对火星轨道变化问题的最后解释 (4 / 5)
        Ineachfrequencydiagramobtainedabove，thestrengthofperiodicitycanbereplacedbyagrey-scale(orcolour)chart.

        Weperformthereplacement，andconnectallthegrey-scale(orcolour)chartsintoonegraphforeachintegration.Thehorizontalaxisofthesenewgraphsshouldbethetime，i.e.thestartingtimesofeachfragmentofdata(ti，wherei=1，…，n).Theverticalaxisrepresentstheperiod(orfrequency)oftheoscillationoforbitalelements.

        WehaveadoptedanFFTbecauseofitsoverwhelmingspeed，sincetheamountofnumericaldatatobedeposedintofrequencyponentsisterriblyhuge(severaltensofGbytes).

        Atypicalexampleofthetime–frequencymapcreatedbytheaboveproceduresisshowninagrey-scalediagramasFig.5，whichshowsthevariationofperiodicityintheeccentricityandinclinationofEarthinN+2integration.InFig.5，thedarkareashowsthatatthetimeindicatedbythevalueontheabscissa，theperiodicityindicatedbytheordinateisstrongerthaninthelighterareaaroundit.WecanrecognizefromthismapthattheperiodicityoftheeccentricityandinclinationofEarthonlychangesslightlyovertheentireperiodcoveredbytheN+2integration.Thisnearlyregulartrendisqualitativelythesameinotherintegrationsandforotherplanets，althoughtypicalfrequenciesdifferplanetbyplanetandelementbyelement.

        4.2Long-termexchangeoforbitalenergyandangularmomentum

        Wecalculateverylong-periodicvariationandexchangeofplanetaryorbitalenergyandangularmomentumusingfilteredDelaunayelementsL，G，H.GandHareequivalenttotheplanetaryorbitalangularmomentumanditsverticalponentperunitmass.LisrelatedtotheplanetaryorbitalenergyEperunitmassasE=?μ2/2L2.Ifthesystemispletelylinear，theorbitalenergyandtheangularmomentumineachfrequencybinmustbeconstant.Non-linearityintheplanetarysystemcancauseanexchangeofenergyandangularmomentuminthefrequencydomain.Theamplitudeofthelowest-frequencyoscillationshouldincreaseifthesystemisunstableandbreaksdowngradually.However，suchasymptomofinstabilityisnotprominentinourlong-termintegrations.

        InFig.7，thetotalorbitalenergyandangularmomentumofthefourinnerplanetsandallnineplanetsareshownforintegrationN+2.Theupperthreepanelsshowthelong-periodicvariationoftotalenergy(denotedasE-E0)，totalangularmomentum(G-G0)，andtheverticalponent(H-H0)oftheinnerfourplanetscalculatedfromthelow-passfilteredDelaunayelements.E0，G0，H0denotetheinitialvaluesofeachquantity.Theabsolutedifferencefromtheinitialvaluesisplottedinthepanels.ThelowerthreepanelsineachfigureshowE-E0，G-G0andH-H0ofthetotalofnineplanets.Thefluctuationshowninthelowerpanelsisvirtuallyentirelyaresultofthemassivejovianplanets.

        Comparingthevariationsofenergyandangularmomentumoftheinnerfourplanetsandallnineplanets，itisapparentthattheamplitudesofthoseoftheinnerplanetsaremuchsmallerthanthoseofallnineplanets:theamplitudesoftheouterfiveplanetsaremuchlargerthanthoseoftheinnerplanets.Thisdoesnotmeanthattheinnerterrestrialplanetarysubsystemismorestablethantheouterone:thisissimplyaresultoftherelativesmallnessofthemassesofthefourterrestrialplanetsparedwiththoseoftheouterjovianplanets.Anotherthingwenoticeisthattheinnerplanetarysubsystemmaybeeunstablemorerapidlythantheouteronebecauseofitsshorterorbitaltime-scales.Thiscanbeseeninthepanelsdenotedasinner4inFig.7wherethelonger-periodicandirregularoscillationsaremoreapparentthaninthepanelsdenotedastotal9.Actually，thefluctuationsintheinner4panelsaretoalargeextentasaresultoftheorbitalvariationoftheMercury.However，wecannotneglectthecontributionfromotherterrestrialplanets，aswewillseeinsubsequentsections.

