AlthoughournumericalintegrationsspanthelifetimeoftheSolarsystem，thenusnecessarytoperformmoreandmorenumericalintegrationstoconfirmandexamineindetailthelong-termstabilityofourplanetarydynamics.

这只是作者君参考的一篇文章，关于太阳系的稳定性。

当然还有其他论文，不过其它论文下载一篇要九美元，作者君下不起，就不贴上来了。

4.4Long-termcouplingofseveralneighbouringplanetpairs

Letusseesomeindividualvariationsofplanetaryorbitalenergyandangularmomentumexpressedbythelow-s10and11showlong-termevolutionoftheorbitalenergyofeachplanetandtheangularmomentuminN+1andN?oticethatsomeplanetsfoarticular，hefigures，theyshownegativecorrelationsinexcnegativecorrelationinexchangeoforbitalenergymeansthattpositivecorrelationinexchangeofangularmomentu，wecanseethatMarsshowsapositivecorrelationintheangularmomentumvariationtotheVenus–curyexhibitscertainnegativecorrelationsintheangularmomentumversustheVenus–Earthsystem，whichseemstobeareactioncausedbytheconservationofangularmomentumintheterrestrialplanetarysubsystem.

ItisnotclearatthemomentwhytheVenus–Earthpairexhibitsanegativecorrelationiaypossiblyexplainthisthroughobservingthegeneralfactthattherearenoseculartermsinplanetarysemimajoraxesuptosecond-orderperturbationtheories(uwer&ClemenceBoccaletti&smeansthattheplanetaryorbitalenergy(whichisdirectlyrelatedtothesemimajoraxisa)mightbemuchlessaffectedbyperturbingplanetsthanistheangularmomentumexchange(whichrelatestoe).Hence，theeccentricitiesofVenusandEarthcanbedisturbedeasilybyJupiterandSaturn，heotherhand，thesemimajorstheenergyexchangemaybelimitedonlywithintheVenus–Earthpair，whichresultsinanegativecorrelationintheexchangeoforbitalenergyinthepair.

ThelongitudeofthenodeofPlutoreferredtothelongitudeofthenodeofNeptune，θ3=ΩP?ΩN，circulatesandtheperiodofthiscirculationisequaltotheperiodofθnθ3beeszero，i.longitudesofascendingnodesofNeptuneandPlutooverlap，theinclinationofPlutobeesmaximum，theeccentricitybeesminimumandtheargumentofperihelionbees90°.Whenθ3bees180°，theinclinationofPlutobeesminimum，theeccentricitybeesmaximumandtheargumentofperihelionbees90°liams&Benson(1971)anticipatedthistypeofresonance，laterconfirmedbyMilani，Nobili&Carpino(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).ThisisaninterestingfactthatKinoshita&Nakai‘s(1995，1996)shorterintegrationswerenotabletodisclose.

Asfortheouterjovianplanetarysubsystem，Jupiter–SaturnandUranus–ever，thestrengthoftheircouplingisnotasstrongparedwiththatoftheVenus–Earthpair.

5±5x1010-yrintegrationsofouterplanetaryorbits

Sincethejovianplanetarymassesaremuchlargerthantheterrestrialplanetarymasses，wetreatthejovianplanetarysystemasanindce，weaddedacoupleoftrialintegrationsthatspan±5x1010yr，includingonlytheouterfiveplanets(thefourjovianplanetsplusPluto).Theresultsexhibittherigorousstabilityoftheouterplanetarysystemoverthislongtime-italconfigurations(Fig.12)，andvariationofeccentricitiesandinclinations(Fig.13)showthisverylong-termsthoughwedonotshowmapshere，thetypicalfrequencyoftheorbitaloscillationofPlutoandtheotherouterplanetsisalmostconstantduringtheseverylong-termintegrationperiods，whichisdemonstratedinthetime–frequencymapsonourwebpage.

Inthesetwointegrations，therelativenumericalerrorinthetotalenergywas?10?6andthatofthetotalangularmomentumwas?10?10.

