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long-tertegrationsaaryorbitsoursorsyste
abstrac
wepresenttheresultsofverylong-ternuricaliaryorbitalotionsover109-yrti-spanscsaquickspectionofournuricaldataryotion，atleastoursipledynaicalodel，seestobequitestableevenoverthisverylongti-spanacloserlookatthelowest-frequencyosciltionsgalow-passfiltershowsthepotentiallydiffivecharacteraryotion，especiallythatofrcurythentricityofrcuryourtegrationsisqualitativelysiirtotheresultsfrojacquesskar'ssecurperturbationtheory(egeax∼035over∼±4gyr)however，therearenoapparentsecentricityorctionanyorbitals，whichayberevealedbystilllonger-ternuricaltegrationswehavealsoperfordaupleoftrialtegrationscdgotionsosoverthedurationof±5x1010yrtheresultdicatesthatthethreeajorresonancestheneptune–ptosystehavebeenataedoverthe1011-yrti-span
1troduction
11defitionoftheproble
thequestionofthestabilityofoursorsystehasbeendebatedoverseveralhundredyears，scetheeraofnewtontheproblehasattractedanyfaoatheaticiansovertheyearsandhaspyedacentralrolethedevelopntofnon-leardynaicsandchaostheoryhowever，wedonotyethaveadefiteanswertothequestionofwhetheroursorsysteisstableornotthisispartlyaresultofthefactthatthedefitionoftheter‘stability’isvaguewhenitisedretiontaryotionthesorsysteactuallyitisnoteasytogiveaclear，rigoroandphysicallyangfuldefitionofthestabilityofoursorsyste
aonganydefitionsofstability，hereweadoptthehilldefition(gdan1993):actuallythisisnotadefitionofstability，butofstabilitywedgunstablewhenurssowherethesyste，startgfroacertaitialnfiguration(chabers，wetherill&boss1996;ito&tanikawa1999)asysteisdefedasexperiencgacloseenunterwhenobodiesapproachoneanotherwithanareaofthergerhillradiotherwisethesysteisdefedasbegstablehenceforwardwarysysteisdynaicallystableifnocloseenunterhappensdurgtheageofoursorsyste，about±5gyrcidentally，thisdefitionayberepceurrenceofanyorbitalcrossgbeeeneistakespcethisisbecaeweknowfroxperiencethatanorbitalcrossgisverylikelytoleadtoacloaryandarysystes(yoshaga，kokubo&ako1999)ofursethisstatentcannotbesiplyappliedtosysteswithstableorbitalresonancessuchastheneptune–ptosyste
12previostudiesandaisofthisresearch
additiontothevaguenessofthenceptofstability，soursorsysteshowacharactertypicalofdynaicalchaos(ssan&wisdo1988，1992)thecaeofthischaoticbehaviourisnowpartlyunderstoodasbegaresultofresonanceoverppg(urray&hon1999;lecar，frankl&hon2001)however，iouldrequiretegratgovearysystescsforaperiodvergseveral10gyrtothoroughlyunderstandthelong-tearyorbits，scechaoticdynaicalsystesarecharacterizedbytheirstrongdependenceonitialnditions
frothatpotofview，anyofthepreviolong-ternuricaltegrationscdedonls(ssan&wisdo1988;koshita&nakai1996)thisisbecaetheorbitalperisaresouchlongerthanthoseosthatitisucheasiertofollowthesysteforagiventegrationperiodatpresent，thelongestnuricaltegrationspublishedjournalsarethoseofduncan&lissauer(1998)althoughtheiratargeastheeffectofpost-a-sequencesorasslossontaryorbits，theyperfordanytegrationsvergupto∼1011yroftheorbitalotionsofstheitialorbitalelensarethesaasthoseofoursorsysteduncan&lissauer'spaper，buttheydecreasetheassofthesungraduallytheirnuricalexpertsthisisbecaetheynsidertheeffectofpost-a-sequencesorasnsequently，theyfoundthatthecrossgti-aryorbits，whichcanbeatypicaldicatorofthestabilityti-scale，isquitesensitivetotherateofassdecreaseofthesunwhentheassofthesunisclosetoitspresentvae，sreastableover1010yr，orperhapslongerduncan&lissaueralsoperfordfoursiirexpertsontheorbitals(ventoneptune)，whichveraspanof∼109yrtheirexperisarenotprehensive，butitseesthattsalsoreastabledurgthetegrationperiod，atagalostregurosciltions
