第76章 对火星轨道变化问题的最后解释

君品相关实已解释问题。

仍质疑——“太含糊”，“火星轨变化比想象！”

吧，既君简单解释够力，咱严肃东西，反正本书写现，嚷嚷本书bug堆，初高物理书挑刺少。

以文章内容：

lng-terintegratinsandstabilityfplanetaryrbitsinurslarsyste

abstrat

epresenttheresultsfverylng-ternuerialintegratinsfplanetaryrbitaltinsver109-yrtie-spansinludingallnineplanets.aquikinspetinfurnuerialdatashsthattheplanetarytin，atleastinursipledynaialdel，seestbequitestableevenverthisverylngtie-span.alserlkatthelest-frequenysillatinsusingal-passfiltershsustheptentiallydiffusiveharaterfterrestrialplanetarytin，espeiallythatferury.thebehaviurftheeentriityferuryinurintegratinsisqualitativelysiilarttheresultsfrjaqueslaskar'sseularperturbatinthery(e.g.eax～0.35ver～±4gyr).hever，therearenapparentseularinreasesfeentriityrinlinatininanyrbitaleleentsftheplanets，hihayberevealedbystilllnger-ternuerialintegratins.ehavealsperfredaupleftrialintegratinsinludingtinsftheuterfiveplanetsvertheduratinf±5x1010yr.theresultindiatesthatthethreeajrresnanesintheneptune–plutsystehavebeenaintainedverthe1011-yrtie-span.

1intrdutin

1.1definitinftheprble

thequestinfthestabilityfurslarsystehasbeendebatedverseveralhundredyears，sinetheerafnetn.theprblehasattratedanyfausatheatiiansvertheyearsandhasplayedaentralrleinthedevelpentfnn-lineardynaisandhasthery.hever，edntyethaveadefiniteansertthequestinfhetherurslarsysteisstablernt.thisispartlyaresultfthefatthatthedefinitinftheter‘stability’isvaguehenitisusedinrelatinttheprblefplanetarytinintheslarsyste.atuallyitisnteasytgivealear，rigrusandphysiallyeaningfuldefinitinfthestabilityfurslarsyste.

anganydefinitinsfstability，hereeadptthehilldefinitin(gladan1993):atuallythisisntadefinitinfstability，butfinstability.edefineasysteasbeingunstablehenalseenunterurssehereinthesyste，startingfraertaininitialnfiguratin(habers，etherillittanikaa1999).asysteisdefinedasexperieningalseenunterhentbdiesapprahneantherithinanareafthelargerhillradius.therisethesysteisdefinedasbeingstable.henefrardestatethaturplanetarysysteisdynaiallystableifnlseenunterhappensduringtheagefurslarsyste，abut±5gyr.inidentally，thisdefinitinaybereplaedbyneinhihanurrenefanyrbitalrssingbeteeneitherfapairfplanetstakesplae.thisisbeauseeknfrexperienethatanrbitalrssingisverylikelytleadtalseenunterinplanetaryandprtplanetarysystes(yshinaga，kkubakin1999).fursethisstateentanntbesiplyappliedtsystesithstablerbitalresnanessuhastheneptune–plutsyste.

1.2previusstudiesandaisfthisresearh

inadditintthevaguenessftheneptfstability，theplanetsinurslarsysteshaharatertypialfdynaialhas(sussanisd1988，1992).theausefthishatibehaviurisnpartlyunderstdasbeingaresultfresnaneverlapping(urraylear，franklinhlan2001).hever，ituldrequireintegratingveranenseblefplanetarysystesinludingallnineplanetsfraperidveringseveral10gyrtthrughlyunderstandthelng-terevlutinfplanetaryrbits，sinehatidynaialsystesareharaterizedbytheirstrngdependeneninitialnditins.

