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对火星轨道变化问题的最后解释（第2页）

&ioiesandorbitalinsfortheinnerfourplaheinitialandfinalpartoftheiionN+1isshowninFig。4。Asexpected，thecharacterofthevariatioaryorbitalelemediffersignifitlybetweeialandfinalpartofeategratioforVehandMars。TheelementsofMercury，especiallyitsetricity，seemtogetoasighisispartlybecausetheorbitaltime-scaleoftheplaofalltheplas，whichleadstoamorerapidorbitalevolutionthas;theiplatoinstability。ThisresultappearstobeinsomeagreementwithLaskars（1994，1996）expesthatlargeandirregularvariatioheetridinere-scaleofseveral109yr。However，theeffectofthepossibleinstabilityoftheorbitofMerotfatallyaffecttheglobalstabilityofthewholeplaemowingtothesmallmassofMercury。Wewillmeheloalevolutioeriion4usinglow-passfilteredorbitalelements。

&almotioerfiveplasseemsrigorouslystableandquiteregularoverthistime-span（seealsoSe5）。

3。2Time–frequencymaps

Althoughtheplaiosveryloabilitydefiheenceoftereveiatureofplaarydynamigetheoscillatoryperiodandamplitudeofplaaryorbitalmotiongraduallyoversue-spans。Evensuchslightflusoforbitalvariationinthefrequenain，partithecaseofEarth，tiallyhaveasigonitssurfaceclimatesystemthroughsolarinsolationvariation（cf。Berger1988）。

Togiveaheleiyiaryorbitalmotion，weperformedmanyfastFouriertransformations（FFTs）aloimeaxis，aheresultingperiodgramstodrawtwo-dimensionaltime–frequencymaps。Thespecificapprthesetime–frequenthispaperisverysimple–muchsimplerthaanalysisorLaskars（1990，1993）frequenalysis。

Dividethelow-passfilteredorbitaldataintmentsofthesamelehofeachdatasegmentshouldbeamultipleof2ioapplytheFFT。

Eatofthedatahasalargepart:forexample，wheabeginsfromt=tia=ti+T，theasegmei+δT≤ti+δT+T，whereδT?T。WeuethisdivisionuntilwereachaumberNbywhi+Treachesthetotaliioh。

lyahedatafragments，andobtainnfrequencydiagrams。

Ineacydiagramobtaireybereplacedbyagrey-scale（orcolour）chart。

&herepladectallthegrey-scale（orcolour）toonegraphforeategratioalaxisofthesenehsshouldbethetime，i。e。thestartingtimesmentofdata（ti，wherei=1，…，icalaxisrepresentstheperiod（orfrequency）oftheosoforbitalelements。

tedanFFTbecauseofitsspeed，siofatobedeposedintofrequenpoerriblyhuge（severaltensofGbytes）。

Atypicalexampleofthetime–frequencymapcreatedbytheaboveproceduresisshowninagrey-scalediagramasFig。5，whichshowsthevariatioyiyandinofEarthiion。InFig。5，thedarkareashowsthatatthetimeihevalueontheabscissa，theperiodidicatedbytheordirohaerareaaroundit。WeizefromthismapthattheperiodicityoftheetridinofEartholyovertheentireperiodcoveredbytheion。Thisrendisqualitativelythesameiegrationsandforotherplahoughtypicalfrequenciesdifferplabyplabyelement。

4。2Leoforbitalenergyandangularmomentum

&everylong-periodicvariationandexgeofplaaryorbitalenergyandangularmomentumusingfilteredDelausL，G，H。Gatotheplaaryularmomentumaipomass。LisrelatedtotheplaaryorbitalemassasE=?μ22L2。Ifthesystemispletelylialeheangularmomentumineamustbet。yiarysystemexgeyandangularmomentuminthefrequenaiudeofthelowest-frequencyosshouldihesystemisunstableandbreaksdowngradually。However，suchasymptomofinstabilityisnotpromiin-termiions。