ontheotherhand，uratesei-analyticalsecurperturbationtheory(skar1988)，skarfdsthatrgeandirregurvariationentricitiesandctionsofts，especiallyofrcuryandarsonati-scaleofseveral109yr(skar1996)theresultsofskar'ssecurperturbationtheoryshouldbenfirdandvestigatedbyfullynuricaltegrations
thispaperwepresentpreliaryresultsofsixlong-ternuricaltegrataryorbits，vergaspanofseveral109yr，andofoothertegrationsvergaspanof±5x1010yrthetotalepsedtiforalltegrationsisorethan5yr，gseveraldedicatedpcsandworkstationsoneofthefundantalncsionsofourlong-tertegrationsistharyotionseestobestabletersofthehillstabilityntionedabove，atleastoverati-spanof±4gyractually，ournuricaltegrationsthesystewasfarorestablethanwhatisdefedbythehillstabilitycriterion:notonlydidnocloseenunterhappendurgthetegrationperiod，aryorbitalelentshavebeennfedanarrowregionbothtiandfrequencydoa，aryotionsarestochasticscethepurposeofthispaperistoexhibitandoverviewtheresultsofourlong-ternuricaltegrations，weshowtypicalexaplefiguresasevidenceoftheverylong-terstabilityaryotionforreaderswhohaveorespecificanddeeperterestsournuricalresults，wehavepreparedawebpage(aess)，whereweshowraworbitalelents，theirlow-passfilteredresults，variationofdeunayelentsandangurontudeficit，andresultsofoursipleti–frequencyanalysisonallofourtegrations
section2webrieflyexpourdynaicalodel，nuricalthodanditialnditionsedourtegrationssection3isdevotedtoadescriptionofthequickresultsofthenuricaltegrationsverylong-terstabilityaryotionisaarypositionsandorbitalelentsaroughestiationofnuricalerrorsisalsogivensection4goesontoadiscsionofthelongest-tearyorbitsgalow-passfilterandcdesadiscsionofangurontudeficitsection5，wepresentasetofnuricaltegrationsfosthatspans±5x1010yrsection6wealsodiscsthelong-teraryotionanditspossiblecae
2descriptionofthenuricaltegrations
（本部分涉及比较复杂的积分计算，作者君就不贴上来了，贴上来了起点也不一定能成功显示。）
23nuricalthod
weutilizeasend-orderwisdo–honsyplectiapasourategrationthod(wisdo&hon1991;koshita，yoshida&nakai1991)withaspecialstart-upproceduretoreducethetruncationerrorofanglevariables，‘warstart’(saha&treae1992，1994)
thestepsizeforthenuricaltegrationsis8dthroughoutalltegrats(n±1，2，3)，whichisabout1/11oftheorbitalperiodo(rcury)asforthedeterationofstepsize，wepartlyfollowtheprevionuricaltegrasssan&wisdo(1988，72d)andsaha&treae(1994，225/32d)weroundedthedecialpartofthetheirstepsizesto8toakethestepsizeaultipleof2ouutionofround-offerrorputationprocessesretiontothis，wisdo&hon(1991)perfordnuricaltegrationsoaryorbitsgthesyplectiaithastepsizeof400d，1/1083oftheorbitalperiodofjupitertheirurateenough，whichpartlyjtifiesourthodofdetergthestepsizehowever，entricityofjupiter(∼005)isuchsallerthanthatofrcury(∼02)，weneparethesetegrationssiplytersofstepsizes
thetegrationos(f±)，wefixedthestepsizeat400d
weadoptgas'fandgfunctionsthesyplectiaptogetherwiththethird-orderhalleythod(danby1992)asasolverforkeplerequationsthenuberofaxiuiterationswesethalley'sthodis15，buttheyneverreachedtheaxiuanyofourtegrations
thetervalofthedataoutputis200000d(∼547yr)forthecalcuts(n±1，2，3)，andabout8000000d(∼21903yr)forthetegrationos(f±)
althoughnooutputfiltergwasdonewhenthenuricaltegrationswereprocess，weappliedalow-passfiltertotheraworbitaldataafterwepletedallthecalcutionsseesection41fororedetail