frthatpintfvie，anyfthepreviuslng-ternuerialintegratinsinludednlytheuterfiveplanets(sussankinshitanakai1996).thisisbeausetherbitalperidsftheuterplanetsaresuhlngerthanthseftheinnerfurplanetsthatitisuheasiertfllthesystefragivenintegratinperid.atpresent，thelngestnuerialintegratinspublishedinjurnalsarethsefdunanlissauer(1998).althughtheiraintargetastheeffetfpst-ain-sequeneslarasslssnthestabilityfplanetaryrbits，theyperfredanyintegratinsveringupt～1011yrftherbitaltinsfthefurjvianplanets.theinitialrbitaleleentsandassesfplanetsarethesaeasthsefurslarsysteindunanlissauer'spaper，buttheydereasetheassfthesungraduallyintheirnuerialexperients.thisisbeausetheynsidertheeffetfpst-ain-sequeneslarasslssinthepaper.nsequently，theyfundthattherssingtie-salefplanetaryrbits，hihanbeatypialindiatrftheinstabilitytie-sale，isquitesensitivettheratefassdereasefthesun.hentheassfthesunislsetitspresentvalue，thejvianplanetsreainstablever1010yr，rperhapslnger.dunanlissaueralsperfredfursiilarexperientsntherbitaltinfsevenplanets(venustneptune)，hihveraspanf～109yr.theirexperientsnthesevenplanetsarentyetprehensive，butitseesthattheterrestrialplanetsalsreainstableduringtheintegratinperid，aintainingalstregularsillatins.

nthetherhand，inhisauratesei-analytialseularperturbatinthery(laskar1988)，laskarfindsthatlargeandirregularvariatinsanappearintheeentriitiesandinlinatinsftheterrestrialplanets，espeiallyferuryandarsnatie-salefseveral109yr(laskar1996).theresultsflaskar'sseularperturbatintheryshuldbenfiredandinvestigatedbyfullynuerialintegratins.

inthispaperepresentpreliinaryresultsfsixlng-ternuerialintegratinsnallnineplanetaryrbits，veringaspanfseveral109yr，andfttherintegratinsveringaspanf±5x1010yr.thettalelapsedtiefrallintegratinsisrethan5yr，usingseveraldediatedpsandrkstatins.nefthefundaentalnlusinsfurlng-terintegratinsisthatslarsysteplanetarytinseestbestableintersfthehillstabilityentinedabve，atleastveratie-spanf±4gyr.atually，inurnuerialintegratinsthesysteasfarrestablethanhatisdefinedbythehillstabilityriterin:ntnlydidnlseenunterhappenduringtheintegratinperid，butalsalltheplanetaryrbitaleleentshavebeennfinedinanarrreginbthintieandfrequenydain，thughplanetarytinsaresthasti.sinethepurpsefthispaperistexhibitandvervietheresultsfurlng-ternuerialintegratins，eshtypialexaplefiguresasevideneftheverylng-terstabilityfslarsysteplanetarytin.frreadershhaverespeifianddeeperinterestsinurnuerialresults，ehavepreparedaebpage(aess)，hereeshrarbitaleleents，theirl-passfilteredresults，variatinfdelaunayeleentsandangularentudefiit，andresultsfursipletie–frequenyanalysisnallfurintegratins.

insetin2ebrieflyexplainurdynaialdel，nuerialethdandinitialnditinsusedinurintegratins.setin3isdevtedtadesriptinfthequikresultsfthenuerialintegratins.verylng-terstabilityfslarsysteplanetarytinisapparentbthinplanetarypsitinsandrbitaleleents.arughestiatinfnuerialerrrsisalsgiven.setin4gesntadisussinfthelngest-tervariatinfplanetaryrbitsusingal-passfilterandinludesadisussinfangularentudefiit.insetin5，epresentasetfnuerialintegratinsfrtheuterfiveplanetsthatspans±5x1010yr.insetin6ealsdisussthelng-terstabilityftheplanetarytinanditspssibleause.

2desriptinfthenuerialintegratins

（本部涉及比较复杂积计算，君贴，贴点功显示。）

2.3nuerialethd

eutilizeasend-rderisd–hlansypletiapasurainintegratinethd(isdkinshita，yshidanakai1991)ithaspeialstart-uppreduretreduethetrunatinerrrfanglevariables，‘arstart’(sahatreaine1992，1994).

