第76章 对火星轨道变化问题的最后解释

oasignifitextent.thisispartlybecausetheorbitaltime-scaleoftheplaheshortestofalltheplas，whichleadstoamorerapidorbitalevolutionthanotherplaheinnermostplamaybeoinstability.thisresultappearstobeinsomeagreementwithlaskar's(1994，1996)expectationsthatlargeandirregularvariationsappearintheetricitiesandinationsofmercuryonatime-scaleofseveral109yr.however，theeffectofthepossibleinstabilityoftheorbitofmercurymaynotfatallyaffecttheglobalstabilityofthewholeplaarysystemowingtothesmallmassofmercury.wewillmentiohelong-termorbitalevolutionofmercurylateriion4usinglow-passfilteredorbitalelements.

    theorbitalmotionoftheouterfiveplasseemsrigorouslystableandquiteregularoverthistime-span(seealsose5).

    3.2time–frequencymaps

    althoughtheplaarymotionexhibitsveryloabilitydefiheenceofcloseenterevents，thechaotiatureofplaarydynamigetheoscillatoryperiodandamplitudeofplaaryorbitalmotiongraduallyoversugtime-spans.evensuchslightfluctuationsoforbitalvariationinthefrequenain，particularlyinthecaseofearth，potentiallyhaveasignifiteffeitssurfaceclimatesystemthroughsolarinsolationvariation(cf.berger1988).

    togiveanoverviewofthelong-termgeinperiodicityinplaaryorbitalmotion，weperformedmanyfastfouriertransformations(ffts)aloimeaxis，andsuperposedtheresultingperiodgramstodrawtwo-dimensionaltime–frequencymaps.thespecificapproachtodrawiime–frequencymapsinthispaperisverysimple–muchsimplerthanthewaveletanalysisorlaskar's(1990，1993)frequenalysis.

    dividethelow-passfilteredorbitaldataintomanyfragmentsofthesamelenh.thelenhofeachdatasegmentshouldbeamultipleof2ioapplythefft.

    eachfragmentofthedatahasalargeoverlappingpart:forexample，whehdatabeginsfromt=tiandendsatt=ti t，thedatasegmentrangesfromti δt≤ti δt t，whereδt?t.wetihisdivisionuntilwereachacertainnumbernbywhi treachesthetotaliionlenh.

    lyanffttoeachofthedatafragments，andobtainnfrequencydiagrams.

    ineachfrequencydiagramobtainedabove，thestrenhofperiodicitybereplacedbyagrey-scale(orcolour)chart.

    weperformthereplat，andectallthegrey-scale(orcolour)chartsintoonegraphforeategration.thehorizontalaxisofthesenehsshouldbethetime，i.e.thestartingtimesofeachfragmentofdata(ti，wherei=1，…，n).theverticalaxisrepresentstheperiod(orfrequency)oftheoscillationoforbitalelements.

    wehaveadoptedanfftbecauseofitsoverwhelmingspeed，siheamountofnumericaldatatobedeposedintofrequenposisterriblyhuge(severaltensofgbytes).

    atypicalexampleofthetime–frequencymapcreatedbytheaboveproceduresisshowninagrey-scalediagramasfig.5，whichshowsthevariationofperiodicityintheetricityandinatiohinn 2iion.infig.5，thedarkareashowsthatatthetimeindicatedbythevalueontheabscissa，theperiodicityindicatedbytheordirohaninthelighterareaaroundit.wereizefromthismapthattheperiodicityoftheetricityandinatiohonlygesslightlyovertheentireperiodcoveredbythen 2iion.thisnearlyregulartrendisqualitativelythesameiegrationsandforotherplas，althoughtypicalfrequenciesdifferplabyplaabyelement.

    4.2long-termexgeoforbitalenergyandangularmomentum

    wecalculateverylong-periodicvariationandexgeofplaaryorbitalenergyandangularmomentumusingfiltereddelaunayelementsl，g，h.gandhareequivalenttotheplaaryorbitalangularmomentumanditsvertipoperunitmass.lisrelatedtotheplaaryorbitalenergyeperunitmassase=?μ22l2.ifthesystemispletelyliheorbitalenergyandtheangularmomentumineachfrequenmustbestant.non-liyintheplaarysystemcauseanexgeofenergyandangularmomentuminthefrequenain.theamplitudeofthelowest-frequencyoscillationshouldincreaseifthesystemisunstableandbreaksdowngradually.however，suchasymptomofinstabilityisnotpromiin-termiions.

