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赤壁小说>死在火星上唐跃麦冬汇合了吗 > 对火星轨道变化问题的最后解释(第2页)

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,theyshownegativecorrelationsinexchangeofenergyandpositivecorrelationsinexchangeofangularmomentuThenegativecorrelationinexchangeoforbitalenergymeansthatthetwoplanetsformacloseddynamicalsystemintermsoftheorbitalenergy.Thepositivecorrelationinexchangeofangularmomentummeansthatthetwoplanetsaresimultaneouslyundercertainlong-termperturbations.CandidatesforperturbersareJupiterandSaturn.AlsoinFig.11,wecanseethatMarsshowsapositivecorrelationintheangularmomentumvariationtotheVenus–EarthsysteMercuryexhibitscertainnegativecorrelationsintheangularmomentumversustheVenus–Earthsystem,whichseemstobeareactioncausedbytheconservationofangularmomentumintheterrestrialplanetarysubsyste

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,buttheyarenotsostrongastodestroythestableplanetarymotionwithinthelifetimeoftherealSolarsysteThesecondfeature,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-termintegrationsandstabilityofplanetaryorbitsinourSolarSysteMon.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。

请勿开启浏览器阅读模式,否则将导致章节内容缺失及无法阅读下一章。

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