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对火星轨道变化问题的最后解释（第2页）

ThevariationofeccentricitiesandorbitalinclinationsfortheinnerfourplanetsintheinitialandfinalpartoftheintegrationN+1isshowninFig。4。Asexpected，thecharacterofthevariationofplanetaryorbitalelementsdoesnotdiffersignificantlybetweentheinitialandfinalpartofeachintegration，atleastforVenus，EarthandMars。TheelementsofMercury，especiallyitseccentricity，seemtochangetoasignificantextent。Thisispartlybecausetheorbitaltime-scaleoftheplanetistheshortestofalltheplanets，whichleadstoamorerapidorbitalevolutionthanotherplanets;theinnermostplanetmaybenearesttoinstability。ThisresultappearstobeinsomeagreementwithLaskars（1994，1996）expectationsthatlargeandirregularvariationsappearintheeccentricitiesandinclinationsofMercuryonatime-scaleofseveral109yr。However，theeffectofthepossibleinstabilityoftheorbitofMercurymaynotfatallyaffecttheglobalstabilityofthewholeplanetarysystemowingtothesmallmassofMercury。Wewillmentionbrieflythelong-termorbitalevolutionofMercurylaterinSection4usinglow-passfilteredorbitalelements。

Theorbitalmotionoftheouterfiveplanetsseemsrigorouslystableandquiteregularoverthistime-span（seealsoSection5）。

3。2Time–frequencymaps

Althoughtheplanetarymotionexhibitsverylong-termstabilitydefinedasthenon-existenceofcloseencounterevents，thechaoticnatureofplanetarydynamicscanchangetheoscillatoryperiodandamplitudeofplanetaryorbitalmotiongraduallyoversuchlongtime-spans。Evensuchslightfluctuationsoforbitalvariationinthefrequencydomain，particularlyinthecaseofEarth，canpotentiallyhaveasignificanteffectonitssurfaceclimatesystemthroughsolarinsolationvariation（cf。Berger1988）。

Togiveanoverviewofthelong-termchangeinperiodicityinplanetaryorbitalmotion，weperformedmanyfastFouriertransformations（FFTs）alongthetimeaxis，andsuperposedtheresultingperiodgramstodrawtwo-dimensionaltime–frequencymaps。Thespecificapproachtodrawingthesetime–frequencymapsinthispaperisverysimple–muchsimplerthanthewaveletanalysisorLaskars（1990，1993）frequencyanalysis。

Dividethelow-passfilteredorbitaldataintomanyfragmentsofthesamelength。Thelengthofeachdatasegmentshouldbeamultipleof2inordertoapplytheFFT。

Eachfragmentofthedatahasalargeoverlappingpart:forexample，whentheithdatabeginsfromt=tiandendsatt=ti+T，thenextdatasegmentrangesfromti+δT≤ti+δT+T，whereδT?T。WecontinuethisdivisionuntilwereachacertainnumberNbywhichtn+Treachesthetotalintegrationlength。

WeapplyanFFTtoeachofthedatafragments，andobtainnfrequencydiagrams。

Ineachfrequencydiagramobtainedabove，thestrengthofperiodicitycanbereplacedbyagrey-scale（orcolour）chart。

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

WehaveadoptedanFFTbecauseofitsoverwhelmingspeed，sincetheamountofnumericaldatatobedecomposedintofrequencycomponentsisterriblyhuge（severaltensofGbytes）。

Atypicalexampleofthetime–frequencymapcreatedbytheaboveproceduresisshowninagrey-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，itisapparentthattheamplitudesofthoseoftheinnerplanetsaremuchsmallerthanthoseofallnineplanets:theamplitudesoftheouterfiveplanetsaremuchlargerthanthoseoftheinnerplanets。Thisdoesnotmeanthattheinnerterrestrialplanetarysubsystemismorestablethantheouterone:thisissimplyaresultoftherelativesmallnessofthemassesofthefourterrestrialplanetscomparedwiththoseoftheouterjovianplanets。Anotherthingwenoticeisthattheinnerplanetarysubsystemmaybecomeunstablemorerapidlythantheouteronebecauseofitsshorterorbitaltime-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，thestrengthoftheircouplingisnotasstrongcomparedwiththatoftheVenus–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。

5。1ResonancesintheNeptune–Plutosystem

Kinoshita&Nakai（1996）integratedtheouterfiveplanetaryorbitsover±5。5×109yr。TheyfoundthatfourmajorresonancesbetweenNeptuneandPlutoaremaintainedduringthewholeintegrationperiod，andthattheresonancesmaybethemaincausesofthestabilityoftheorbitofPluto。Themajorfourresonancesfoundinpreviousresearchareasfollows。Inthefollowingdescription，λdenotesthemeanlongitude，Ωisthelongitudeoftheascendingnodeand?isthelongitudeofperihelion。SubscriptsPandNdenotePlutoandNeptune。

MeanmotionresonancebetweenNeptuneandPluto（3:2）。Thecriticalargumentθ1=3λP?2λN??Plibratesaround180°withanamplitudeofabout80°andalibrationperiodofabout2×104yr。

