheorbitaloscillationofptoandtheotherouterplasisalostnstantdurgtheseverylong-tertegrationperiods,whichisdeonstratedtheti–freencyapsonourwebpa
theseotegrations,therelativenuricalerrorthetotalenergywas&si;10&;6andthatofthetotalangularontuwas&si;10&;10
51renancestheneptune–ptosyste
koshita≈ap;nakai(1996)tegratedtheouterfiveplaaryorbitsover&psn;55&tis;109yrtheyfoundthatfourajorrenancesbeeenneptuneandptoareataeddurgthewholetegrationperiod,andthattherenancesaybetheacaesofthestabilityoftheorbitofptotheajorfourrenancesfoundprevioresearchareasfollowsthefollogdescription,&labda;denotestheanlongitude,&oga;isthelongitudeoftheascendgnodeandϖisthelongitudeofperihelionsubscriptspandndenoteptoandneptune
anotionrenancebeeenneptuneandpto(3:2)thecriticalarguntθ1=3&labda;p&;2&labda;n&;ϖplibratesaround180°withanaplitudeofabout80°andalibrationperiodofabout2&tis;104yr
thearguntofperihelionofpto&oga;p=θ2=ϖp&;&oga;plibratesaround90°withaperiodofabout38&tis;106yrthedoantperiodicvariationsoftheeentricityandclationofptoaresynchronizedwiththelibrationofitsarguntofperihelionthisisanticipatedthesecularperturbationtheorynstructedbykozai(1962)