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第76章 对火星轨道变化问题的最后解释（第2页）

&esp;&esp;verticalviewofthefournerparyorbits（frothez-axisdirection）attheitiandfalpartfthetegrationsn±1theaxesunitsareauthexy-pneissettothevariantpneofrsystetotangurontu（a）theitialpartofn+1（t=0to005x109yr）（b）thefalpartofn+1（t=49339x108to49886x109yr）（c）theitialpartofn?1（t=0to?005x109yr）（d）thefalpartofn?1（t=?39180x109to?39727x109yr）eachpanel，atotalof23684potsareplottedwithantervalofabout2190yrover5x107yrlidleseachpaneldenotethepresentorbitfthefourterrestrialps（takenfrode245）

&esp;&esp;thevariationofeentricitiesandorbitalclationsforthenerfourpstheitiandfalpartofthetegrationn+1isshownfig4asexpected，thecharacterofthevariationofparyorbitalelentsdoesnotdiffersignificantlybeeentheitiandfalpartofeachtegration，atleastforven，earthandarstheelentfrcury，especiallyitseentricity，seetochantoasignificantextentthisispartlybecaetheorbitalti-scaleofthepistheshortestofalltheps，whichleadstoaorerapidorbitalevotionthanotherps;thenerostpaybenearesttostabilitythisresultappearstobeagreenithskar&039;s（1994，1996）expectationsthatrandirregurvariationsappeartheeentricitiesandclationfrcuryonati-scaleofseveral109yrhowever，theeffectofthepossiblestabilityoftheorbitofrcuryaynotfatallyaffecttheglobalstabilityofthewholeparysysteogtothesallasfrcurywewillntionbrieflythelong-terorbitalevotionofrcurytersection4glow-passfilteredorbitalelents

&esp;&esp;theorbitalotionoftheouterfivepsseesrigorolystableanditereguroverthisti-span（seealsection5）

&esp;&esp;32ti–freencyaps

&esp;&esp;althoughtheparyotionexhibitsverylong-terstabilitydefedasthenon-existenceofcloseenunterevents，thechaotiatureofparydynaicscanchantheosciltoryperiodandaplitudeofparyorbitalotiongraduallyoversuchlongti-spansevensuchslightfctuationforbitalvariationthefreencydoa，particurlythecaseofearth，canpotentiallyhaveasignificanteffectonitssurfacecliatesystethroughrtionvariation（cfberr1988）

&esp;&esp;togiveanoverviewofthelong-terchanperiodicityparyorbitalotion，weperfordanyfastfouriertransforations（ffts）alongthetiaxis，andsuperposedtheresultgperiodgrastodrawo-dinsionalti–freencyapsthespecificapproachtodragtheseti–freencyapsthispaperisverysiple–uchsiplerthanthewaveletanalysirskar&039;s（1990，1993）freencyanalysis

&esp;&esp;dividethelow-passfilteredorbitaldatatoanyfragntfthesalenhthelenhofeachdatasegntshouldbeaultipleof2ordertoapplythefft

&esp;&esp;eachfragntofthedatahasaroverppgpart:forexaple，whentheithdatabegsfrot=tiandendsatt=ti+t，thenextdatasegntransfroti+δt≤ti+δt+t，whereδt?twentuethisdivisionuntilwereachacertanubernbywhichtn+treachesthetotaltegrationlenh

&esp;&esp;weapplyanffttoeachofthedatafragnts，andobtanfreencydiagras

&esp;&esp;eachfreencydiagraobtaedabove，thestrenhofperiodicitycanberepcedbyagrey-scale（orlour）chart

&esp;&esp;weperfortherepcent，andnnectallthegrey-scale（orlour）chartstoonegraphforeachtegrationthehorizontaxifthesenewgraphsshouldbetheti，iethestartgtifeachfragntofdata（ti，wherei=1，…，n）theverticaxisrepresentstheperiod（orfreency）oftheosciltionoforbitalelents