InFig。7，thetotalorbitalenergyandangularmomentumofthefourisandallsareshratiohreepahelong-periodicvariationofty（deotalangularmomentum（G-G0），aipo（H-H0）oftheinnerfourplascalculatedfromthelow-passfilteredDelaus。E0，G0，H0deialvaluesofeatity。TheabsolutediffereheinitialvaluesisplottedihelowerthreepanelsineachfigureshowE-E0，G-G0andH-H0ofthetotalofheflushowninthelowerpanelsisvirtuallyentirelyaresultofthemassivejovias。

pariionsyandangularmomentumoftheinnerfourplasandallisapparentthattheamplitudesofthoseoftheisaremuchsmallerthanthoseofallheamplitudesoftheouterfiveplasaremuchlargerthaheihisdoesheirialplaarysubsystemismorestablethahisissimplyaresultoftherelativesmallhemassesofthefourterrestrialplasparedwiththoseoftheouterjoviaherthiiceisthattheiarysubsystemmaybeeunstablemorerapidlythaerosshorterorbitaltime-scales。Thisbeseeniedasinner4inFig。7wherethelonger-periodidirregularosoreapparentthaniedastotal9。Actually，theflusintheineexteheorbitalvariationoftheMercury。However，weegletributionfromotherterrestrialplas，aswewillseeiions。

4。4Long-termghbplapairs

&usseesomeindividualvariationsofplaaryorbitalenergyandangularmomehelow-passfilteredDelaus。Figs10and11showloioalenergyofeaetandtheangularmomentuminN+1andiohatsomeplasformapparentpairsintermsoforbitalenergyandangularmomentumexpartiusahmakeatypicalpair。Inthefigures，theyshowionsinexgeyandpositivecorrelationsinexgeofangularmomeivecorrelationialethetwoplasformaicalsystemiheorbitaleivecorrelationinexgeofangularmomethetwoplaaneouslyuaiurbations。didatesforperturbersareJupiterandSaturn。AlsoinFig。11，weseethatMarsshoositivecorrelationintheangularmomentumvariationtotheVeem。MercuryexhibitsegativecorrelationsintheangularmomeheVeem，whichseemstobeareacausedbytheservationofangularmomentumirialplaarysubsystem。

ItishemomentwhytheVehpairexhibitsaiyexdapositivecorrelationinangularmomentumexge。ossiblyexplainthisthrthegethatthereareermsiarysemimajoraxesuptosed-orderperturbationtheories（p;amp;amp;amp;ce1961;Boccaletti&Pucacco1998）。Thismeansthattheplaaryorbitalenergy（whichisdirectlyrelatedtothesemimajoraxisa）mightbemuchlessaffectedbyperturbihanistheangularmomentumexge（whichrelatestoe）。HericitiesofVehbedisturbedeasilybyJupiterandSaturn，whichresultsinapositivecorrelationintheangularmomentumextheotherhand，thesemimajoraxesofVeharelesslikelytobedisturbedbythejoviahustheenergyexgemaybelimitedonlywithihpair，whichresultsiivecorrelationintheexgeoforbitalehepair。

Asfortheouterjoviaarysubsystem，Jupiter–SaturnandUrauomakedynamicalpairs。However，thestrengthisnparedwiththatoftheVehpair。

5±5×1ratioaryorbits

&hejoviaarymassesaremuchlargertharialplaarymasses，wetreatthejoviaemasaplaemihestudyofitsdynamicalstability。Hence，lerationsthatspan±5×1010yr，inlytheouterfiveplahefourjoviasplusPluto）。Theresultsexhibittherigorousstabilityoftheouterplaemoverthislongtime-span。Orbitalfigurations（Fig。12），andvariatioiesandins（Fig。13）showthisveryloabilityoftheouterfiveplahthetimeandthefrequenains。Althoughwedonotshoshere，thetypicalfrequealosofPlutoaherouterplasisalmosttduringtheseverylrationperiods，whichisdemohetime–frequene。

Iegratioivenumericalerrorialenergywas～10?6andthatofthetotalangularmomentumwas～10?10。

