对火星轨道变化问题的最后解释 (第2/2页)
　　
　　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–muchsimplerthanthewaveletanalysisorLaskar's(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=−μ2/2L2.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，Ωisthelongitudeoftheascendingnodeandϖisthelongitudeofperihelion.SubscriptsPandNdenotePlutoandNeptune.
　　
　　MeanmotionresonancebetweenNeptuneandPluto(3:2).Thecriticalargumentθ1=3λP−2λN−ϖPlibratesaround180°withanamplitudeofabout80°andalibrationperiodofabout2×104yr.
　　
　　TheargumentofperihelionofPlutoωP=θ2=ϖP−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=ϖP−ϖN+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&Nakai's(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。