24errorestiation
241retiveerrorstotalenergyandangurontu
aordgtooneofthebasicpropertiesofsyplectictegrators，whichnservethephysicallynservativequantitieswell(totalorbitalenergyandangurontu)，ourlong-ternuricaltegrationsseetohavebeenperfordwithverysallerrorstheaveragedretiveerrorsoftotalenergy(∼10−9)andoftotangurontu(∼10−11)havereaednearlynstantthroughoutthetegrationperiod(fig1)thespecialstartupprocedure，warstart，wouldhavereducedtheaveragedretiveerrortotalenergybyaboutoneorderofagnitudeorore
retivenuricalerrorofthetotangurontuδa/a0andthetotalenergyδe/e0ournuricaltegrationsn±1，2，3，whereδeandδaaretheabsotechangeofthetotalenergyandtotangurontu，respectively，ande0anda0aretheiritialvaesthehorizontanitisgyr
notethatdiffereneratgsystes，differentatheaticallibraries，anddifferenthardwarearchitecturesresultdifferentnuricalerrors，throughthevariationsround-offerrorhandlgandnuricalgorithstheupperpaneloffig1，wecanregnizethissituationthesecurnuricalerrorthetotangurontu，whichshouldberigorolypreserveduptoache-eprecision
24arylongitudes
scethesyplectiapspreservetotalenergyandtotangurontuofn-bodydynaicalsystesherentlywell，thedegreeoftheirpreservationaynotbeaguracyofnuricaltegrations，especiallyasaasureoftheposis，iearylongitudestoestiatethenuriarylongitudes，weperfordthefollogproceduresweparedtheresultofouralong-tertegrationswithsotesttegrations，whichspanuchshorterperiodsbuuracythantheategrationsforthispurpose，weperuratetegrationwithastepsizeof0125d(1/64oftheategrations)spanng3x105yr，startgwiththesaitialnditionsasthen−1tegrationwensiderthatthistesttegrationprovideswitha‘pseudo-true’aryorbitalevotionnext，weparethetesttegrationwiththeategration，n−1fortheperiodof3x105yr，weseeadifferenceananoaliesoftheearthbeeentheotegrationsof∼052°(thecaseofthen−1tegration)thisdifferencecanbeextrapotedtothevae∼8700°，about25rotationsofearthafter5gyr，scetheerroroflongitudescreaseslearlywithtithesyplectiapsiirly，thelongitudeerrorofptocanbeestiatedas∼12°thisvaeforptoisuchbetterthantheresultkoshita&nakai(1996)wherethedifferenceisestiatedas∼60°
3nuricalresults–ignceattherawdata
thissectionwebrieflyreviewthelong-tearyorbitalotionthroughsosnapshotsofrawnuricaldatatheorsdicateslong-terstabilityallofournuricaltegrations:noorbitalcrossgsnorcloseenuntersbetstookpce
31generaldescriptionoftaryorbits
first，webrieflylookatthegeneralcharacterofthelong-tearyorbitsourterestherefocesparticurlyonthenerfosforwhichtheorbitalti-scalesareuchshorterthanthoseosaswecanseeclearlyfrothepnarorbitalnfigurationsshownfigs2and3，orbitalpositionsoftsdifferlittlebeeentheitiandfalpartofeachnuricaltegration，whichspansseveralgyrthesolidlesdenotgthepresesliealosiththeswarofdotseventhefalpartoftegrations(b)and(d)thisdicatesthatthroughouttheentiretegrationperiodthealostreguaryorbitalotionreanearlythesaastheyareatpresen
verticalviewoaryorbits(frothez-axisdirection)attheitiandfalpartsofthetegrationsn±1theaxesunitsareauthexy-pneissettothevariantpneofsorsystetotangurontu(a)theitialpartofn+1(t=0to00547x109yr)(b)thefalpartofn+1(t=49339x108to49886x109yr)(c)theitialpartofn−1(t=0to−00547x109yr)(d)thefalpartofn−1(t=−39180x109to−39727x109yr)eachpanel，atotalof23684potsareplottedwithantervalofabout2190yrover547x107yrsolidleseachpaneldenotethepresentorbitsofthefos(takenfrode245)