    infig.7，thetotalorbitalenergyandangularmomentumofthefourinnerplasandallnineplasareshownforiionn 2.theupperthreepanelsshowthelong-periodicvariationoftotalenergy(denotedase-e0)，totalangularmomentum(g-g0)，aipo(h-h0)oftheinnerfourplascalculatedfromthelow-passfiltereddelaunayelements.e0，g0，h0deheinitialvaluesofeachquantity.theabsolutedifferentheinitialvaluesisplottedinthepahelowerthreepanelsineachfigureshowe-e0，g-g0andh-h0ofthetotalofnineplahefluctuationshowninthelowerpanelsisvirtuallyentirelyaresultofthemassivejovianplas.

    paringthevariationsofenergyandangularmomentumoftheinnerfourplasandallnineplas，itisapparentthattheamplitudesofthoseoftheinnerplasaremuchsmallerthanthoseofallnineplaheamplitudesoftheouterfiveplasaremuchlargerthanthoseoftheinnerplahisdoeshattheierrestrialplaarysubsystemismorestablethaerohisissimplyaresultoftherelativesmallnessofthemassesofthefourterrestrialplasparedwiththoseoftheouterjovianplas.ahiiceisthattheinnerplaarysubsystemmaybeeunstablemorerapidlythaeronebecauseofitsshorterorbitaltime-scales.thisbeseeninthepanelsdenotedasinner4infig.7wherethelonger-periodidirregularoscillationsaremoreapparentthaninthepanelsdeotal9.actually，thefluctuationsintheinner4panelsareteextentasaresultoftheorbitalvariationofthemercury.however，weothetributionfromotherterrestrialplas，aswewillseeinsubsequeions.

    4.4long-termcouplingofseveralneighbplapairs

    letusseesomeindividualvariationsofplaaryorbitalenergyandangularmomentumexpressedbythelow-passfiltereddelaunayelements.figs10and11showlong-termevolutionoftheorbitalenergyofeachplaandtheangularmomentuminn 1andn?2iioicethatsomeplasformapparentpairsintermsoforbitalenergyandangularmomentumexge.inparticular，venusahmakeatypicalpair.inthefigures，theyshowivecorrelationsinexgeofenergyandpositivecorrelationsinexgeofangularmomentum.theivecorrelationinexgeoforbitalenergymeansthatthetwoplasformacloseddynamicalsystemintermsoftheorbitalenergy.thepositivecorrelationinexgeofangularmomentummeansthatthetwoplasaresimultaneouslyundercertainlong-termperturbations.didatesforperturbersarejupiterandsaturn.alsoinfig.11，weseethatmarsshows'itivecorrelationintheangularmomentumvariationtothevehsystem.mercuryexhibitscertaiivecorrelationsintheangularmomentumversusthevehsystem，whichseemstobeareacausedbytheservationofangularmomentumierrestrialplaarysubsystem.

    itisnotclearatthemomentwhythevehpairexhibitsaivecorrelationinenergyexgeand'itivecorrelationinangularmomentumexge.ossiblyexplainthisthroughthegeneralfactthattherearenoseculartermsinplaarysemimajoraxesuptosed-orderperturbationtheories(cf.brouwerboccalettipucacco1998).thismeansthattheplaaryorbitalenergy(whichisdirectlyrelatedtothesemimajoraxisa)mightbemuchlessaffectedbyperturbingplahanistheangularmomentumexge(whichrelatestoe).heheetricitiesofvenusahbedisturbedeasilybyjupiterandsaturn，whichresultsin'itivecorrelationintheangularmomentumexge.oherhand，thesemimajoraxesofvenusaharelesslikelytobedisturbedbythejovianplahustheenergyexgemaybelimitedonlywithinthevehpair，whichresultsiivecorrelationintheexgeoforbitalenergyinthepair.

    asfortheouterjovianplaarysubsystem，jupiter–saturnanduranus–uomakedynamicalpairs.however，thestrenhoftheircouplingisnotasstrongparedwiththatofthevehpair.

    5±5x1010-yriionsofouterplaaryorbits

    sihejovianplaarymassesaremuchlargerthaerrestrialplaarymasses，wetreatthejovianplaarysystemasanindepeplaarysystemintermsofthestudyofitsdynamicalstability.hence，weaddedacoupleoftrialiionsthatspan±5x1010yr，includingonlytheouterfiveplahefourjovianplaspluspluto).theresultsexhibittherigorousstabilityoftheouterplaarysystemoverthislongtime-span.orbitalfigurations(fig.12)，andvariationofetricitiesandinations(fig.13)showthisveryloabilityoftheouterfiveplasinboththetimeandthefrequenains.althoughwedonotshoshere，thetypicalfrequencyoftheorbitaloscillationofplutoaherouterplasisalmoststantduringtheseverylong-termiionperiods，whichisdemonstratediime–frequencymapsonourwebpage.

    iwoiions，therelativenumericalerrorialenergywas～10?6andthatofthetotalangularmomentumwas～10?10.