thestepsizefrthenuerialintegratinsis8dthrughutallintegratinsfthenineplanets(n±1，2，3)，hihisabut111ftherbitalperidftheinnerstplanet(erury).asfrthedeterinatinfstepsize，epartlyfllthepreviusnuerialintegratinfallnineplanetsinsussanisd(1988，7.2d)andsahatreaine(1994，22532d).erundedthedeialpartfthetheirstepsizest8takethestepsizeaultiplef2inrdertreduetheauulatinfrund-fferrrintheputatinpresses.inrelatintthis，isdhlan(1991)perfrednuerialintegratinsftheuterfiveplanetaryrbitsusingthesypletiapithastepsizef400d，110.83ftherbitalperidfjupiter.theirresultseestbeaurateenugh，hihpartlyjustifiesurethdfdeteriningthestepsize.hever，sinetheeentriityfjupiter(～0.05)isuhsallerthanthatferury(～0.2)，eneedseareheneparetheseintegratinssiplyintersfstepsizes.

intheintegratinftheuterfiveplanets(f±)，efixedthestepsizeat400d.

eadptgauss'fandgfuntinsinthesypletiaptgetheriththethird-rderhalleyethd(danby1992)asaslverfrkeplerequatins.thenuberfaxiuiteratinsesetinhalley'sethdis15，buttheyneverreahedtheaxiuinanyfurintegratins.

theintervalfthedatautputis200000d(～547yr)frthealulatinsfallnineplanets(n±1，2，3)，andabut8000000d(～21903yr)frtheintegratinftheuterfiveplanets(f±).

althughnutputfilteringasdnehenthenuerialintegratinsereinpress，eappliedal-passfilterttherarbitaldataafterehadpletedallthealulatins.seesetin4.1frredetail.

2.4errrestiatin

2.4.1relativeerrrsinttalenergyandangularentu

ardingtnefthebasiprpertiesfsypletiintegratrs，hihnservethephysiallynservativequantitiesell(ttalrbitalenergyandangularentu)，urlng-ternuerialintegratinsseethavebeenperfredithverysallerrrs.theaveragedrelativeerrrsfttalenergy(～10?9)andfttalangularentu(～10?11)havereainednearlynstantthrughuttheintegratinperid(fig.1).thespeialstartuppredure，arstart，uldhavereduedtheaveragedrelativeerrrinttalenergybyabutnerderfagnituderre.

relativenuerialerrrfthettalangularentuδaa0andthettalenergyδee0inurnuerialintegratinsn±1，2，3，hereδeandδaaretheabslutehangefthettalenergyandttalangularentu，respetively，ande0anda0aretheirinitialvalues.thehrizntalunitisgyr.

ntethatdifferentperatingsystes，differentatheatiallibraries，anddifferenthardarearhiteturesresultindifferentnuerialerrrs，thrughthevariatinsinrund-fferrrhandlingandnuerialalgriths.intheupperpanelffig.1，eanregnizethissituatinintheseularnuerialerrrinthettalangularentu，hihshuldberigruslypreserveduptahine-epreisin.

2.4.2errrinplanetarylngitudes

sinethesypletiapspreservettalenergyandttalangularentufn-bdydynaialsystesinherentlyell，thedegreeftheirpreservatinayntbeagdeasureftheaurayfnuerialintegratins，espeiallyasaeasurefthepsitinalerrrfplanets，i.e.theerrrinplanetarylngitudes.testiatethenuerialerrrintheplanetarylngitudes，eperfredthefllingpredures.eparedtheresultfurainlng-terintegratinsithsetestintegratins，hihspanuhshrterperidsbutithuhhigherauraythantheainintegratins.frthispurpse，eperfredauhreaurateintegratinithastepsizef0.125d(164ftheainintegratins)spanning3x105yr，startingiththesaeinitialnditinsasinthen?1integratin.ensiderthatthistestintegratinprvidesusitha‘pseud-true’slutinfplanetaryrbitalevlutin.next，eparethetestintegratiniththeainintegratin，n?1.frtheperidf3x105yr，eseeadiffereneineananaliesftheearthbeteenthetintegratinsf～0.52°(intheasefthen?1integratin).thisdiffereneanbeextraplatedtthevalue～8700°，abut25rtatinsfearthafter5gyr，sinetheerrrflngitudesinreaseslinearlyithtieinthesypletiap.siilarly，thelngitudeerrrfplutanbeestiatedas～12°.thisvaluefrplutisuhbetterthantheresultinkinshitanakai(1996)herethediffereneisestiatedas～60°.

