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第76章 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.thisresultappearstobeinsomeagreementwithlaskar's（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–muchsimplerthanthewaveletanalysisorlaskar's（1990，1993）frequencyanalysis.

dividethelow-passfilteredorbitaldataintomanyfragmentsofthesamelenh.thelenhofeachdatasegmentshouldbeamultipleof2inordertoapplythefft.

eachfragmentofthedatahasalargeoverlappingpart:forexample，whentheithdatabeginsfromt=tiandendsatt=ti+t，thenextdatasegmentrangesfromti+δt≤ti+δt+t，whereδt?t.wecontinuethisdivisionuntilwereachacertainnumbernbywhichtn+treachesthetotalintegrationlenh.

weapplyanffttoeachofthedatafragments，andobtainnfrequencydiagrams.

ineachfrequencydiagramobtainedabove，thestrenhofperiodicitycanbereplacedbyagrey-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，theyshownegativecorrelationsinexchangeofenergyandpositivecorrelationsinexchangeofangularmomentuthenegativecorrelationinexchangeoforbitalenergymeansthatthetwoplanetsformacloseddynamicalsystemintermsoftheorbitalenergy.thepositivecorrelationinexchangeofangularmomentummeansthatthetwoplanetsaresimultaneouslyundercertainlong-termperturbations.candidatesforperturbersarejupiterandsaturn.alsoinfig.11，wecanseethatmarsshows'itivecorrelationintheangularmomentumvariationtothevenus–earthsystemercuryexhibitscertainnegativecorrelationsintheangularmomentumversusthevenus–earthsystem，whichseemstobeareactioncausedbytheconservationofangularmomentumintheterrestrialplanetarysubsyste

itisnotclearatthemomentwhythevenus–earthpairexhibitsanegativecorrelationinenergyexchangeand'itivecorrelationinangularmomentumexchange.wemaypossiblyexplainthisthroughobservingthegeneralfactthattherearenoseculartermsinplanetarysemimajoraxesuptosecond-orderperturbationtheories（cf.brouwerboccalettipucacco1998）.thismeansthattheplanetaryorbitalenergy（whichisdirectlyrelatedtothesemimajoraxisa）mightbemuchlessaffectedbyperturbingplanetsthanistheangularmomentumexchange（whichrelatestoe）.hence，theeccentricitiesofvenusandearthcanbedisturbedeasilybyjupiterandsaturn，whichresultsin'itivecorrelationintheangularmomentumexchange.ontheotherhand，thesemimajoraxesofvenusandeartharelesslikelytobedisturbedbythejovianplanets.thustheenergyexchangemaybelimitedonlywithinthevenus–earthpair，whichresultsinanegativecorrelationintheexchangeoforbitalenergyinthepair.

asfortheouterjovianplanetarysubsystem，jupiter–saturnanduranus–neptuneseemtomakedynamicalpairs.however，thestrenhoftheircouplingisnotasstrongcomparedwiththatofthevenus–earthpair.

5±5x1010-yrintegrationsofouterplanetaryorbits

sincethejovianplanetarymassesaremuchlargerthantheterrestrialplanetarymasses，wetreatthejovianplanetarysystemasanindependentplanetarysystemintermsofthestudyofitsdynamicalstability.hence，weaddedacoupleoftrialintegrationsthatspan±5x1010yr，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

kinoshitanakai（1996）integratedtheouterfiveplanetaryorbitsover±5.5x109yr.theyfoundthatfourmajorresonancesbetweenneptuneandplutoaremaintainedduringthewholeintegrationperiod，andthattheresonancesmaybethemaincausesofthestabilityoftheorbitofpluto.themajorfourresonancesfoundinpreviousresearchareasfollows.inthefollowingdescription，λdenotesthemeanlongitude，Ωisthelongitudeoftheascendingnodeand?isthelongitudeofperihelion.subscriptspandndenoteplutoandneptune.

meanmotionresonancebetweenneptuneandpluto（3:2）.thecriticalargumentθ1=3λp?2λn??plibratesaround180°withanamplitudeofabout80°andalibrationperiodofabout2x104yr.

theargumentofperihelionofplutowp=θ2=?p?Ωplibratesaround90°withaperiodofabout3.8x106yr.thedominantperiodicvariationsoftheeccentricityandinclinationofplutoaresynchronizedwiththelibrationofitsargumentofperihelion.thisisanticipatedinthesecularperturbationtheoryconstructedbykozai（1962）.

thelongitudeofthenodeofplutoreferredtothelongitudeofthenodeofneptune，θ3=Ωp?Ωn，circulatesandtheperiodofthiscirculationisequaltotheperiodofθ2libration.whenθ3becomeszero，i.e.thelongitudesofascendingnodesofneptuneandplutooverlap，theinclinationofplutobecomesmaximum，theeccentricitybecomesminimumandtheargumentofperihelionbecomes90°.whenθ3becomes180°，theinclinationofplutobecomesminimum，theeccentricitybecomesmaximumandtheargumentofperihelionbecomes90°again.williamsbenson（1971）anticipatedthistypeofresonance，laterconfirmedbymilani，nobilicarpino（1989）.

anargumentθ4=?p??n+3（Ωp?Ωn）libratesaround180°withalongperiod，～5.7x108yr.

inournumericalintegrations，theresonances（i）–（iii）arewellmaintained，andvariationofthecriticalargumentsθ1，θ2，θ3remainsimilarduringthewholeintegrationperiod（figs14–16）.however，thefourthresonance（iv）appearstobedifferent:thecriticalargumentθ4alternateslibrationandcirculationovera1010-yrtime-scale（fig.17）.thisisaninterestingfactthatkinoshitanakai's（1995，1996）shorterintegrationswerenotabletodisclose.

6discussion

whatkindofdynamicalmechanismmaintainsthislong-termstabilityoftheplanetarysystem?wecanimmediatelythinkoftwomajorfeaturesthatmayberesponsibleforthelong-termstability.first，thereseemtobenosignificantlower-orderresonances（meanmotionandsecular）betweenanypairamongthenineplanets.jupiterandsaturnareclosetoa5:2meanmotionresonance（thefamous‘greatinequality’），butnotjustintheresonancezone.higher-orderresonancesmaycausethechaoticnatureoftheplanetarydynamicalmotion，buttheyarenotsostrongastodestroythestableplanetarymotionwithinthelifetimeoftherealsolarsystethesecondfeature，whichwethinkismoreimportantforthelong-termstabilityofourplanetarysystem，isthedifferenceindynamicaldistancebetweenterrestrialandjovianplanetarysubsystems（itotanikawa1999，2001）.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-termintegrationsandstabilityofplanetaryorbitsinoursolarsystemon.not.r.astron.soc.336，483–500（2002）

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