TheargumentofperihelionofPlutoωP=θ2=?P?ΩPlibratesaround90°withaperiodofabout3。8×106yr。ThedominantperiodicvariationsoftheeccentricityandinclinationofPlutoaresynchronizedwiththelibrationofitsargumentofperihelion。ThisisanticipatedinthesecularperturbationtheoryconstructedbyKozai（1962）。

ThelongitudeofthenodeofPlutoreferredtothelongitudeofthenodeofNeptune，θ3=ΩP?ΩN，circulatesandtheperiodofthiscirculationisequaltotheperiodofθ2libration。Whenθ3becomeszero，i。e。thelongitudesofascendingnodesofNeptuneandPlutooverlap，theinclinationofPlutobecomesmaximum，theeccentricitybecomesminimumandtheargumentofperihelionbecomes90°。Whenθ3becomes180°，theinclinationofPlutobecomesminimum，theeccentricitybecomesmaximumandtheargumentofperihelionbecomes90°again。Williams&Benson（1971）anticipatedthistypeofresonance，laterconfirmedbyMilani，Nobili&Carpino（1989）。

Anargumentθ4=?P??N+3（ΩP?ΩN）libratesaround180°withalongperiod，～5。7×108yr。

Inournumericalintegrations，theresonances（i）–（iii）arewellmaintained，andvariationofthecriticalargumentsθ1，θ2，θ3remainsimilarduringthewholeintegrationperiod（Figs14–16）。However，thefourthresonance（iv）appearstobedifferent:thecriticalargumentθ4alternateslibrationandcirculationovera1010-yrtime-scale（Fig。17）。ThisisaninterestingfactthatKinoshita&Nakais（1995，1996）shorterintegrationswerenotabletodisclose。

6Discussion

Whatkindofdynamicalmechanismmaintainsthislong-termstabilityoftheplanetarysystem?Wecanimmediatelythinkoftwomajorfeaturesthatmayberesponsibleforthelong-termstability。First，thereseemtobenosignificantlower-orderresonances（meanmotionandsecular）betweenanypairamongthenineplanets。JupiterandSaturnareclosetoa5:2meanmotionresonance（thefamous‘greatinequality’），butnotjustintheresonancezone。Higher-orderresonancesmaycausethechaoticnatureoftheplanetarydynamicalmotion，buttheyarenotsostrongastodestroythestableplanetarymotionwithinthelifetimeoftherealSolarsystem。Thesecondfeature，whichwethinkismoreimportantforthelong-termstabilityofourplanetarysystem，isthedifferenceindynamicaldistancebetweenterrestrialandjovianplanetarysubsystems（Ito&Tanikawa1999，2001）。WhenwemeasureplanetaryseparationsbythemutualHillradii（R_），separationsamongterrestrialplanetsaregreaterthan26RH，whereasthoseamongjovianplanetsarelessthan14RH。Thisdifferenceisdirectlyrelatedtothedifferencebetweendynamicalfeaturesofterrestrialandjovianplanets。Terrestrialplanetshavesmallermasses，shorterorbitalperiodsandwiderdynamicalseparation。Theyarestronglyperturbedbyjovianplanetsthathavelargermasses，longerorbitalperiodsandnarrowerdynamicalseparation。Jovianplanetsarenotperturbedbyanyothermassivebodies。

Thepresentterrestrialplanetarysystemisstillbeingdisturbedbythemassivejovianplanets。However，thewideseparationandmutualinteractionamongtheterrestrialplanetsrendersthedisturbanceineffective;thedegreeofdisturbancebyjovianplanetsisO（eJ）（orderofmagnitudeoftheeccentricityofJupiter），sincethedisturbancecausedbyjovianplanetsisaforcedoscillationhavinganamplitudeofO（eJ）。Heighteningofeccentricity，forexampleO（eJ）～0。05，isfarfromsufficienttoprovokeinstabilityintheterrestrialplanetshavingsuchawideseparationas26RH。Thusweassumethatthepresentwidedynamicalseparationamongterrestrialplanets（>26RH）isprobablyoneofthemostsignificantconditionsformaintainingthestabilityoftheplanetarysystemovera109-yrtime-span。Ourdetailedanalysisoftherelationshipbetweendynamicaldistancebetweenplanetsandtheinstabilitytime-scaleofSolarsystemplanetarymotionisnowon-going。

AlthoughournumericalintegrationsspanthelifetimeoftheSolarsystem，thenumberofintegrationsisfarfromsufficienttofilltheinitialphasespace。Itisnecessarytoperformmoreandmorenumericalintegrationstoconfirmandexamineindetailthelong-termstabilityofourplanetarydynamics。

——以上文段引自Ito，T。&Tanikawa，K。Long-termintegrationsandstabilityofplanetaryorbitsinourSolarSystem。Mon。Not。R。Astron。Soc。336，483–500（2002）

这只是作者君参考的一篇文章，关于太阳系的稳定性。

还有其他论文，不过也都是英文的，相关课题的中文文献很少，那些论文下载一篇要九美元（《Nature》真是暴利），作者君写这篇文章的时候已经回家，不在检测中心，所以没有数据库的使用权，下不起，就不贴上来了。