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long-tertegrationsandstabilityofparyorbitsoursorsyste

abstract

wepresenttheresultsofverylong-ternuricaltegrationsofparyorbitalotionsover109-yrti-spanscdgallnepsaquickspectionofournuricaldatashowsthattheparyotion，atleastoursipledynaicalodel，seestobequitestableevenoverthisverylongti-spanacloserlookatthelowest-frequencyosciltionsgalow-passfiltershowsthepotentiallydiffivecharacterofterrestrialparyotion，especiallythatofrcurythebehaviouroftheeentricityofrcuryourtegrationsisqualitativelysiirtotheresultsfrojacquesskar&039;ssecurperturbationtheory(egeax～035over～±4gyr)however，therearenoapparentsecurcreasesofeentricityorclationanyorbitalelentsoftheps，whichayberevealedbystilllonger-ternuricaltegrationswehavealsoperfordaupleoftrialtegrationscdgotionsoftheouterfivepsoverthedurationof±5x1010yrtheresultdicatesthatthethreeajorresonancestheneptune–ptosystehavebeenataedoverthe1011-yrti-span

1troduction

11defitionoftheproble

thequestionofthestabilityofoursorsystehasbeendebatedoverseveralhundredyears，scetheeraofnewtontheproblehasattractedanyfaoatheaticiansovertheyearsandhaspyedacentralrolethedevelopntofnon-leardynaicsandchaostheoryhowever，wedonotyethaveadefiteanswertothequestionofwhetheroursorsysteisstableornotthisispartlyaresultofthefactthatthedefitionoftheter‘stability’isvaguewhenitisedretiontotheprobleofparyotionthesorsysteactuallyitisnoteasytogiveaclear，rigoroandphysicallyangfuldefitionofthestabilityofoursorsyste

aonganydefitionsofstability，hereweadoptthehilldefition(gdan1993):actuallythisisnotadefitionofstability，butofstabilitywedefeasysteasbegunstablewhenacloseenunterourssowherethesyste，startgfroacertaitialnfiguration(chabers，wetherill≈ap;ap;ap;boss1996;ito≈ap;ap;ap;tanikawa1999)asysteisdefedasexperiencgacloseenunterwhenobodiesapproachoneanotherwithanareaofthergerhillradiotherwisethesysteisdefedasbegstablehenceforwardwestatethatourparysysteisdynaicallystableifnocloseenunterhappensdurgtheageofoursorsyste，about±5gyrcidentally，thisdefitionayberepcedbyonewhichanourrenceofanyorbitalcrossgbeeeneitherofapairofpstakespcethisisbecaeweknowfroexperiencethatanorbitalcrossgisverylikelytoleadtoacloseenunterparyandproarysystes(yoshaga，kokubo≈ap;ap;ap;ako1999)ofursethisstatentcannotbesiplyappliedtosysteswithstableorbitalresonancessuchastheneptune–ptosyste

12previostudiesandaisofthisresearch

additiontothevaguenessofthenceptofstability，thepsoursorsysteshowacharactertypicalofdynaicalchaos(ssan≈ap;ap;ap;wisdo1988，1992)thecaeofthischaoticbehaviourisnowpartlyunderstoodasbegaresultofresonanceoverppg(urray≈ap;ap;ap;holan1999;lecar，frankl≈ap;ap;ap;holan2001)however，itwouldrequiretegratgoveranensebleofparysystescdgallnepsforaperiodvergseveral10gyrtothoroughlyunderstandthelong-terevotionofparyorbits，scechaoticdynaicalsystesarecharacterizedbytheirstrongdependenceonitialnditions

frothatpotofview，anyofthepreviolong-ternuricaltegrationscdedonlytheouterfiveps(ssan≈ap;ap;ap;wisdo1988;koshita≈ap;ap;ap;nakai1996)thisisbecaetheorbitalperiodsoftheouterpsaresouchlongerthanthoseofthenerfourpsthatitisucheasiertofollowthesysteforagiventegrationperiodatpresent，thelongestnuricaltegrationspublishedjournalsarethoseofduncan≈ap;ap;ap;lissauer(1998)althoughtheiratargetwastheeffectofpost-a-sequencesorasslossonthestabilityofparyorbits，theyperfordanytegrationsvergupto～1011yroftheorbitalotionsofthefourjovianpstheitialorbitalelentsandassesofpsarethesaasthoseofoursorsysteduncan≈ap;ap;ap;lissauer&039;spaper，buttheydecreasetheassofthesungraduallytheirnuricalexperintsthisisbecaetheynsidertheeffectofpost-a-sequencesorasslossthepapernsequently，theyfoundthatthecrossgti-scaleofparyorbits，whichcanbeatypicaldicatorofthestabilityti-scale，isquitesensitivetotherateofassdecreaseofthesunwhentheassofthesunisclosetoitspresentvae，thejovianpsreastableover1010yr，orperhapslongerduncan≈ap;ap;ap;lissaueralsoperfordfoursiirexperintsontheorbitalotionofsevenps(ventoneptune)，whichveraspanof～109yrtheirexperintsonthesevenpsarenotyetprehensive，butitseesthattheterrestrialpsalsoreastabledurgthetegrationperiod，atagalostregurosciltions