3nuerialresults–i.glaneattheradata

inthissetinebrieflyreviethelng-terstabilityfplanetaryrbitaltinthrughsesnapshtsfranuerialdata.therbitaltinfplanetsindiateslng-terstabilityinallfurnuerialintegratins:nrbitalrssingsnrlseenuntersbeteenanypairfplanetstkplae.

3.1generaldesriptinfthestabilityfplanetaryrbits

first，ebrieflylkatthegeneralharaterfthelng-terstabilityfplanetaryrbits.urinterestherefusespartiularlyntheinnerfurterrestrialplanetsfrhihtherbitaltie-salesareuhshrterthanthseftheuterfiveplanets.aseanseelearlyfrtheplanarrbitalnfiguratinsshninfigs2and3，rbitalpsitinsftheterrestrialplanetsdifferlittlebeteentheinitialandfinalpartfeahnuerialintegratin，hihspansseveralgyr.theslidlinesdentingthepresentrbitsftheplanetsliealstithinthesarfdtseveninthefinalpartfintegratins(b)and(d).thisindiatesthatthrughuttheentireintegratinperidthealstregularvariatinsfplanetaryrbitaltinreainnearlythesaeastheyareatpresent.

vertialviefthefurinnerplanetaryrbits(frthez-axisdiretin)attheinitialandfinalpartsftheintegratinsn±1.theaxesunitsareau.thexy-planeissetttheinvariantplanefslarsystettalangularentu.(a)theinitialpartfn+1(t=0t0.0547x109yr).(b)thefinalpartfn+1(t=4.9339x108t4.9886x109yr).()theinitialpartfn?1(t=0t?0.0547x109yr).(d)thefinalpartfn?1(t=?3.9180x109t?3.9727x109yr).ineahpanel，attalf23684pintsareplttedithanintervalfabut2190yrver5.47x107yr.slidlinesineahpaneldentethepresentrbitsfthefurterrestrialplanets(takenfrde245).

thevariatinfeentriitiesandrbitalinlinatinsfrtheinnerfurplanetsintheinitialandfinalpartftheintegratinn+1isshninfig.4.asexpeted，theharaterfthevariatinfplanetaryrbitaleleentsdesntdiffersignifiantlybeteentheinitialandfinalpartfeahintegratin，atleastfrvenus，earthandars.theeleentsferury，espeiallyitseentriity，seethangetasignifiantextent.thisispartlybeausetherbitaltie-saleftheplanetistheshrtestfalltheplanets，hihleadstarerapidrbitalevlutinthantherplanets;theinnerstplanetaybenearesttinstability.thisresultappearstbeinseagreeentithlaskar's(1994，1996)expetatinsthatlargeandirregularvariatinsappearintheeentriitiesandinlinatinsferurynatie-salefseveral109yr.hever，theeffetfthepssibleinstabilityftherbitferuryayntfatallyaffettheglbalstabilityfthehleplanetarysysteingtthesallassferury.eillentinbrieflythelng-terrbitalevlutinferurylaterinsetin4usingl-passfilteredrbitaleleents.

therbitaltinftheuterfiveplanetsseesrigruslystableandquiteregularverthistie-span(seealssetin5).

3.2tie–frequenyaps

althughtheplanetarytinexhibitsverylng-terstabilitydefinedasthenn-existeneflseenunterevents，thehatinaturefplanetarydynaisanhangethesillatryperidandaplitudefplanetaryrbitaltingraduallyversuhlngtie-spans.evensuhslightflutuatinsfrbitalvariatininthefrequenydain，partiularlyintheasefearth，anptentiallyhaveasignifianteffetnitssurfaeliatesystethrughslarinslatinvariatin(f.berger1988).

tgiveanverviefthelng-terhangeinperidiityinplanetaryrbitaltin，eperfredanyfastfuriertransfratins(ffts)alngthetieaxis，andsuperpsedtheresultingperidgrastdrat-diensinaltie–frequenyaps.thespeifiapprahtdraingthesetie–frequenyapsinthispaperisverysiple–uhsiplerthantheaveletanalysisrlaskar's(1990，1993)frequenyanalysis.

dividethel-passfilteredrbitaldataintanyfragentsfthesaelenh.thelenhfeahdatasegentshuldbeaultiplef2inrdertapplythefft.