    5.1resoheune–plutosystem

    kinoshitanakai(1996)iedtheouterfiveplaaryorbitsover±5.5x109yr.theyfoundthatfourmajorresoweeuneandplutoaremaintainedduringthewholeiionperiod，andthattheresonancesmaybethemaincausesofthestabilityoftheorbitofpluto.themajorfourresonancesfoundinpreviousresearchareasfollows.inthefollowingdescriptioesthemeanlongitude，Ωisthelongitudeoftheasdingnodeand?isthelongitudeofperihelion.subscriptspaeplutoaune.

    meanmotionresoweeuneandpluto(3:2).thecriticalargumentθ1=3λp?2λn??plibratesaround180°litudeofabout80°andalibrationperiodofabout2x104yr.

    theargumentofperihelionofplutowp=θ2=?p?Ωplibratesaround90°eriodofabout3.8x106yr.thedominantperiodicvariationsoftheetricityandinationofplutoaresynizedwiththelibrationofitsargumentofperihelion.thisisanticipatedinthesecularperturbationtheorystructedbykozai(1962).

    thelongitudeofthenodeofplutoreferredtothelongitudeofthenodeofune，θ3=Ωp?Ωn，circulatesandtheperiodofthiscirculationisequaltotheperiodofθ2libration.whenθ3beeszero，i.e.thelongitudesofasdingnodesofuneandplutooverlap，theinationofplutobeaximum，theetricitybeinimumandtheargumentofperihelionbees90°.whenθ3bees180°，theinationofplutobeinimum，theetricitybeaximumandtheargumentofperihelionbees90°again.williamsbenson(1971)anticipatedthistypeofresoerfirmedbymilani，nobilicarpino(1989).

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

    inournumeritegrations，theresonances(i)–(iii)arewellmaintained，andvariationofthecriticalargumentsθ1，θ2，θ3remainsimilarduringthewholeiionperiod(figs14–16).however，thefourthresonance(iv)appearstobedifferent:thecriticalargumentθ4alternateslibrationandcirculationovera1010-yrtime-scale(fig.17).thisisaingfactthatkinoshitanakai's(1995，1996)shorteriionswerenotabletodisclose.

    6discussion

    whatkindofdynamicalmeismmaintainsthisloabilityoftheplaarysystem?weimmediatelythinkoftwomajorfeaturesthatmayberesponsiblefortheloability.first，thereseemtobenosignifitlower-orderresonances(meanmotionandsecular)betairamongthenineplas.jupiterandsaturnareclosetoa5:2meanmotionresohefamous‘greatinequality’)，butnotjustintheresonane.higher-orderresonancesmaycausethechaotiatureoftheplaarydynamicalmotion，buttheyarenotsastodestroythestableplaarymotionwithinthelifetimeoftherealsolarsystem.thesedfeature，whichwethinkismoreimportantfortheloabilityofourplaarysystem，isthedifferendynamicaldistaweenterrestrialandjovianplaarysubsystems(itotanikawa1999，2001).whenwemeasureplaaryseparationsbythemutualhillradii(r_)，separationsamongterrestrialplasaregreaterthan26rh，whereasthoseamongjovianplasarelessthan14rh.thisdifferenceisdirectlyrelatedtothediffereweendynamicalfeaturesofterrestrialandjovianplaerrestrialplashavesmallermasses，shorterorbitalperiodsandwiderdynamicalseparation.theyarestronglyperturbedbyjovianplahathavelargermasses，longerorbitalperiodsandnarrowerdynamicalseparation.jovianplasareurbedbyanyothermassivebodies.

    thepresentterrestrialplaarysystemisstillbeingdisturbedbythemassivejovianplas.however，thewideseparationandmutualiionamoerrestrialplasrehedisturbaneffective;thedegreeofdisturbancebyjovianplasiso(ej)(orderofmagnitudeoftheetricityofjupiter)，sihedisturbancecausedbyjovianplasisaforcedoscillationhavinganamplitudeofo(ej).heighteningofetricity，forexampleo(ej)～0.05，isfarfromsuffittoprovokeinstabilityierrestrialplashavingsuchawideseparationas26rh.thusweassumethatthepresentwidedynamicalseparationamongterrestrialplas(;26rh)isprobablyohemostsignifitditionsformaintainiabilityoftheplaarysystemovera109-yrtime-span.ourdetailedanalysisoftherelationshipbetweendynamicaldistaweenplasaabilitytime-scaleofsolarsystemplaarymotionisnowon-going.

    althoughournumeritegrationsspanthelifetimeofthesolarsystem，thenumberofiionsisfarfromsuffittofilltheinitialphasespace.itisnecessarytoperformmoreandmorenumeritegrationstofirmandexamineiailtheloabilityofourplaarydynamics.

    ——以上文段引自ito，t.tanikawa，k.long-termiionsandstabilityofplaaryorbitsinoursolarsystem.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。