ontheotherhand，hisauratesei-analyticalsecurperturbationtheory(skar1988)，skarfdsthatrgeandirregurvariationscanappeartheeentricitiesandclationsoftheterrestrialps，especiallyofrcuryandarsonati-scaleofseveral109yr(skar1996)theresultsofskar&039;ssecurperturbationtheoryshouldbenfirdandvestigatedbyfullynuricaltegrations

thispaperwepresentpreliaryresultsofsixlong-ternuricaltegrationsonallneparyorbits，vergaspanofseveral109yr，andofoothertegrationsvergaspanof±5x1010yrthetotalepsedtiforalltegrationsisorethan5yr，gseveraldedicatedpcsandworkstationsoneofthefundantalncsionsofourlong-tertegrationsisthatsorsysteparyotionseestobestabletersofthehillstabilityntionedabove，atleastoverati-spanof±4gyractually，ournuricaltegrationsthesystewasfarorestablethanwhatisdefedbythehillstabilitycriterion:notonlydidnocloseenunterhappendurgthetegrationperiod，butalsoalltheparyorbitalelentshavebeennfedanarrowregionbothtiandfrequencydoa，thoughparyotionsarestochasticscethepurposeofthispaperistoexhibitandoverviewtheresultsofourlong-ternuricaltegrations，weshowtypicalexaplefiguresasevidenceoftheverylong-terstabilityofsorsysteparyotionforreaderswhohaveorespecificanddeeperterestsournuricalresults，wehavepreparedawebpage(aess)，whereweshowraworbitalelents，theirlow-passfilteredresults，variationofdeunayelentsandangurontudeficit，andresultsofoursipleti–frequencyanalysisonallofourtegrations

section2webrieflyexpourdynaicalodel，nuricalthodanditialnditionsedourtegrationssection3isdevotedtoadescriptionofthequickresultsofthenuricaltegrationsverylong-terstabilityofsorsysteparyotionisapparentbothparypositionsandorbitalelentsaroughestiationofnuricalerrorsisalsogivensection4goesontoadiscsionofthelongest-tervariationofparyorbitsgalow-passfilterandcdesadiscsionofangurontudeficitsection5，wepresentasetofnuricaltegrationsfortheouterfivepsthatspans±5x1010yrsection6wealsodiscsthelong-terstabilityoftheparyotionanditspossiblecae

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–holansyplecticapasourategrationthod(wisdo≈ap;ap;ap;holan1991;koshita，yoshida≈ap;ap;ap;nakai1991)withaspecialstart-upproceduretoreducethetruncationerrorofanglevariables，‘warstart’(saha≈ap;ap;ap;treae1992，1994)

thestepsizeforthenuricaltegrationsis8dthroughoutalltegrationsoftheneps(n±1，2，3)，whichisabout1/11oftheorbitalperiodofthenerostp(rcury)asforthedeterationofstepsize，wepartlyfollowtheprevionuricaltegrationofallnepsssan≈ap;ap;ap;wisdo(1988，72d)andsaha≈ap;ap;ap;treae(1994，225/32d)weroundedthedecialpartofthetheirstepsizesto8toakethestepsizeaultipleof2ordertoreducetheauutionofround-offerrortheputationprocessesretiontothis，wisdo≈ap;ap;ap;holan(1991)perfordnuricaltegrationsoftheouterfiveparyorbitsgthesyplecticapwithastepsizeof400d，1/1083oftheorbitalperiodofjupitertheirresultseestobeaurateenough，whichpartlyjtifiesourthodofdetergthestepsizehowever，scetheeentricityofjupiter(～005)isuchsallerthanthatofrcury(～02)，weneedsocarewhenweparethesetegrationssiplytersofstepsizes

thetegrationoftheouterfiveps(f±)，wefixedthestepsizeat400d

weadoptgas&039;fandgfunctionsthesyplecticaptogetherwiththethird-orderhalleythod(danby1992)asasolverforkeplerequationsthenuberofaxiuiterationswesethalley&039;sthodis15，buttheyneverreachedtheaxiuanyofourtegrations

thetervalofthedataoutputis200000d(～547yr)forthecalcutionsofallneps(n±1，2，3)，andabout8000000d(～21903yr)forthetegrationoftheouterfiveps(f±)

althoughnooutputfiltergwasdonewhenthenuricaltegrationswereprocess，weappliedalow-passfiltertotheraworbitaldataafterwehadpletedallthecalcutionsseesection41fororedetail