eahfragentfthedatahasalargeverlappingpart:frexaple，hentheithdatabeginsfrt=tiandendsatt=ti+t，thenextdatasegentrangesfrti+δt≤ti+δt+t，hereδt?t.entinuethisdivisinuntilereahaertainnubernbyhihtn+treahesthettalintegratinlenh.

eapplyanfftteahfthedatafragents，andbtainnfrequenydiagras.

ineahfrequenydiagrabtainedabve，thestrenhfperidiityanbereplaedbyagrey-sale(rlur)hart.

eperfrthereplaeent，andnnetallthegrey-sale(rlur)hartsintnegraphfreahintegratin.thehrizntalaxisfthesenegraphsshuldbethetie，i.e.thestartingtiesfeahfragentfdata(ti，herei=1，…，n).thevertialaxisrepresentstheperid(rfrequeny)fthesillatinfrbitaleleents.

ehaveadptedanfftbeausefitsverhelingspeed，sinetheauntfnuerialdatatbedepsedintfrequenypnentsisterriblyhuge(severaltensfgbytes).

atypialexaplefthetie–frequenyapreatedbytheabvepreduresisshninagrey-salediagraasfig.5，hihshsthevariatinfperidiityintheeentriityandinlinatinfearthinn+2integratin.infig.5，thedarkareashsthatatthetieindiatedbythevaluentheabsissa，theperidiityindiatedbytherdinateisstrngerthaninthelighterareaarundit.eanregnizefrthisapthattheperidiityftheeentriityandinlinatinfearthnlyhangesslightlyvertheentireperidveredbythen+2integratin.thisnearlyregulartrendisqualitativelythesaeintherintegratinsandfrtherplanets，althughtypialfrequeniesdifferplanetbyplanetandeleentbyeleent.

4.2lng-terexhangefrbitalenergyandangularentu

ealulateverylng-peridivariatinandexhangefplanetaryrbitalenergyandangularentuusingfiltereddelaunayeleentsl，g，h.gandhareequivalentttheplanetaryrbitalangularentuanditsvertialpnentperunitass.lisrelatedttheplanetaryrbitalenergyeperunitassase=?μ22l2.ifthesysteispletelylinear，therbitalenergyandtheangularentuineahfrequenybinustbenstant.nn-linearityintheplanetarysysteanauseanexhangefenergyandangularentuinthefrequenydain.theaplitudefthelest-frequenysillatinshuldinreaseifthesysteisunstableandbreaksdngradually.hever，suhasyptfinstabilityisntprinentinurlng-terintegratins.

infig.7，thettalrbitalenergyandangularentufthefurinnerplanetsandallnineplanetsareshnfrintegratinn+2.theupperthreepanelsshthelng-peridivariatinfttalenergy(dentedase-e0)，ttalangularentu(g-g0)，andthevertialpnent(h-h0)ftheinnerfurplanetsalulatedfrthel-passfiltereddelaunayeleents.e0，g0，h0dentetheinitialvaluesfeahquantity.theabslutedifferenefrtheinitialvaluesisplttedinthepanels.thelerthreepanelsineahfigureshe-e0，g-g0andh-h0fthettalfnineplanets.theflutuatinshninthelerpanelsisvirtuallyentirelyaresultftheassivejvianplanets.

paringthevariatinsfenergyandangularentuftheinnerfurplanetsandallnineplanets，itisapparentthattheaplitudesfthseftheinnerplanetsareuhsallerthanthsefallnineplanets:theaplitudesftheuterfiveplanetsareuhlargerthanthseftheinnerplanets.thisdesnteanthattheinnerterrestrialplanetarysubsysteisrestablethantheuterne:thisissiplyaresultftherelativesallnessftheassesfthefurterrestrialplanetsparediththseftheuterjvianplanets.antherthingentieisthattheinnerplanetarysubsysteaybeeunstablererapidlythantheuternebeausefitsshrterrbitaltie-sales.thisanbeseeninthepanelsdentedasinner4infig.7herethelnger-peridiandirregularsillatinsarereapparentthaninthepanelsdentedasttal9.atually，theflutuatinsintheinner4panelsaretalargeextentasaresultftherbitalvariatinftheerury.hever，eanntnegletthentributinfrtherterrestrialplanets，aseillseeinsubsequentsetins.

