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thevariationofeentricitiesandorbitalclationsforthenerfourplastheitialandfalpartofthetegrationn+1isshownfig4asexpected,the插racterofthevariationofplaaryorbitalelentsdoesnotdiffersignificantlybeeentheitialandfalpartofeachtegration,atleastforven,earthandarstheelentsofrcury,especiallyitseentricity,seeto插ntoasignificantextentthisispartlybecaetheorbitalti-scaleoftheplaistheshortestofalltheplas,whichleadstoaorerapidorbitalevotionthanotherplas;thenerostplaaybenearesttostabilitythisresultappearstobeagreentwithlaskar&39;s(1994,1996)expectationsthatlarandirregularvariationsappeartheeentricitiesandclationsofrcuryonati-scaleofseveral109yrhowever,theeffectofthepossiblestabilityoftheorbitofrcuryaynotfatallyaffecttheglobalstabilityofthewholeplaarysysteogtothesallassofrcurywewillntionbrieflythelong-terorbitalevotionofrcurylatersection4glow-passfilteredorbitalelents
theorbitalotionoftheouterfiveplasseesrigorolystableanditeregularoverthisti-span(seealsection5)
32ti–freencyaps
althoughtheplaaryotionexhibitsverylong-terstabilitydefedasthenon-existenceofcloseenunterevents,the插oticnatureofplaarydynaicscan插ntheoscillatoryperiodandaplitudeofplaaryorbitalotiongraduallyoversuchlongti-spansevensuchslightfctuationsoforbitalvariationthefreencydoa,particularlythecaseofearth,canpotentiallyhaveasignificanteffectonitssurfacecliatesystethroughlarlationvariation(cfberr1988)
togiveanoverviewofthelong-ter插nperiodicityplaaryorbitalotion,weperfordanyfastfouriertransforations(ffts)alongthetiaxis,andsuperposedtheresultgperiodgrastodrawo-dinsionalti–freencyapsthespecificapproachtodragtheseti–freencyapsthispaperisverysiple–uchsiplerthanthewaveletanalysisorlaskar&39;s(1990,1993)freencyanalysis
dividethelow-passfilteredorbitaldatatoanyfragntsofthesalengththelengthofeachdatasegntshouldbeaultipleof2ordertoapplythefft
eachfragntofthedatahasalaroverlappgpart:forexaple,whentheithdatabegsfrot=tiandendsatt=ti+t,thenextdatasegntransfroti+δt≤ti+δt+t,whereδt?twentuethisdivisionuntilwereachacertanubernbywhichtn+treachesthetotaltegrationlength
weapplyanffttoeachofthedatafragnts,andobtanfreencydiagras
eachfreencydiagraobtaedabove,thestrengthofperiodicitycanbereplacedbyagrey-scale(orlour)插rt
weperforthereplacent,andnnectallthegrey-scale(orlour)插rtstoonegraphforeachtegrationthehorizontalaxisofthesenewgraphsshouldbetheti,iethestartgtisofeachfragntofdata(ti,wherei=1,…,n)theverticalaxisrepresentstheperiod(orfreency)oftheoscillationoforbitalelents
wehaveadoptedanfftbecaeofitsoverwhelgspeed,scetheaountofnuricaldatatobedeposedtofreencyponentsisterriblyhu(severaltensofgbytes)
atypicalexapleoftheti–freencyapcreatedbytheaboveproceduresisshownagrey-scalediagraasfig5,whichshowsthevariationofperiodicitytheeentricityandclationofearthn+2tegrationfig5,thedarkareashowsthatatthetidicatedbythevaeontheabscissa,theperiodicitydicatedbytheordateisstronrthanthelighterareaarounditwecanregnizefrothisapthattheperiodicityoftheeentricityandclationofearthonly插nsslightlyovertheentireperiodveredbythen+2tegrationthisnearlyregulartrendisalitativelythesaothertegrationsandforotherplas,althoughtypicalfreenciesdifferplabyplaandelentbyelent
42long-terex插noforbitalenergyandangularontu
wecalculateverylong-periodicvariationandex插nofplaaryorbitalenergyandangularontugfiltereddelaunayelentsl,g,hgandhareeivalenttotheplaaryorbitalangularontuanditsverticalponentperunitasslisrelatedtotheplaaryorbitalenergyeperunitassase=&;&u;22l2ifthesysteispletelylear,theorbitalenergyandtheangularontueachfreencybtbenstantnon-learitytheplaarysystecancaeanex插nofenergyandangularontuthefreencydoatheaplitudeofthelowest-freencyoscillationshouldcreaseifthesysteisunstableandbreaksdowngraduallyhowever,suchasyptoofstabilityisnotproentourlong-tertegrations
fig7,thetotalorbitalenergyandangularontuofthefournerplasandallneplasareshownfortegrationn+2theupperthreepanelsshowthelong-periodicvariationoftotalenergy(denotedase-e0),totalangularontu(g-g0),andtheverticalponent(h-h0)ofthenerfourplascalculatedfrothelow-passfiltereddelaunayelentse0,g0,h0denotetheitialvaesofeachantitytheabtedifferencefrotheitialvaesisplottedthepanelsthelowerthreepanelseachfigureshowe-e0,g-g0andh-h0ofthetotalofneplasthefctuationshownthelowerpanelsisvirtuallyentirelyaresultoftheassivejovianplas