entricitiesandorbitalctionsfostheitiandfalpartofthetegrationn+1isshownfig4asexpected，thecharacteroftaryorbitalelentsdoesnotdiffersignificantlybeeentheitiandfalpartofeachtegration，atleastforven，earthandarstheelentsofrcury，entricity，seetochangetoasignificantextentthisispartlybecaetheorbitalti-istheshos，whichleadstoaorerapidorbitalevos;aybenearesttostabilitythisresultappearstobesoagreenithskar's(1994，1996)expectationsthatrgeandirregurvariatentricitiesandctionsofrcuryonati-scaleofseveral109yrhowever，theeffectofthepossiblestabilityoftheorbitofrcuryaynotfatallyaffecttheglobalstabirysysteogtothesalssofrcurywewillntionbrieflythelong-terorbitalevotionofrcurytersection4glow-passfilteredorbitalelents
theorbitalotionosseesrigorolystableandquitereguroverthisti-span(seealsosection5)
32ti–frequencyaps
aryotionexhibitsverylong-terstabilitydefedasthenon-existenceofcloseenunterevents，arydynaicscanchangetheosciltoryperiodaaryorbitalotiongraduallyoversuchlongti-spansevensuchslightfctuationsoforbitalvariationthefrequencydoa，particurlythecaseofearth，canpotentiallyhaveasignificanteffectonitssurfacecliatesystethroughsorsotionvariation(cfberger198
togiveanoverviewofthelong-terchangearyorbitalotion，weperfordanyfastfouriertransforations(ffts)alongthetiaxis，andsuperposedtheresultgperiodgrastodrawo-dsionalti–frequencyapsthespecificapproachtodragtheseti–frequencyapsthispaperisverysiple–uchsiplerthanthewaveletanalysisorskar's(1990，1993)frequencyanalysis
dividethelow-passfilteredorbitaldatatoanyfragntsofthesalengththelengthofeachdatasegntshouldbeaultipleof2ordertoapplytheff
eachfragntofthedatahasargeoverppgpart:forexaple，whentheithdatabegsfrot=tiandendsatt=ti+t，thenextdatasegntrangesfroti+δt≤ti+δt+t，whereδt?entuethisdivisionuntilwereachacertanubernbywhichtn+treachesthetotaltegrationlength
weapplyanffttoeachofthedatafragnts，andobtanfrequencydiagras
eachfrequencydiagraobtaedabove，thestrengthofperiodicitycanberepcedbyagrey-scale(orlour)char
weperfortherepcent，andnnectallthegrey-scale(orlour)chartstoonegraphforeachtegrationthehorizontaxisofthesenewgraphsshouldbetheti，iethestartgtisofeachfragntofdata(ti，wherei=1，…，n)theverticaxisrepresentstheperiod(orfrequency)oftheosciltionoforbitalelents
wehaveadoptedanfftbecaeofitsoverwhelgspeed，scetheaountofnuposedonentsisterriblyhuge(severaltensofgbytes)
atypicalexapleoftheti–frequencyapcreatedbytheaboveproceduresisshownagrey-scalediagraasfig5，whichshowsthevariationofpentricityandctionofearthn+2tegrationfig5，thedarkareashowsthatatthetidicatedbythevaeontheabscissa，theperiodicitydicatedbytheordateisstrongerthanthelighterareaaroundiecanregnizefrothisapthatthepentricityandctionofearthonlychangesslightlyovertheentireperiodveredbythen+2tegrationthisnearlyregurtrendisqualitativelythesaothertegratios，althoughtypicalfreqandelentbyelen
42long-terxchangeoforbitalenergyandangurontu
wecalcuteverylong-periodicvariationaryorbitalenergyandangurontugfiltereddeunayelentsl，g，hgandhareeqaryorbitangurontuanditsponentperunitassliaryorbitalenergyeperunitassase=−μ2/2l2pletelylear，theorbitalenergyandtheangurontueachfrequencybtbenstantnon-rysystecancaeanexchangeofenergyandangurontuthefrequencydoatheaplitudeofthelowest-frequencyosciltionshouldcreaseifthesysteisunstableandbreaksdowngraduallyhowever，suchasyptoofstabilityisnotproentourlong-tertegrations