4.4lng-teruplingfseveralneighburingplanetpairs

letusseeseindividualvariatinsfplanetaryrbitalenergyandangularentuexpressedbythel-passfiltereddelaunayeleents.figs10and11shlng-terevlutinftherbitalenergyfeahplanetandtheangularentuinn+1andn?2integratins.entiethatseplanetsfrapparentpairsintersfrbitalenergyandangularentuexhange.inpartiular，venusandearthakeatypialpair.inthefigures，theyshnegativerrelatinsinexhangefenergyandpsitiverrelatinsinexhangefangularentu.thenegativerrelatininexhangefrbitalenergyeansthatthetplanetsfralseddynaialsysteintersftherbitalenergy.thepsitiverrelatininexhangefangularentueansthatthetplanetsaresiultaneuslyunderertainlng-terperturbatins.andidatesfrperturbersarejupiterandsaturn.alsinfig.11，eanseethatarsshs'itiverrelatinintheangularentuvariatintthevenus–earthsyste.eruryexhibitsertainnegativerrelatinsintheangularentuversusthevenus–earthsyste，hihseestbeareatinausedbythenservatinfangularentuintheterrestrialplanetarysubsyste.

itisntlearattheenthythevenus–earthpairexhibitsanegativerrelatininenergyexhangeand'itiverrelatininangularentuexhange.eaypssiblyexplainthisthrughbservingthegeneralfatthattherearenseulartersinplanetaryseiajraxesuptsend-rderperturbatintheries(f.bruerbalettipua1998).thiseansthattheplanetaryrbitalenergy(hihisdiretlyrelatedttheseiajraxisa)ightbeuhlessaffetedbyperturbingplanetsthanistheangularentuexhange(hihrelateste).hene，theeentriitiesfvenusandearthanbedisturbedeasilybyjupiterandsaturn，hihresultsin'itiverrelatinintheangularentuexhange.nthetherhand，theseiajraxesfvenusandeartharelesslikelytbedisturbedbythejvianplanets.thustheenergyexhangeaybeliitednlyithinthevenus–earthpair，hihresultsinanegativerrelatinintheexhangefrbitalenergyinthepair.

asfrtheuterjvianplanetarysubsyste，jupiter–saturnanduranus–neptuneseetakedynaialpairs.hever，thestrenhftheiruplingisntasstrngparediththatfthevenus–earthpair.

5±5x1010-yrintegratinsfuterplanetaryrbits

sinethejvianplanetaryassesareuhlargerthantheterrestrialplanetaryasses，etreatthejvianplanetarysysteasanindependentplanetarysysteintersfthestudyfitsdynaialstability.hene，eaddedaupleftrialintegratinsthatspan±5x1010yr，inludingnlytheuterfiveplanets(thefurjvianplanetsplusplut).theresultsexhibittherigrusstabilityftheuterplanetarysysteverthislngtie-span.rbitalnfiguratins(fig.12)，andvariatinfeentriitiesandinlinatins(fig.13)shthisverylng-terstabilityftheuterfiveplanetsinbththetieandthefrequenydains.althughedntshapshere，thetypialfrequenyftherbitalsillatinfplutandthetheruterplanetsisalstnstantduringtheseverylng-terintegratinperids，hihisdenstratedinthetie–frequenyapsnurebpage.

inthesetintegratins，therelativenuerialerrrinthettalenergyas～10?6andthatfthettalangularentuas～10?10.

5.1resnanesintheneptune–plutsyste

kinshitanakai(1996)integratedtheuterfiveplanetaryrbitsver±5.5x109yr.theyfundthatfurajrresnanesbeteenneptuneandplutareaintainedduringthehleintegratinperid，andthattheresnanesaybetheainausesfthestabilityftherbitfplut.theajrfurresnanesfundinpreviusresearhareasflls.inthefllingdesriptin，λdentestheeanlngitude，Ωisthelngitudeftheasendingndeand?isthelngitudefperihelin.subsriptspandndenteplutandneptune.

eantinresnanebeteenneptuneandplut(3:2).theritialarguentθ1=3λp?2λn??plibratesarund180°ithanaplitudefabut80°andalibratinperidfabut2x104yr.