5.1ResonancesintheNeptune–Plutosystem

6Discussion

Whatkindofdynamicalmechanismmaintainsthislong-termstabilityoftheplanetarysystemWecanimmediatelythinkoftwomajorfeaturesthatmayberesponsibleforthelong-st，thereseemtobenosignificantlower-orderresonances(meanmotionandsecular)iterandSaturnareclosetoa5:2meanmotionresonance(thefamous‘greatinequality’)，her-orderresonancesmaycausethechaoticnatureoftheplanetarydynamicalmotion，buttheyarenotsostrongastodestrsecondfeature，whichwethinkismoreimportantforthelong-termstabilityofourplanetarysystem，isthedifferenceindynamicaldistancebetweenterrestrialandjovianplanetarysubsystems(Ito&Tanikawa1999，2001).WhenwemeasureplanetaryseparationsbythemutualHillradii(R_)，separationsamongterrestrialplanetsaregreaterthan26RH，sdifferenceisdirectlyrelatedtotherestrialplanetshavesmallermasses，yarestronglyperturbedbyjovianplanetsthathavelargermasses，ianplanetsarenotperturbedbyanyothermassivebodies.

Thepresentterrestriever，thewideseparationandmutualinteractionamongtheterrestrialplanetsrendersthedisturbanceineffecthedegreeofdisturbancebyjovianplanetsisO(eJ)(orderofmagnitudeoftheeccentricityofJupiter)，sincethedisturbancecausedbyjovianplanetsisaforcedoscillationhavinganamplitudeofO(eJ).Heighteningofeccentricity，forexampleO(eJ)?0.05，isfarfromsufficienttoprovokeinsweassumethatthepresentwidedynamicalseparationamongterrestrialplanets(>26RH)isprobablyoneofthemostsignificantconditionsformaintainingthestabilityoftheplanetarysystemovera109-yrtime-detailedanalysisoftherelationshipbetweendynamicaldistancebetweenplanetsandtheinstabilitytime-scaleofSolarsystemplanetarymotionisnowon-going.

对火星轨道变化问题的最后解释 (第3/3页)

lrelatedtotheplanetaryorbitalenergyEperunitmassasE=?μ2/hesystemispletelylinear，theorb-linearityintheplanetarysystemcamplitudeofthelowest-frequencyoscilever，suchasymptomofinstabilityisnotprominentinourlong-termintegrations.

，thetotalorbitalenergyandangularmomentumofthefourinnerplanetsandallnineplanetsareshownforintegrationN+upperthreepanelsshowthelong-periodicvariationoftotalenergy(denotedasE-E0)，totalangularmomentum(G-G0)，andtheverticalponent(H-H0)ofthein，G0，lowerthreepanelsineachfigureshowE-E0，G-G0andH-fluctuationshowninthelowerpanelsisvirtuallyentirelyaresultofthemassivejovianplanets.

Comparingthevariationsofenergyandangularmomentumoftheinnerfourplanetsandallnineplanets，itisapparentthattheamplitudesofthoseoftheinnerplanetsaremuchsmallerthanthoseofallnineplanets:theamplitusdoesnotmeanthattheinnerterrestrialplanetarysubsystemismorestablethantheouterone:thisissimplyaresultoftherelativesmallnessofthemassesofththerthingwenoticeisthattheinnerplanetarysubsystemmaybeeunstablemorerapidlythantheouteronebecauseof，thefluctuationsintheinner4paever，wecannotneglectthecontributionfromotherterrestrialplanets，aswewillseeinsubsequentsections.

Kinoshita&Nakai(1996)integratedtheouterfiveplanetaryorbitsover±5.5xyfoundthatfourmajorresonancesbetweenNeptuneandPlutoaremaintainedduringthewholeintegrationperiod，andthahefollowingdescription，λdenotesthemeanlongitude，Ωisthelongitudeoftheascendingnodeand?scriptsPandNdenotePlutoandNeptune.

MeanmotionresonancebetweenNeptuneandPluto(3:2).Thecriticalargumentθ1=3λP?2λN??Plibratesaround180°withanamplitudeofabout80°andalibrationperiodofabout2x104yr.

TheargumentofperihelionofPlutowP=θ2=?P?ΩPlibratesaround90°nticipatedinthesecularperturbationtheoryconstructedbyKozai(1962).

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