对火星轨道变化问题的最后解释[2/2页]

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    elementsofMercury，especiallyitseccentricity，seemtochangetoasignificantextent.Thisispartlybecausetheorbitaltimescaleoftheplanetistheshortestofalltheplanets，whichleadstoamorerapidorbitalevolutionthanotherplanets;theinnermostplanetmaybenearesttoinstability.ThisresultappearstobeinsomeagreementwithLaskar's(1994，1996)expectationsthatlargeandirregularvariationsappearintheeccentricitiesandinclinationsofMercuryonatimescaleofseveral109yr.However，theeffectofthepossibleinstabilityoftheorbitofMercurymaynotfatallyaffecttheglobalstabilityofthewholeplanetarysystemowingtothesmallmassofMercury.WewillmentionbrieflythelongtermorbitalevolutionofMercurylaterinSection4usinglowpassfilteredorbitalelements.
    Theorbitalmotionoftheouterfiveplanetsseemsrigorouslystableandquiteregularoverthistimespan(seealsoSection5).
    3.2Time?frequencymaps
    Althoughtheplanetarymotionexhibitsverylongtermstabilitydefinedasthenonexistenceofcloseencounterevents，thechaoticnatureofplanetarydynamicscanchangetheoscillatoryperiodandamplitudeofplanetaryorbitalmotiongraduallyoversuchlongtimespans.Evensuchslightfluctuationsoforbitalvariationinthefrequencydomain，particularlyinthecaseofEarth，canpotentiallyhaveasignificanteffectonitssurfaceclimatesystemthroughsolarinsolationvariation(cf.Berger1988).
    Togiveanoverviewofthelongtermchangeinperiodicityinplanetaryorbitalmotion，weperformedmanyfastFouriertransformations(FFTs)alongthetimeaxis，andsuperposedtheresultingperiodgramstodrawtwodimensionaltime?frequencymaps.Thespecificapproachtodrawingthesetime?frequencymapsinthispaperisverysimple?muchsimplerthanthewaveletanalysisorLaskar's(1990，1993)frequencyanalysis.
    Dividethelowpassfilteredorbitaldataintomanyfragmentsofthesamelength.Thelengthofeachdatasegmentshouldbeamultipleof2inordertoapplytheFFT.
    Eachfragmentofthedatahasalargeoverlappingpart:forexample，whentheithdatabeginsfromt=tiandendsatt=ti+T，thenextdatasegmentrangesfromti+δT≤ti+δT+T，whereδT?T.WecontinuethisdivisionuntilwereachacertainnumberNbywhichtn+Treachesthetotalintegrationlength.
    WeapplyanFFTtoeachofthedatafragments，andobtainnfrequencydiagrams.
    Ineachfrequencydiagramobtainedabove，thestrengthofperiodicitycanbereplacedbyagreyscale(orcolour)chart.
    Weperformthereplacement，andconnectallthegreyscale(orcolour)chartsintoonegraphforeachintegration.Thehorizontalaxisofthesenewgraphsshouldbethetime，i.e.thestartingtimesofeachfragmentofdata(ti，wherei=1，…，n).Theverticalaxisrepresentstheperiod(orfrequency)oftheoscillationoforbitalelements.
    WehaveadoptedanFFTbecauseofitsoverwhelmingspeed，sincetheamountofnumericaldatatobedecomposedintofrequencycomponentsisterriblyhuge(severaltensofGbytes).
    Atypicalexampleofthetime?frequencymapcreatedbytheaboveproceduresisshowninagreyscalediagramasFig.5，whichshowsthevariationofperiodicityintheeccentricityandinclinationofEarthinN+2integration.InFig.5，thedarkareashowsthatatthetimeindicatedbythevalueontheabscissa，theperiodicityindicatedbytheordinateisstrongerthaninthelighterareaaroundit.WecanrecognizefromthismapthattheperiodicityoftheeccentricityandinclinationofEarthonlychangesslightlyovertheentireperiodcoveredbytheN+2integration.Thisnearlyregulartrendisqualitativelythesameinotherintegrationsandforotherplanets，althoughtypicalfrequenciesdifferplanetbyplanetandelementbyelement.