24errorestiation

241retiveerrorstotalenergyandangurontu

aordgtooneofthebasicpropertiesofsyplectictegrators，whichnservethephysicallynservativequantitieswell(totalorbitalenergyandangurontu)，ourlong-ternuricaltegrationsseetohavebeenperfordwithverysallerrorstheaveragedretiveerrorsoftotalenergy(～10?9)andoftotalangurontu(～10?11)havereaednearlynstantthroughoutthetegrationperiod(fig1)thespecialstartupprocedure，warstart，wouldhavereducedtheaveragedretiveerrortotalenergybyaboutoneorderofagnitudeorore

retivenuricalerrorofthetotalangurontuδa/a0andthetotalenergyδe/e0ournuricaltegrationsn±1，2，3，whereδeandδaaretheabsotechangeofthetotalenergyandtotalangurontu，respectively，ande0anda0aretheiritialvaesthehorizontalunitisgyr

notethatdifferentoperatgsystes，differentatheaticallibraries，anddifferenthardwarearchitecturesresultdifferentnuricalerrors，throughthevariationsround-offerrorhandlgandnuricalalgorithstheupperpaneloffig1，wecanregnizethissituationthesecurnuricalerrorthetotalangurontu，whichshouldberigorolypreserveduptoache-eprecision

242errorparylongitudes

scethesyplecticapspreservetotalenergyandtotalangurontuofn-bodydynaicalsystesherentlywell，thedegreeoftheirpreservationaynotbeagoodasureoftheauracyofnuricaltegrations，especiallyasaasureofthepositionalerrorofps，ietheerrorparylongitudestoestiatethenuricalerrortheparylongitudes，weperfordthefollogproceduresweparedtheresultofouralong-tertegrationswithsotesttegrations，whichspanuchshorterperiodsbutwithuchhigherauracythantheategrationsforthispurpose，weperfordauchoreauratetegrationwithastepsizeof0125d(1/64oftheategrations)spanng3x105yr，startgwiththesaitialnditionsasthen?1tegrationwensiderthatthistesttegrationprovideswitha‘pseudo-true’sotionofparyorbitalevotionnext，weparethetesttegrationwiththeategration，n?1fortheperiodof3x105yr，weseeadifferenceananoaliesoftheearthbeeentheotegrationsof～052°(thecaseofthen?1tegration)thisdifferencecanbeextrapotedtothevae～8700°，about25rotationsofearthafter5gyr，scetheerroroflongitudescreaseslearlywithtithesyplecticapsiirly，thelongitudeerrorofptocanbeestiatedas～12°thisvaeforptoisuchbetterthantheresultkoshita≈ap;ap;ap;nakai(1996)wherethedifferenceisestiatedas～60°

3nuricalresults–ignceattherawdata

thissectionwebrieflyreviewthelong-terstabilityofparyorbitalotionthroughsosnapshotsofrawnuricaldatatheorbitalotionofpsdicateslong-terstabilityallofournuricaltegrations:noorbitalcrossgsnorcloseenuntersbeeenanypairofpstookpce

31generaldescriptionofthestabilityofparyorbits

first，webrieflylookatthegeneralcharacterofthelong-terstabilityofparyorbitsourterestherefocesparticurlyonthenerfourterrestrialpsforwhichtheorbitalti-scalesareuchshorterthanthoseoftheouterfivepsaswecanseeclearlyfrothepnarorbitalnfigurationsshownfigs2and3，orbitalpositionsoftheterrestrialpsdifferlittlebeeentheitialandfalpartofeachnuricaltegration，whichspansseveralgyrthesolidlesdenotgthepresentorbitsofthepsliealostwiththeswarofdotseventhefalpartoftegrations(b)and(d)thisdicatesthatthroughouttheentiretegrationperiodthealostregurvariationsofparyorbitalotionreanearlythesaastheyareatpresent

verticalviewofthefournerparyorbits(frothez-axisdirection)attheitialandfalpartsofthetegrationsn±1theaxesunitsareauthexy-pneissettothevariantpneofsorsystetotalangurontu(a)theitialpartofn+1(t=0to00547x109yr)(b)thefalpartofn+1(t=49339x108to49886x109yr)(c)theitialpartofn?1(t=0to?00547x109yr)(d)thefalpartofn?1(t=?39180x109to?39727x109yr)eachpanel，atotalof23684potsareplottedwithantervalofabout2190yrover547x107yrsolidleseachpaneldenotethepresentorbitsofthefourterrestrialps(takenfrode245)