&esp;&esp;wehaveadoptedanfftbecaeofitverwhelgspeed，scetheaountofnuricaldatatobedeposedtofreencyponentsisterriblyhu（severaltenfgbytes）

&esp;&esp;atypicalexapleoftheti–freencyapcreatedbytheaboveproceduresisshownagrey-scalediagraasfig5，whichshowsthevariationofperiodicitytheeentricityandclationofearthn+2tegrationfig5，thedarkareashowsthatatthetidicatedbythevaeontheabscissa，theperiodicitydicatedbytheordateisstronrthanthelighterareaarounditwecanregnizefrothisapthattheperiodicityoftheeentricityandclationofearthonlychansslightlyovertheentireperiodveredbythen+2tegrationthisnearlyregurtrendisalitativelythesaothertegrationsandforotherps，althoughtypicalfreenciesdifferpbypandelentbyelent

&esp;&esp;42long-terxchanoforbitalenergyandangurontu

&esp;&esp;wecalcuteverylong-periodicvariationandexchanofparyorbitalenergyandangurontugfiltereddeunayelentsl，g，hgandhareeivalenttotheparyorbitangurontuanditsverticalponentperunitasslisretedtotheparyorbitalenergyeperunitassase=?μ22l2ifthesysteispletelylear，theorbitalenergyandtheangurontueachfreencybtbenstantnon-learitytheparysystecancaeanexchanofenergyandangurontuthefreencydoatheaplitudeofthelowest-freencyosciltionshouldcreaseifthesysteisunstableandbreaksdowngraduallyhowever，suchasyptoofstabilityisnotproentourlong-tertegrations

&esp;&esp;fig7，thetotalorbitalenergyandangurontuofthefournerpsandallnepsareshownfortegrationn+2theupperthreepanelsshowthelong-periodicvariationoftotalenergy（denotedase-e0），totangurontu（g-g0），andtheverticalponent（h-h0）ofthenerfourpscalcutedfrothelow-passfiltereddeunayelentse0，g0，h0denotetheitialvaefeachantitytheabtedifferencefrotheitialvaesisplottedthepanelsthelowerthreepanelseachfigureshowe-e0，g-g0andh-h0ofthetotalofnepsthefctuationshownthelowerpanelsisvirtuallyentirelyaresultoftheassivejovianps

&esp;&esp;pargthevariationfenergyandangurontuofthenerfourpsandallneps，itisapparentthattheaplitudefthoseofthenerpsareuchsallerthanthoseofallneps:theaplitudeftheouterfivepsareuchrrthanthoseofthenerpsthisdoesnotanthatthenerterrestrialparysubsysteisorestablethantheouterone:thisissiplyaresultoftheretivesallnesftheassefthefourterrestrialpsparedwiththoseoftheouterjovianpsanotherthgwenoticeisthatthenerparysubsysteaybeeunstableorerapidlythantheouteronebecaeofitsshorterorbitalti-scalesthiscanbeseenthepanelsdenotedasner4fig7wherethelonr-periodicandirregurosciltionsareoreapparentthanthepanelsdenotedastotal9actually，thefctuationsthener4panelsaretoarextentasaresultoftheorbitalvariationofthercuryhowever，wecannotneglectthentributionfrootherterrestrialps，aswewillseesubseentsections

&esp;&esp;44long-teruplgofseveralneighbourgppairs

&esp;&esp;letseedividualvariationfparyorbitalenergyandangurontuxpressedbythelow-passfiltereddeunayelentsfigs10and11showlong-tervotionoftheorbitalenergyofeachpandtheangurontun+1andn?2tegrationswenoticethatpsforapparentpairsterforbitalenergyandangurontuxchanparticur，venandearthakeatypicalpairthefigures，theyshownegativerretionsexchanofenergyandpositiverretionsexchanofangurontuthenegativerretionexchanoforbitalenergyansthattheopsforacloseddynaicalsysteterftheorbitalenergythepositiverretionexchanofangurontuansthattheopsaresiultaneoldercertalong-terperturbationscandidatesforperturbersarejupiterandsaturnalfig11，wecanseethatarsshows&039;itiverretiontheangurontuvariationtotheven–earthsystercuryexhibitscertanegativerretionstheangurontuverstheven–earthsyste，whichseestobeareactioncaedbythenservationofangurontutheterrestrialparysubsyste