        4.4Long-termcouplingofseveralneighbouringplanetpairs

        Letusseesomeindividualvariationsofplanetaryorbitalenergyandangularmomentumexpressedbythelow-passfilteredDelaunayelements.Figs10and11showlong-termevolutionoftheorbitalenergyofeachplanetandtheangularmomentuminN+1andN?2integrations.Wenoticethatsomeplanetsformapparentpairsintermsoforbitalenergyandangularmomentumexchange.Inparticular，VenusandEarthmakeatypicalpair.Inthefigures，theyshownegativecorrelationsinexchangeofenergyandpositivecorrelationsinexchangeofangularmomentum.Thenegativecorrelationinexchangeoforbitalenergymeansthatthetwoplanetsformacloseddynamicalsystemintermsoftheorbitalenergy.Thepositivecorrelationinexchangeofangularmomentummeansthatthetwoplanetsaresimultaneouslyundercertainlong-termperturbations.CandidatesforperturbersareJupiterandSaturn.AlsoinFig.11，wecanseethatMarsshowsapositivecorrelationintheangularmomentumvariationtotheVenus–Earthsystem.MercuryexhibitscertainnegativecorrelationsintheangularmomentumversustheVenus–Earthsystem，whichseemstobeareactioncausedbytheconservationofangularmomentumintheterrestrialplanetarysubsystem.

        ItisnotclearatthemomentwhytheVenus–Earthpairexhibitsanegativecorrelationinenergyexchangeandapositivecorrelationinangularmomentumexchange.Wemaypossiblyexplainthisthroughobservingthegeneralfactthattherearenoseculartermsinplanetarysemimajoraxesuptosecond-orderperturbationtheories(cf.Brouwer&Clemence1961;Boccaletti&Pucacco1998).Thismeansthattheplanetaryorbitalenergy(whichisdirectlyrelatedtothesemimajoraxisa)mightbemuchlessaffectedbyperturbingplanetsthanistheangularmomentumexchange(whichrelatestoe).Hence，theeccentricitiesofVenusandEarthcanbedisturbedeasilybyJupiterandSaturn，whichresultsinapositivecorrelationintheangularmomentumexchange.Ontheotherhand，thesemimajoraxesofVenusandEartharelesslikelytobedisturbedbythejovianplanets.ThustheenergyexchangemaybelimitedonlywithintheVenus–Earthpair，whichresultsinanegativecorrelationintheexchangeoforbitalenergyinthepair.

        Asfortheouterjovianplanetarysubsystem，Jupiter–SaturnandUranus–Neptuneseemtomakedynamicalpairs.However，thestrengthoftheircouplingisnotasstrongparedwiththatoftheVenus–Earthpair.

        5±5×1010-yrintegrationsofouterplanetaryorbits

        Sincethejovianplanetarymassesaremuchlargerthantheterrestrialplanetarymasses，wetreatthejovianplanetarysystemasanindependentplanetarysystemintermsofthestudyofitsdynamicalstability.Hence，weaddedacoupleoftrialintegrationsthatspan±5×1010yr，includingonlytheouterfiveplanets(thefourjovianplanetsplusPluto).Theresultsexhibittherigorousstabilityoftheouterplanetarysystemoverthislongtime-span.Orbitalconfigurations(Fig.12)，andvariationofeccentricitiesandinclinations(Fig.13)showthisverylong-termstabilityoftheouterfiveplanetsinboththetimeandthefrequencydomains.Althoughwedonotshowmapshere，thetypicalfrequencyoftheorbitaloscillationofPlutoandtheotherouterplanetsisalmostconstantduringtheseverylong-termintegrationperiods，whichisdemonstratedinthetime–frequencymapsonourwebpage.

        Inthesetwointegrations，therelativenumericalerrorinthetotalenergywas～10?6andthatofthetotalangularmomentumwas～10?10.

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