第76章 对火星轨道变化问题的最后解释(9/14)
rey-scalediagramasfig.5，whichshowsthevariationofperiodicityintheeccentricityandinclinationofearthinn+2integration.infig.5，thedarkareashowsthatatthetimeindicatedbythevalueontheabscissa，theperiodicityindicatedbytheordinateisstrongerthaninthelighterareaaroundit.wecanrecognizefromthismapthattheperiodicityoftheeccentricityandinclinationofearthonlychangesslightlyovertheentireperiodcoveredbythen+2integration.thisnearlyregulartrendisqualitativelythesameinotherintegrationsandforotherplanets，althoughtypicalfrequenciesdifferplanetbyplanetandelementbyelement.
4.2long-termexchangeoforbitalenergyandangularmomentum
wecalculateverylong-periodicvariationandexchangeofplanetaryorbitalenergyandangularmomentumusingfiltereddelaunayelementsl，g，h.gandhareequivalenttotheplanetaryorbitalangularmomentumanditsverticalcomponentperunitmass.lisrelatedtotheplanetaryorbitalenergyeperunitmassase=?μ22l2.ifthesystemiscompletelylinear，theorbitalenergyandtheangularmomentumineachfrequencybinmustbeconstant.non-linearityintheplanetarysystemcancauseanexchangeofenergyandangularmomentuminthefrequencydomain.theamplitudeofthelowest-frequencyoscillationshouldincreaseifthesystemisunstableandbreaksdowngradually.however，suchasymptomofinstabilityisnotprominentinourlong-termintegrations.
infig.7，thetotalorbitalenergyandangularmomentumofthefourinnerplanetsandallnineplanetsareshownforintegrationn+2.theupperthreepanelsshowthelong-periodicvariationoftotalenergy(denotedase-e0)，totalangularmomentum(g-g0)，andtheverticalcomponent(h-h0)oftheinnerfourplanetscalculatedfromthelow-passfiltereddelaunayelements.e0，g0，h0denotetheinitialvaluesofeachquantity.theabsolutedifferencefromtheinitialvaluesisplottedinthepanels.thelowerthreepanelsineachfigureshowe-e0，g-g0andh-h0ofthetotalofnineplanets.thefluctuationshowninthelowerpanelsisvirtuallyentirelyaresultofthemassivejovianplanets.
comparingthevariationsofenergyandangularmomentumoftheinnerfourplanetsandallnineplanets，itisapparentthattheamplitudesofthoseoftheinnerplane
4.2long-termexchangeoforbitalenergyandangularmomentum
wecalculateverylong-periodicvariationandexchangeofplanetaryorbitalenergyandangularmomentumusingfiltereddelaunayelementsl，g，h.gandhareequivalenttotheplanetaryorbitalangularmomentumanditsverticalcomponentperunitmass.lisrelatedtotheplanetaryorbitalenergyeperunitmassase=?μ22l2.ifthesystemiscompletelylinear，theorbitalenergyandtheangularmomentumineachfrequencybinmustbeconstant.non-linearityintheplanetarysystemcancauseanexchangeofenergyandangularmomentuminthefrequencydomain.theamplitudeofthelowest-frequencyoscillationshouldincreaseifthesystemisunstableandbreaksdowngradually.however，suchasymptomofinstabilityisnotprominentinourlong-termintegrations.
infig.7，thetotalorbitalenergyandangularmomentumofthefourinnerplanetsandallnineplanetsareshownforintegrationn+2.theupperthreepanelsshowthelong-periodicvariationoftotalenergy(denotedase-e0)，totalangularmomentum(g-g0)，andtheverticalcomponent(h-h0)oftheinnerfourplanetscalculatedfromthelow-passfiltereddelaunayelements.e0，g0，h0denotetheinitialvaluesofeachquantity.theabsolutedifferencefromtheinitialvaluesisplottedinthepanels.thelowerthreepanelsineachfigureshowe-e0，g-g0andh-h0ofthetotalofnineplanets.thefluctuationshowninthelowerpanelsisvirtuallyentirelyaresultofthemassivejovianplanets.
comparingthevariationsofenergyandangularmomentumoftheinnerfourplanetsandallnineplanets，itisapparentthattheamplitudesofthoseoftheinnerplane
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