    4.2Longtermexchangeoforbitalenergyandangularmomentum
    WecalculateverylongperiodicvariationandexchangeofplanetaryorbitalenergyandangularmomentumusingfilteredDelaunayelementsL，G，H.GandHareequivalenttotheplanetaryorbitalangularmomentumanditsverticalcomponentperunitmass.LisrelatedtotheplanetaryorbitalenergyEperunitmassasE=?μ2\/2L2.Ifthesystemiscompletelylinear，theorbitalenergyandtheangularmomentumineachfrequencybinmustbeconstant.Nonlinearityintheplanetarysystemcancauseanexchangeofenergyandangularmomentuminthefrequencydomain.Theamplitudeofthelowestfrequencyoscillationshouldincreaseifthesystemisunstableandbreaksdowngradually.However，suchasymptomofinstabilityisnotprominentinourlongtermintegrations.
    InFig.7，thetotalorbitalenergyandangularmomentumofthefourinnerplanetsandallnineplanetsareshownforintegrationN+2.Theupperthreepanelsshowthelongperiodicvariationoftotalenergy(denotedasEE0)，totalangularmomentum(GG0)，andtheverticalcomponent(HH0)oftheinnerfourplanetscalculatedfromthelowpassfilteredDelaunayelements.E0，G0，H0denotetheinitialvaluesofeachquantity.Theabsolutedifferencefromtheinitialvaluesisplottedinthepanels.ThelowerthreepanelsineachfigureshowEE0，GG0andHH0ofthetotalofnineplanets.Thefluctuationshowninthelowerpanelsisvirtuallyentirelyaresultofthemassivejovianplanets.
    Comparingthevariationsofenergyandangularmomentumoftheinnerfourplanetsandallnineplanets，itisapparentthattheamplitudesofthoseoftheinnerplanetsaremuchsmallerthanthoseofallnineplanets:theamplitudesoftheouterfiveplanetsaremuchlargerthanthoseoftheinnerplanets.Thisdoesnotmeanthattheinnerterrestrialplanetarysubsystemismorestablethantheouterone:thisissimplyaresultoftherelativesmallnessofthemassesofthefourterrestrialplanetscomparedwiththoseoftheouterjovianplanets.Anotherthingwenoticeisthattheinnerplanetarysubsystemmaybecomeunstablemorerapidlythantheouteronebecauseofitsshorterorbitaltimescales.Thiscanbeseeninthepanelsdenotedasinner4inFig.7wherethelongerperiodicandirregularoscillationsaremoreapparentthaninthepanelsdenotedastotal9.Actually，thefluctuationsintheinner4panelsaretoalargeextentasaresultoftheorbitalvariationoftheMercury.However，wecannotneglectthecontributionfromotherterrestrialplanets，aswewillseeinsubsequentsections.
    4.4Longtermcouplingofseveralneighbouringplanetpairs
    LetusseesomeindividualvariationsofplanetaryorbitalenergyandangularmomentumexpressedbythelowpassfilteredDelaunayelements.Figs10and11showlongtermevolutionoftheorbitalenergyofeachplanetandtheangularmomentuminN+1andN?2integrations.Wenoticethatsomeplanetsformapparentpairsintermsoforbitalenergyandangularmomentumexchange.Inparticular，VenusandEarthmakeatypicalpair.Inthefigures，theyshownegativecorrelationsinexchangeofenergyandpositivecorrelationsinexchangeofangularmomentum.Thenegativecorrelationinexchangeoforbitalenergymeansthatthetwoplanetsformacloseddynamicalsystemintermsoftheorbitalenergy.Thepositivecorrelationinexchangeofangularmomentummeansthatthetwoplanetsaresimultaneouslyundercertainlongtermperturbations.CandidatesforperturbersareJupiterandSaturn.AlsoinFig.11，wecanseethatMarsshowsapositivecorrelationintheangularmomentumvariationtotheVenus?Earthsystem.MercuryexhibitscertainnegativecorrelationsintheangularmomentumversustheVenus?Earthsystem，whichseemstobeareactioncausedbytheconservationofangularmomentumintheterrestrialplanetarysubsystem.