thearguentfperihelinfplutp=θ2=?p?Ωplibratesarund90°ithaperidfabut3.8x106yr.thedinantperidivariatinsftheeentriityandinlinatinfplutaresynhrnizediththelibratinfitsarguentfperihelin.thisisantiipatedintheseularperturbatintherynstrutedbykzai(1962).

thelngitudefthendefplutreferredtthelngitudefthendefneptune，θ3=Ωp?Ωn，irulatesandtheperidfthisirulatinisequalttheperidfθ2libratin.henθ3beeszer，i.e.thelngitudesfasendingndesfneptuneandplutverlap，theinlinatinfplutbeesaxiu，theeentriitybeesiniuandthearguentfperihelinbees90°.henθ3bees180°，theinlinatinfplutbeesiniu，theeentriitybeesaxiuandthearguentfperihelinbees90°again.illiasbensn(1971)antiipatedthistypefresnane，laternfiredbyilani，nbiliarpin(1989).

anarguentθ4=?p??n+3(Ωp?Ωn)libratesarund180°ithalngperid，～5.7x108yr.

inurnuerialintegratins，theresnanes(i)–(iii)areellaintained，andvariatinftheritialarguentsθ1，θ2，θ3reainsiilarduringthehleintegratinperid(figs14–16).hever，thefurthresnane(iv)appearstbedifferent:theritialarguentθ4alternateslibratinandirulatinvera1010-yrtie-sale(fig.17).thisisaninterestingfatthatkinshitanakai's(1995，1996)shrterintegratinserentabletdislse.

6disussin

hatkindfdynaialehanisaintainsthislng-terstabilityftheplanetarysyste?eaniediatelythinkftajrfeaturesthatayberespnsiblefrthelng-terstability.first，thereseetbensignifiantler-rderresnanes(eantinandseular)beteenanypairangthenineplanets.jupiterandsaturnarelseta5:2eantinresnane(thefaus‘greatinequality’)，butntjustintheresnanezne.higher-rderresnanesayausethehatinatureftheplanetarydynaialtin，buttheyarentsstrngastdestrythestableplanetarytinithinthelifetieftherealslarsyste.thesendfeature，hihethinkisreiprtantfrthelng-terstabilityfurplanetarysyste，isthediffereneindynaialdistanebeteenterrestrialandjvianplanetarysubsystes(ittanikaa1999，2001).heneeasureplanetaryseparatinsbytheutualhillradii(r_)，separatinsangterrestrialplanetsaregreaterthan26rh，hereasthseangjvianplanetsarelessthan14rh.thisdiffereneisdiretlyrelatedtthedifferenebeteendynaialfeaturesfterrestrialandjvianplanets.terrestrialplanetshavesallerasses，shrterrbitalperidsandiderdynaialseparatin.theyarestrnglyperturbedbyjvianplanetsthathavelargerasses，lngerrbitalperidsandnarrerdynaialseparatin.jvianplanetsarentperturbedbyanytherassivebdies.

thepresentterrestrialplanetarysysteisstillbeingdisturbedbytheassivejvianplanets.hever，theideseparatinandutualinteratinangtheterrestrialplanetsrendersthedisturbaneineffetive;thedegreefdisturbanebyjvianplanetsis(ej)(rderfagnitudeftheeentriityfjupiter)，sinethedisturbaneausedbyjvianplanetsisafredsillatinhavinganaplitudef(ej).heighteningfeentriity，frexaple(ej)～0.05，isfarfrsuffiienttprvkeinstabilityintheterrestrialplanetshavingsuhaideseparatinas26rh.thuseassuethatthepresentidedynaialseparatinangterrestrialplanets(;26rh)isprbablynefthestsignifiantnditinsfraintainingthestabilityftheplanetarysystevera109-yrtie-span.urdetailedanalysisftherelatinshipbeteendynaialdistanebeteenplanetsandtheinstabilitytie-salefslarsysteplanetarytinisnn-ging.

althughurnuerialintegratinsspanthelifetieftheslarsyste，thenuberfintegratinsisfarfrsuffiienttfilltheinitialphasespae.itisneessarytperfrreandrenuerialintegratinstnfirandexaineindetailthelng-terstabilityfurplanetarydynais.

——以文段引it，t.tanikaa，k.lng-terintegratinsandstabilityfplanetaryrbitsinurslarsyste.n.nt.r.astrn.s.336，483–500(2002)

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