fig7，thetotalorbitalenergyandangurontareshownfortegrationn+2theupperthreepanelsshowthelong-periodicvariationoftotalenergy(denotedase-e0)，totangurontu(g-g0)，andtheponent(h-h0)oscalcutedfrothelow-passfiltereddeunayelentse0，g0，h0denotetheitialvaesofeachquantitytheabsotedifferencefrotheitialvaesisplottedthepanelsthelowerthreepanelseachfigureshowe-e0，g-g0andh-h0oftsthefctuationshownthelowerpanelsisvirtuallyentirelyaresultofthes
&pargthevariationsofenergyandanguront，itisapparentthattheaplitudesofthsareuchsallerthants:theaplitudesosareuchrgerthanthsthisdoesnotanthatthenarysubsysteisorestablethantheouterone:thisissiplyaresultoftheretivesallnessoftheassesofthefourterrestrialparedwiththoseoftsanotherthgwenoticeiaeunstableorerapidlythantheouteronebecaeofitsshorterorbitalti-scalesthiscanbeseenthepanelsdenotedasner4fig7wherethelonger-periodicandirregurosciltionsareoreapparentthanthepanelsdenotedastotal9actually，thefctuationsthener4panelsaretoargeextentasaresultoftheorbitalvariationofthercuryhowever，wecannotneglectthentributionfrooths，aswewillseesubsequentsections
44long-teruplgofseverapai
letseesodividuaaryorbitalenergyandangurontuxpressedbythelow-passfiltereddeunayelentsfigs10and11showlong-tervotionoftheorbitaandtheangurontun+1andn−2tegrationswensforapparentpairstersoforbitalenergyandangurontuxchangeparticur，venandearthakeatypicalpairthefigures，theyshownegativerretionsexchangeofenergyandpositiverretionsexchangeofangurontuthenegativerretionexchangeoforbitalenergysforacloseddynaicalsystetersoftheorbitalenergythepositiverretionexchangeofangurontaresiultaneolyundercertalong-terperturbationscandidatesforperturbersarejupiterandsaturnalsofig11，wecanseethatarsshowsapositiverretiontheangurontuvariationtotheven–earthsystercuryexhibitscertanegativerretionstheangurontuverstheven–earthsyste，whichseestobeareactioncaedbythenservationofangurontutarysubsyste
itisnotclearattheonhytheven–earthpairexhibitsanegativerretionenergyexchangeandapositiverretionangurontuxchangeweaypossiblyexpthisthroughobservgthegeneralfactthattherearenosaryseiajoraxesuptosend-orderperturbationtheories(cfbrouwer&clence1961;boaletti&o1998)thiaryorbitalenergy(whichisdirectlyretedtotheseiajoraxisa)ightbeuchlessaffectesthanistheangurontuxchange(whichretestoe)hence，entricitiesofvenandearthcanbedisturbedeasilybyjupiterandsaturn，whichresultsapositiverretiontheangurontuxchangeontheotherhand，theseiajoraxesofvenandeartharelesslikelytobedisturbsththeenergyexchangeaybeliitedonlywiththeven–earthpair，whichresultsanegativerretiontheexchangeoforbitalenergythepair
asfortarysubsyste，jupiter–saturnanduran–neptuneseetoakedynaicalpairshowever，thestrengthoftheiruplgisnotasparedwiththatoftheven–earthpair
5±5x1010-yrtegraryorbits
saryassesareuchrgerthantaryasses，wetarysysteasarysystetersofthestudyofitsdynaicalstabilityhence，weaddedaupleoftrialtegrationsthatspan±5x1010yr，cdgonls(spspto)theresultsexhibittherigorostabirysysteoverthislongti-spanorbitalnfigurations(fig12)，entricitiesandctions(fig13)showthisverylong-terstabilityosboththetiandthefrequencydoasalthoughwedonotshowapshere，thetypicalfrequencyoftheorbitalosciltionofptoandsisalostnstantdurgtheseverylong-tertegrationperiods，whichisdeonstratedtheti–frequencyapsonourwebpage
theseotegrations，theretivenuricalerrorthetotalenergywas∼10−6andthatofthetotangurontuwas∼10−10
51resonancestheneptune–ptosyste