&esp;&esp;itisnotclearattheonhytheven–earthpairexhibitsanegativerretionenergyexchanand&039;itiverretionangurontuxchanweaypossiblyexpthisthroughobservgtheneralfactthattherearenosecurtersparyseiajoraxesuptosend-orderperturbationtheories（cfbrouwerboalettipucao1998）thisansthattheparyorbitalenergy（whichisdirectlyretedtotheseiajoraxisa）ightbeuchlessaffectedbyperturbgpsthanistheangurontuxchan（whichretestoe）hence，theeentricitiefvenandearthcanbedisturbedeasilybyjupiterandsaturn，whichresults&039;itiverretiontheangurontuxchanontheotherhand，theseiajoraxefvenandeartharelesslikelytobedisturbedbythejovianpsththeenergyexchanaybeliitedonlywiththeven–earthpair，whichresultsanegativerretiontheexchanoforbitalenergythepair

&esp;&esp;asfortheouterjovianparysubsyste，jupiter–saturnanduran–neptuneseetoakedynaicalpairshowever，thestrenhoftheiruplgisnotasstrongparedwiththatoftheven–earthpair

&esp;&esp;5±5x1010-yrtegrationfouterparyorbits

&esp;&esp;scethejovianparyassesareuchrrthantheterrestrialparyasses，wetreatthejovianparysysteasandependentparysysteterfthestudyofitsdynaicalstabilityhence，weaddedaupleoftrialtegrationsthatspan±5x1010yr，cdgonlytheouterfiveps（thefourjovianpspspto）theresultsexhibittherigorostabilityoftheouterparysysteoverthislongti-spanorbitalnfigurations（fig12），andvariationofeentricitiesandclations（fig13）showthisverylong-terstabilityoftheouterfivepsboththetiandthefreencydoasalthoughwedonotshowapshere，thetypicalfreencyoftheorbitalosciltionofptoandtheotherouterpsisalostnstantdurgtheseverylong-tertegrationperiods，whichisdeonstratedtheti–freencyapnourwebpa

&esp;&esp;theseotegrations，theretivenuricalerrorthetotalenergywas～10?6andthatofthetotangurontuwas～10?10

&esp;&esp;51renancestheneptune–ptosyste

&esp;&esp;koshitanakai（1996）tegratedtheouterfiveparyorbitver±55x109yrtheyfoundthatfourajorrenancesbeeenneptuneandptoareataeddurgthewholetegrationperiod，andthattherenancesaybetheacaefthestabilityoftheorbitofptotheajorfourrenancesfoundprevioresearchareasfollowsthefollogdescription，λdenotestheanlongitude，Ωisthelongitudeoftheascendgnodeand?isthelongitudeofperihelionsubscriptspandndenoteptoandneptune

&esp;&esp;anotionrenancebeeenneptuneandpto（3:2）thecriticarguntθ1=3λp?2λn??plibratesaround180°withanaplitudeofabout80°andalibrationperiodofabout2x104yr

&esp;&esp;thearguntofperihelionofptowp=θ2=?p?Ωplibratesaround90°withaperiodofabout38x106yrthedoantperiodicvariationftheeentricityandclationofptoaresynchronizedwiththelibrationofitsarguntofperihelionthisisanticipatedthesecurperturbationtheorynstructedbykozai（1962）