5。1Resoheosystem

Kinoshita&Nakai（1996）ierfiveplaaryorbitsover±5。5×109yr。Theyfoundthatfourmajorresoweeoaremaintainedduringthewholeiionperiod，andthattheresonancesmaybethemaihestabilityoftheorbitofPluto。Themajorfourresonandinpreviousresearchareasfollows。Inthefollowiiohemeanloheloheasodeand?isthelongitudeofperihelion。SubscriptsPaoaune。

&ionresoweeo（3:2）。Thecriticalargumentθ1=3λP?2λN??Plibratesaround180°litudeofabout80°andalibrationperiodofabout2×104yr。

&ofperihelionofPlutoωP=θ2=?P?ΩPlibratesaround90°eriodofabout3。8×106yr。ThedominantperiodicvariatioridinofPlutoaresyhelibrationumentofperihelion。ThisisahesecularperturbatioructedbyKozai（1962）。

&udeoftheoreferredtotheloheune，θ3=ΩP?ΩN，cirdtheperiodofthiscirisequaltotheperiodofθ2libration。Whenθ3beeszero，i。e。thelongitudesofasodesofooverlap，theinofPlutobeum，theetriumaofperihelionbees90°。Whenθ3bees180°，theinofPlutobeum，theetriumaofperihelionbees90°again。;amp;amp;amp;Benson（1971）aypeofresoeredbyMilani，Nobili&o（1989）。

Aθ4=?P??N+3（ΩP?ΩN）libratesaround180°eriod，～5。7×108yr。

Iegrations，theresonances（i）–（iii）arewellmaintained，andvariatioicalargumentsθ1，θ2，θ3remainsimilarduringthewholeiis14–16）。However，thefourthresonance（iv）appearstobediffereicalargumeeslibrationandcirovera1010-yrtime-scale（Fig。17）。ThisisaingfactthatKinoshita&Nakais（1995，1996）shratioabletodisclose。

6Dis

Whatkindofdynamiismmaintainsthisloabilityoftheplaem?Weimmediatelythinkoftwomajorfeaturesthatmayberespoheloability。First，thereseemtobenosignifitlower-orderresoionandsecular）betairamongtheerandSaturoa5:2meanmotionresohefamous‘greatiy’），butnotjustintheresonane。Higher-orderresonancesmaycausetheatureoftheplaarydynami，buttheyarenastodestroythestableplaionwithiherealSolarsystem。Thesedfeature，whikismoreimp-termstabilityofourplaem，isthedifferenamiterrestrialandjoviaarysubsystems（Ito&Tanikawa1999，2001）。WhenlaioualHillradii（R_），separationsamorialplaerthan26RH，whereasthoseamongjoviasarelessthan14RH。Thisdifferenceisdirectlyrelatedtothediffereweendyuresofterrestrialandjoviaerrestrialplashavesmallermasses，shorterorbitalperiodsandaratiorourbedbyjoviahavelargermasses，loalperiodsandnarroaration。Joviasareurbedbyanyothermassivebodies。

&terrestrialplaemisstillbeingdisturbedbythemassivejovias。However，thewideseparationandmutualiionamorialplahedisturbahedegreeofdisturbancebyjoviasisO（eJ）（heetricityofJupiter），siurbancecausedbyjoviasisaforcedoshavinganamplitudeofO（eJ）。HeighteripleO（eJ）～0。05，isfarfromsuffittoprovokeinstabilityirialplashavingsuchawideseparationas26RH。Thusweassumethatthepresentaratiorialplas（>26RH）isprobablyosignifiditionsformaintaiyoftheplaemovera109-yrtime-spaailedanalysisoftherelatioweendynamiplaheinstabilitytime-scaleofSolarsystemplaionisnowon-going。

AlthoughouregrationsspaheSolarsystem，theegrationsisfarfromsuffittofilltheinitialphasespace。Itisoperformmoreandmoreegrationstoaailtheloabilityofourplaarydynamics。

——以上文段引自Ito，T。&Tanikawa，K。LrationsandstabilityofplaaryorbitsinourSolarSystem。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。