对火星轨道变化问题的最后解释（2 / 2）

ThevariatioricitiesandorbitalinationsfortheinnerfourpsiiandfinalpartoftheiionN+1isshowninFig.4.Asexpected，thecharacterofthevariationofparyorbitalelementsdoesnotdiffersignifitlybetweeiandfinalpartofeategration，atleastforVehandMars.TheelementsofMercury，especiallyitsetricity，seemtogetoasignifitextent.Thisispartlybecausetheorbitaime-scaleofthepistheshortestofalheps，whichleadstoamorerapidorbitalevolutionthanotherpstheinnermostpmaybeoinstability.ThisresuappearstobeinsomeagreementwithLaskar''''''''s1994，1996expectationsthatrgeandirregurvariationsappeariricitiesandinationsofMercuryonatime-scaleofseveral109yr.However，theeffectofthepossibleinstabilityoftheorbitofMercurymaynotfatallyaffecttheglobalstabilityofthewholeparysystemowingtothesmallmassofMercury.Wewillmentiohelong-termorbitalevolutionofMercuryteriion4usinglow-passfieredorbitalelements.

Theorbitalmotionoftheouterfivepsseemsrigorouslystableandquitereguroverthistime-spanseealsoSe5.

3.2Time–frequencymaps

Ahoughtheparymotionexhibitsveryloabilitydefiheenceofclosees，thechaotiatureofparydynamibgetheoscioryperiodandamplitudeofparyorbitalmotiongraduallytime-spans.Evensuchslightfluctuationsoforbitalvariationinthefrequenain，particurlyinthecaseofEarth，potentiallyhaveasignifiteffeitssurfaceclimatesystemthroughsorinsotionvariationcf.Berger1988.

Togiveanoverviewofthelong-termgeinperiodicityiaryorbitalmotion，weperformedmanyfastFouriertransformationsFFTsaloimeaxis，andsuperposedtheresuingperiodgramstodrawtwo-dimensionaime–frequencymaps.Thespecificapproachtodrawiime–frequencymapsinthispaperisverysimple–muchsimplerthanthewaveletanalysisorLaskar''''''''s1990，1993frequenalysis.

Dividethelow-passfieredorbitaldataintomanyfragmentsofthesamelength.Thelengthofeachdatasegmentshouldbeamuipleof2ioapplytheFFT.

内容未完，下一页继续阅读 </p><p style="font-size:16px">Eachfragmentofthedatahasargeoverppingpart:forexample，whehdatabeginsfromt=tiandendsatt=ti+T，thedatasegmentrangesfromti+δT≤ti+δT+T，whereδT?T.WetihisdivisionuntilwereachacertainnumberNbywhi+Treachesthetotaliioh.

lyanFFTtoeachofthedatafragments，andobtainnfrequencydiagrams.

Ineachfrequencydiagramobtainedabove，thestrengthofperiodicityberepcedbyagrey-scaleorcolourchart.

Weperformtherept，aalhegrey-scaleorcolourchartsintoonegraphforeategration.Thehorizontaxisofthesenehsshouldbethetime，i.e.thestartingtimesofeachfragmentofdatati，wherei=1，…，icaxisrepresentstheperiodorfrequencyoftheosciionoforbitalelements.

WehaveadoptedanFFTbecauseofitsoverwhelmingspeed，siheamountofnumericaldatatobedeposedintofrequenposisterriblyhugeseveraensofGbytes.

Atypicalexampleofthetime–frequencymapcreatedbytheaboveproceduresisshowninagrey-scalediagramasFig.5，whichshowsthevariationofperiodicityiricityandinatiohinN+2iion.InFig.5，thedarkareashowsthatatthetimeindicatedbythevalueontheabscissa，theperiodicityindicatedbytheordirohaninthelighterareaaroundit.WereizefromthismapthattheperiodicityoftheetricityandinatiohonlygesslightlyovertheentireperiodcoveredbytheN+2iion.Thisnearlyregurtrendisqualitativelythesameinotheriionsandforotherps，ahoughtypicalfrequenciesdifferpbypabyelement.

4.2Long-termexgeoforbitalenergyandangurmomentum

Wecalcuteverylong-periodicvariationandexgeofparyorbitalenergyandangurmomentumusingfieredDeunayelementsL，G，H.GandHareequivalenttotheparyorbitangurmomentumanditsverticalpoperunitmass.LisretedtotheparyorbitalenergyEperunitmassasE=?μ2/2L2.Ifthesystemispletelyliheorbitalenergyandtheangurmomentumineachfrequenmustbestant.Non-liyintheparysystemcauseanexgeofenergyandangurmomentuminthefrequenain.Theamplitudeofthelowest-frequencyosciionshouldincreaseifthesystemisunstableandbreaksdowngradually.However，suchasymptomofinstabilityisnotpromiin-termiions.

