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