koshita&nakai(1996)tegratearyorbitsover±55x109yrtheyfoundthatfourajorresonancesbeeenneptuneandptoareataeddurgthewholetegrationperiod，andthattheresonancesaybetheacaesofthestabilityoftheorbitofptotheajorfourresonancesfoundprevioresearchareasfollowsthefollogdescription，λdenotestheanlongitude，Ωisthelongitudeoftheascendgnodeandϖisthelongitudeofperihelionsubscriptspandndenoteptoandneptune
anotionresonancebeeenneptuneandpto(3:2)thecriticarguntθ1=3λp−2λn−ϖplibratesaround180°withanaplitudeofabout80°andalibrationperiodofabout2x104yr
thearguntofperihelionofptowp=θ2=ϖp−Ωplibratesaround90°withaperiodofabout38x106yrthedoantperiodicentricityandctionofptoaresynchronizedwiththelibrationofitsarguntofperihelionthisisanticipatedthesecurperturbationtheorynstructedbykozai(1962)
thelongitudeofthenodeofptoreferredtothelongitudeofthenodeofneptune，θ3=Ωp−Ωn，circutesandtheperiodofthiscircutionisequaltotheperiodofθ2librationwhenθ3beeszero，iethelongitudesofascendgnodesofneptuneandptooverp，theclesaxiu，esiuandtheargues90°whenθ3bees180°，theclesiu，esaxiuandtheargues90°agawillias&benson(1971)anticipatedthistypeofresonance，ternfirdbyi，nobili&carpo(1989)
anarguntθ4=ϖp−ϖn+3(Ωp−Ωn)libratesaround180°withalongperiod，∼57x108yr
ournuricaltegrations，theresonances(i)–(iii)areweltaed，andvariationofthecriticarguntsθ1，θ2，θ3reasiirdurgthewholetegrationperiod(figs14–16)however，thefourthresonance(iv)appearstobedifferent:thecriticarguntθ4alternateslibrationandcircutionovera1010-yrti-scale(fig17)thisisanterestgfactthatkoshita&nakai's(1995，1996)shortertegrationswerenotabletodisclose
6discsion
whatkdofdynaicalchanisatasthislong-terarysyste?wecanidiatelythkofoajorfeaturesthatayberesponsibleforthelong-terstabilityfirst，thereseetobenosignificantlower-orderresonances(anotionandsecur)beeenanypaisjupiterandsaturnareclosetoa5:2anotionresonance(thefao‘greatequality’)，butnotjttheresonancezonehigher-orderresonancesarydynaicalotion，buttheyarenotsostrongastodesaryotionwiththelifetioftherealsorsystethesendfeature，whichwethkisoreiportantforthelong-terarysyste，isthedifferencedynaicaldistancebeeenterrestarysubsystes(ito&tanikawa1999，2001)aryseparationsbytheutualhillradii(r_)，separationsasaregreaterthan26rh，whereasthosarelessthan14rhthisdifferenceisdirectlyretedtothedifferencebeeendynaicalfeaturesofterrestsshavesallerasses，shorterorbitalperiodsandwiderdynaicalseparationtheyarestronglypertsthathavergerasses，longerorbitalperiodsandnarrowerdynaicalseparationsarenotperturbedbyanyotherassivebodies
thepresearysysteisstillbegdisturbedbytheshowever，thewideseparationandutualteractionaongtsrendersthedisturbanceeffective;thedegreeofdistursiso(ej)(orderoentricityofjupiter)，scethedisturbancecsisaforcedosciltionhavganaplitudeofo(ej)entricity，forexapleo(ej)∼005，isfarfrosufficientrovokestabilitytshavgsuchawideseparationas26rhthweassuthatthepresenidedynaicalseparationas(>26rh)isprobablyoneoftheostsignificantnditionsforatagthesarysysteovera109-yrti-spanourdetailedanalysisoftheretionshipbeeendynaicaldisandthestabilityti-scalearyotionisnowon-gog
althoughournuricaltegrationsspanthelifetiofthesorsyste，thenuberoftegrationsisfarfrosufficienttofilltheitialphasespaceitisnecessaryerfororeandorenuricaltegrationstonfirandexaedetailthelong-terarydynaics
——以上文段引自ito，t&tanikawa，klong-tertegrationsaaryorbitsoursorsysteonnotrastronsoc336，483–500(2002)
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