InFig.7，thetotalorbitalenergyandangurmomentumofthefourinnerpsandallninepsareshownforiionN+2.Theupperthreepanelsshowthelong-periodicvariationoftotalenergydenotedasE-E0，totangurmomentumG-G0，aicalpoH-H0oftheinnerfourpscalcutedfromthelow-passfieredDeunayelements.E0，G0，H0deheinitialvaluesofeachquantity.TheabsolutediffereialvaluesisplottedinthepahelowerthreepanelsineachfigureshowE-E0，G-G0andH-H0ofthetotalofhefluctuationshowninthelowerpanelsisvirtuallyentirelyaresuofthemassivejovias.

paringthevariationsofenergyandangurmomentumoftheinnerfourpsandallnineps，itisapparentthattheamplitudesofthoseoftheinnerpsaremuchsmallerthanthoseofallheamplitudesoftheouterfivepsaremuchrgerthanthoseoftheinnerps.Thisdoeshattheierrestrialparysubsystemismorestablethaerohisissimplyaresuoftheretivesmallnessofthemassesofthefourterrestrialpsparedwiththoseoftheouterjovias.Ahiiceisthattheinnerparysubsystemmaybeeunstablemorerapidlythaeronebecauseofitsshorterorbitaime-scales.Thisbeseeninthepanelsdenotedasinner4inFig.7wherethelonger-periodidirregurosciionsaremoreapparentthaninthepanelsdeotal9.Actually，thefluctuationsintheinner4panelsareteextentasaresuoftheorbitalvariationoftheMercury.However，weothetributionfromotherterrestrialps，aswewillseeinsubsequeions.

4.4Long-termcouplingofseveralneighbppairs

内容未完，下一页继续阅读 </p><p style="font-size:16px">Letusseesomeindividualvariationsofparyorbitalenergyandangurmomentumexpressedbythelow-passfieredDeunayelements.Figs10and11showlong-termevolutionoftheorbitalenergyofeaetandtheangurmomentuminN+1andN?2iioicethatsomepsformapparentpairsintermsoforbitalenergyandangurmomentumexge.Inparticur，Venusahmakeatypicalpair.Inthefigures，theyshowivecorretionsinexgeofenergyandpositivecorretionsinexgeofangurmomentuhepositivecorretioninexgeofangurmomentummeansthatthetwopsaresimuaneouslyundercertainlong-termperturbations.didatesforperturbersareJupiterandSaturn.AlsoinFig.11，weseethatMarsshoositivecorretionintheangurmomentumvariationtotheVehsystem.MercuryexhibitscertaiivecorretionsintheangurmomentumversustheVehsystem，whichseemstobeareacausedbytheservationofangurmomentumierrestrialparysubsystem.

ItisnotclearatthemomentwhytheVehpairexhibitsaivecorretioninenergyexgeandapositivecorretioninangurmomentumexge.ossiblyexpinthisthroughthegeneralfactthatthereareermsiarysemimajoraxesuptosed-orderperturbationtheoriescf.BrouClemence1961Boccaletti&Pucacco1998.Thismeansthattheparyorbitalenergywhichisdirectlyretedtothesemimajoraxisamightbemuchlessaffectedbyperturbingphanistheangurmomentumexgewhichretestoe.HeheetricitiesofVenusahbedisturbedeasilybyJupiterandSaturn，whichresusinapositivecorretionintheangurmomentumexge.Oherhand，thesemimajoraxesofVenusaharelesslikelytobedisturbedbythejovias.ThustheenergyexgemaybelimitedonlywithintheVehpair，whichresusinaivecorretionintheexgeoforbitalenergyinthepair.

Asfortheouterjoviaarysubsystem，Jupiter–SaturnandUranus–uomakedynamicalpairs.However，thestrengthoftheircouplingisnotasstrongparedwiththatoftheVehpair.

5±5×1010-yriionsofouterparyorbits

Sihejoviaarymassesaremuchrgerthaerrestrialparymasses，wetreatthejoviaarysystemasanindepeparysystemintermsofthestudyofitsdynamicalstability.Hence，weaddedacoupleoftrialiionsthatspan±5×1010yr，includingonlytheouterfivepsthefourjoviasplusPluto.Theresusexhibittherigorousstabilityoftheouterparysystemoverthislongtime-span.OrbitalfigurationsFig.12，andvariatioricitiesandinationsFig.13showthisveryloabilityoftheouterfivepsinboththetimeandthefrequenains.Ahoughwedonotshoshere，thetypicalfrequencyoftheorbitalosciionofPlutoaherouterpsisalmoststantduringtheseverylong-termiionperiods，whichisdemonstratediime–frequencymapsonourwebpage.

Iwoiions，theretivenumericalerrorialenergywas～10?6andthatofthetotangurmomentumwas～10?10.