&esp;&esp;thelongitudeofthenodeofptoreferredtothelongitudeofthenodeofneptune，θ3=Ωp?Ωn，circutesandtheperiodofthiscircutionisealtotheperiodofθ2librationwhenθ3beeszero，iethelongitudefascendgnodefneptuneandptooverp，theclationofptobeesaxiu，theeentricitybeesiuandthearguntofperihelionbees90°whenθ3bees180°，theclationofptobeesiu，theeentricitybeesaxiuandthearguntofperihelionbees90°agawilliasbenn（1971）anticipatedthistypeofrenance，ternfirdbyini，nobilicarpo（1989）

&esp;&esp;anarguntθ4=?p??n+3（Ωp?Ωn）libratesaround180°withalongperiod，～57x108yr

&esp;&esp;ournuricaltegrations，therenances（i）–（iii）arewellataed，andvariationofthecriticarguntsθ1，θ2，θ3reasiirdurgthewholetegrationperiod（figs14–16）however，thefourthrenance（iv）appearstobedifferent:thecriticarguntθ4alternateslibrationandcircutionovera1010-yrti-scale（fig17）thisisanterestgfactthatkoshitanakai&039;s（1995，1996）shortertegrationswerenotabletodisclose

&esp;&esp;6discsion

&esp;&esp;whatkdofdynaicalchanisatasthislong-terstabilityoftheparysyste?wecanidiatelythkofoajorfeaturesthatayberesponsibleforthelong-terstabilityfirst，thereseetobenosignificantlower-orderrenances（anotionandsecur）beeenanypairaongthenepsjupiterandsaturnareclosetoa5:2anotionrenance（thefao‘greateality’），butnotjttherenancezonehigher-orderrenancesaycaethechaotiatureoftheparydynaicalotion，buttheyarenotstrongastodestroythestableparyotionwiththelifetioftherealrsystethesendfeature，whichwethkisoreiportantforthelong-terstabilityofourparysyste，isthedifferencedynaicaldistancebeeenterrestriandjovianparysubsystes（itotanikawa1999，2001）whenweasureparyseparationsbytheutualhillradii（r_），separationsaongterrestrialpsaregreaterthan26rh，whereasthoseaongjovianpsarelessthan14rhthisdifferenceisdirectlyretedtothedifferencebeeendynaicalfeaturefterrestriandjovianpsterrestrialpshavesallerasses，shorterorbitalperiodsandwiderdynaicalseparationtheyarestronglyperturbedbyjovianpsthathaverrasses，lonrorbitalperiodsandnarrowerdynaicalseparationjovianpsarenotperturbedbyanyotherassivebodies

&esp;&esp;thepresentterrestrialparysysteisstillbegdisturbedbytheassivejovianpshowever，thewideseparationandutualteractionaongtheterrestrialpsrendersthedisturbanceeffective;thedegreeofdisturbancebyjovianpsi（ej）（orderofagnitudeoftheeentricityofjupiter），scethedisturbancecaedbyjovianpsisaforcedosciltionhavganaplitudeofo（ej）heightengofeentricity，forexapleo（ej）～005，isfarfrosufficientrovokestabilitytheterrestrialpshavgsuchawideseparationas26rhthweassuthatthepresenidedynaicalseparationaongterrestrialps（;26rh）isprobablyoneoftheostsignificantnditionsforatagthestabilityoftheparysysteovera109-yrti-spanourdetailedanalysiftheretionshipbeeendynaicaldistancebeeenpsandthestabilityti-scaleofrsysteparyotionisnowon-gog

&esp;&esp;althoughournuricaltegrationsspanthelifetiofthersyste，thenuberoftegrationsisfarfrosufficienttofilltheitialphasespaceitisnecessaryerfororeandorenuricaltegrationstonfirandexaedetailthelong-terstabilityofourparydynaics

&esp;&esp;——以上文段引自ito，ttanikawa，klong-tertegrationsandstabilityofparyorbitsourrsysteonnotrastronc336，483–500（2002）

&esp;&esp;这只是作者君参考的一篇文章，关于太阳系的稳定性。

&esp;&esp;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。

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