    ItisnotclearatthemomentwhytheVenus?Earthpairexhibitsanegativecorrelationinenergyexchangeandapositivecorrelationinangularmomentumexchange.Wemaypossiblyexplainthisthroughobservingthegeneralfactthattherearenoseculartermsinplanetarysemimajoraxesuptosecondorderperturbationtheories(cf.Brouweramp;amp;Clemence1961;Boccalettiamp;amp;Pucacco1998).Thismeansthattheplanetaryorbitalenergy(whichisdirectlyrelatedtothesemimajoraxisa)mightbemuchlessaffectedbyperturbingplanetsthanistheangularmomentumexchange(whichrelatestoe).Hence，theeccentricitiesofVenusandEarthcanbedisturbedeasilybyJupiterandSaturn，whichresultsinapositivecorrelationintheangularmomentumexchange.Ontheotherhand，thesemimajoraxesofVenusandEartharelesslikelytobedisturbedbythejovianplanets.ThustheenergyexchangemaybelimitedonlywithintheVenus?Earthpair，whichresultsinanegativecorrelationintheexchangeoforbitalenergyinthepair.
    Asfortheouterjovianplanetarysubsystem，Jupiter?SaturnandUranus?Neptuneseemtomakedynamicalpairs.However，thestrengthoftheircouplingisnotasstrongcomparedwiththatoftheVenus?Earthpair.
    5±5×1010yrintegrationsofouterplanetaryorbits
    Sincethejovianplanetarymassesaremuchlargerthantheterrestrialplanetarymasses，wetreatthejovianplanetarysystemasanindependentplanetarysystemintermsofthestudyofitsdynamicalstability.Hence，weaddedacoupleoftrialintegrationsthatspan±5×1010yr，includingonlytheouterfiveplanets(thefourjovianplanetsplusPluto).Theresultsexhibittherigorousstabilityoftheouterplanetarysystemoverthislongtimespan.Orbitalconfigurations(Fig.12)，andvariationofeccentricitiesandinclinations(Fig.13)showthisverylongtermstabilityoftheouterfiveplanetsinboththetimeandthefrequencydomains.Althoughwedonotshowmapshere，thetypicalfrequencyoftheorbitaloscillationofPlutoandtheotherouterplanetsisalmostconstantduringtheseverylongtermintegrationperiods，whichisdemonstratedinthetime?frequencymapsonourwebpage.
    Inthesetwointegrations，therelativenumericalerrorinthetotalenergywas～10?6andthatofthetotalangularmomentumwas～10?10.
    5.1ResonancesintheNeptune?Plutosystem
    Kinoshitaamp;amp;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.Williamsamp;amp;Benson(1971)anticipatedthistypeofresonance，laterconfirmedbyMilani，Nobiliamp;amp;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:thecriticalargumentθ4alternateslibrationandcirculationovera1010yrtimescale(Fig.17).ThisisaninterestingfactthatKinoshitaamp;amp;Nakai's(1995，1996)shorterintegrationswerenotabletodisclose.
    6Discussion
    Whatkindofdynamicalmechanismmaintainsthislongtermstabilityoftheplanetarysystem?Wecanimmediatelythinkoftwomajorfeaturesthatmayberesponsibleforthelongtermstability.First，thereseemtobenosignificantlowerorderresonances(meanmotionandsecular)betweenanypairamongthenineplanets.JupiterandSaturnareclosetoa5:2meanmotionresonance(thefamous‘greatinequality)，butnotjustintheresonancezone.Higherorderresonancesmaycausethechaoticnatureoftheplanetarydynamicalmotion，buttheyarenotsostrongastodestroythestableplanetarymotionwithinthelifetimeoftherealSolarsystem.Thesecondfeature，whichwethinkismoreimportantforthelongtermstabilityofourplanetarysystem，isthedifferenceindynamicaldistancebetweenterrestrialandjovianplanetarysubsystems(Itoamp;amp;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(amp;gt;26RH)isprobablyoneofthemostsignificantconditionsformaintainingthestabilityoftheplanetarysystemovera109yrtimespan.OurdetailedanalysisoftherelationshipbetweendynamicaldistancebetweenplanetsandtheinstabilitytimescaleofSolarsystemplanetarymotionisnowongoing.
    AlthoughournumericalintegrationsspanthelifetimeoftheSolarsystem，thenumberofintegrationsisfarfromsufficienttofilltheinitialphasespace.Itisnecessarytoperformmoreandmorenumericalintegrationstoconfirmandexamineindetailthelongtermstabilityofourplanetarydynamics.
    ——以上文段引自Ito，T.amp;Tanikawa，K.LongtermintegrationsandstabilityofplanetaryorbitsinourSolarSystem.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。

对火星轨道变化问题的最后解释[2/2页]

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