5.1Resoheune–Plutosystem

Kinoshita&Nakai1996iedtheouterfiveparyorbitsover±5.5×109yr.TheyfoundthatfourmajorresoweeuneandPlutoaremaintainedduringthewholeiionperiod，andthattheresonancesmaybethemaincausesofthestabilityoftheorbitofPluto.Themajorfourresonancesfoundinpreviousresearchareasfollows.Inthefollowingdescription，λdehemeanlongitude，Ωisthelongitudeoftheasdingnodeand?isthelongitudeofperihelion.SubscriptsPaoaune.

MeanmotionresoweeuneandPluto3:2.Thecriticargumentθ1=3λP?2λN??Plibratesaround180°litudeofabout80°andalibrationperiodofabout2×104yr.

TheargumentofperihelionofPlutoωP=θ2=?P?ΩPlibratesaround90°eriodofabout3.8×106yr.ThedominantperiodicvariationsoftheetricityandinationofPlutoaresyhthelibrationofitsargumentofperihelion.ThisisanticipatedinthesecurperturbationtheorystructedbyKozai1962.

ThelongitudeofthenodeofPlutoreferredtothelongitudeofthenodeofune，θ3=ΩP?ΩN，circutesandtheperiodofthiscircutionisequaotheperiodofθ2libration.Whenθ3beeszero，i.e.thelongitudesofasdingnodesofuneandPlutooverp，theinationofPlutobeesmaximum，theetricitybeesminimumandtheargumentofperihelionbees90°.Whenθ3bees180°，theinationofPlutobeesminimum，theetricitybeesmaximumandtheargumentofperihelionbees90°again.Williams&Benson1971anticipatedthistypeofresoerfirmedbyMini，Nobili&Carpino1989.

内容未完，下一页继续阅读 </p><p style="font-size:16px">Anargumentθ4=?P??N+3ΩP?ΩNlibratesaround180°withalongperiod，～5.7×108yr.

Inournumericaliions，theresonancesi–iiiarewellmaintained，andvariationofthecriticargumentsθ1，θ2，θ3remainsimirduringthewholeiionperiodFigs14–16.However，thefourthresonanceivappearstobedifferent:thecriticargumentθ4aernateslibrationandcircutionovera1010-yrtime-scaleFig.17.ThisisaniingfactthatKinoshita&Nakai''''''''s1995，1996shorteriionswerenotabletodisclose.

6Discussion

Whatkindofdynamicalmeismmaintainsthisloabilityoftheparysystem?Weimmediatelythinkoftwomajorfeaturesthatmayberesponsiblefortheloability.First，thereseemtobenosignifitlower-orderresonancesmeanmotionandsecurbetairamongthenineps.JupiterandSaturnareclosetoa5:2meanmotionresohefamous‘greatinequality’，butnotjustintheresonane.Higher-orderresonancesmaycausethechaotiatureoftheparydynamicalmotion，buttheyarenotsastodestroythestableparymotionwithinthelifetimeoftherealSorsystem.Thesedfeature，whichwethinkismoreimportantfortheloabilityofourparysystem，isthedifferendynamicaldistaweenterrestriandjoviaarysubsystemsIto&Tanikawa1999，2001.WhenwemeasureparyseparationsbythemutualHillradiiR_，separationsamongterrestrialpsaregreaterthan26RH，whereasthoseamongjoviasarelessthan14RH.Thisdifferenceisdirectlyretedtothediffereweendynamicalfeaturesofterrestriandjovias.Terrestrialpshavesmallermasses，shorterorbitalperiodsandwiderdynamicalseparation.Theyarestronglyperturbedbyjoviahathavergermasses，longerorbitalperiodsandnarrowerdynamicalseparation.Joviasareurbedbyanyothermassivebodies.

Thepresentterrestrialparysystemisstillbeingdisturbedbythemassivejovias.However，thewideseparationandmutualiionamoerrestrialpsrehedisturbaivethedegreeofdisturbancebyjoviasisOeJorderofmagnitudeoftheetricityofJupiter，sihedisturbancecausedbyjoviasisaforcedosciionhavinganamplitudeofOeJ.Heighteniricity，forexampleOeJ～0.05，isfarfromsuffittoprovokeinstabilityierrestrialpshavingsuchawideseparationas26RH.Thusweassumethatthepresentwidedynamicalseparationamongterrestrialps>26RHisprobablyohemostsignifitditionsformaintainiabilityoftheparysystemovera109-yrtime-span.Ourdetailedanalysisoftheretionshipbetweendynamicaldistaweesaabilitytime-scaleofSorsystemparymotionisnowon-going.

AhoughournumericaliionsspanthelifetimeoftheSorsystem，thenumberofiionsisfarfromsuffittofilheinitialphasespace.Itisnecessarytoperformmoreandmorenumericaliionstofirmandexamineiaiheloabilityofourparydynamics.

——以上文段引自Ito，T.&Tanikawa，K.Long-termiionsandstabilityofparyorbitsinourSorSyste.R.Astron.Soc.336，483–5002002

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