对火星轨道变化问题的最后解释（3/4）

thevariationofeccentricitiesandorbitalinclinationsfortheinnerfourplanetsintheinitialandfinalpartoftheintegrationn+，thecharacterofthevariationofplanetaryorbitalelementsdoesnotdiffersignificantlybetweentheinitialandfinalpartofeachintegration，atleastforvenus，，especiallyitseccentricity，-scaleoftheplanetistheshortestofalltheplanets，whichleadstoamorerapidorbitalevolutionthanotherplanets;'s(1994，1996)expectationsthatlargeandirregularvariationsappearintheeccentricitiesandinclinationsofmercuryonatime-，-termorbitalevolutionofmercurylaterinsection4usinglow-

theorbitalmotionoftheouterfiveplanetsseemsrigorouslystableandquiteregularoverthistime-span(seealsosection5).

–frequencymaps

althoughtheplanetarymotionexhibitsverylong-termstabilitydefinedasthenon-existenceofcloseencounterevents，thechaoticnatureofplanetarydynamicscanchangetheoscillatoryperiodandamplitudeofplanetaryorbitalmotiongraduallyoversuchlongtime-，particularlyinthecaseofearth，canpotentiallyhaveasignificanteffectonitssurfaceclimatesystemthroughsolarinsolationvariation().

togiveanoverviewofthelong-termchangeinperiodicityinplanetaryorbitalmotion，weperformedmanyfastfouriertransformations(ffts)alongthetimeaxis，andsuperposedtheresultingperiodgramstodrawtwo-dimensionaltime––frequencymapsinthispaperisverysimple–muchsimplerthanthewaveletanalysisorlaskar's(1990，1993)

dividethelow-

eachfragmentofthedatahasalargeoverlappingpart:forexample，whentheithdatabeginsfromt=tiandendsatt=ti+t，thenextdatasegmentrangesfromti+δt≤ti+δt+t，whereδt?+

weapplyanffttoeachofthedatafragments，

ineachfrequencydiagramobtainedabove，thestrengthofperiodicitycanbereplacedbyagrey-scale(orcolour)

weperformthereplacement，andconnectallthegrey-scale(orcolour)，(ti，wherei=1，…，n).theverticalaxisrepresentstheperiod(orfrequency)

wehaveadoptedanfftbecauseofitsoverwhelmingspeed，sincetheamountofnumericaldatatobedecomposedintofrequencycomponentsisterriblyhuge(severaltensofgbytes).

atypicalexampleofthetime–frequencymapcreatedbytheaboveproceduresisshowninagrey-，whichshowsthevariationofperiodicityintheeccentricityandinclinationofearthinn+，thedarkareashowsthatatthetimeindicatedbythevalueontheabscissa，+，

-termexchangeoforbitalenergyandangularmomentum

wecalculateverylong-periodicvariationandexchangeofplanetaryorbitalenergyandangularmomentumusingfiltereddelaunayelementsl，g，=?μ2/，--，suchasymptomofinstabilityisnotprominentinourlong-

，thetotalorbitalenergyandangularmomentumofthefourinnerplanetsandallnineplanetsareshownforintegrationn+-periodicvariationoftotalenergy(denotedase-e0)，totalangularmomentum(g-g0)，andtheverticalcomponent(h-h0)oftheinnerfourplanetscalculatedfromthelow-，g0，-e0，g-g0andh-

comparingthevariationsofenergyandangularmomentumoftheinnerfourplanetsandallnineplanets，itisapparentthattheamplitudesofthoseoftheinnerplanetsaremuchsmallerthanthoseofallnineplanets::--，，wecannotneglectthecontributionfromotherterrestrialplanets，

-termcouplingofseveralneighbouringplanetpairs

letusseesomeindividualvariationsofplanetaryorbitalenergyandangularmomentumexpressedbythelow--termevolutionoftheorbitalenergyofeachplanetandtheangularmomentuminn+1andn?，，-，wecanseethatmarsshowsapositivecorrelationintheangularmomentumvariationtothevenus––earthsystem，

itisnotclearatthemomentwhythevenus–-orderperturbationtheories(&clemence1961;boccaletti&pucacco1998).thismeansthattheplanetaryorbitalenergy(whichisdirectlyrelatedtothesemimajoraxisa)mightbemuchlessaffectedbyperturbingplanetsthanistheangularmomentumexchange(whichrelatestoe).hence，theeccentricitiesofvenusandearthcanbedisturbedeasilybyjupiterandsaturn，，–earthpair，