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long-tertegrationsandstabilityofparyorbitsoursorsyste
abstract
&nbsepresenttheresultsofverylong-ternuricaltegrationsofparyorbitalotionsover109-yrti-spanscludgallnepsaquickspectionofournuricaldatashowsthattheparyotion，atleastoursipledynaicalodel，seestobequitestableevenoverthisverylongti-spanacloserlookatthelowest-frequencyosciltionsgalow-passfiltershowsthepotentiallydiffivecharacterofterrestrialparyotion，especiallythatofrcurythebehaviouroftheeentricityofrcuryourtegrationsisqualitativelysiirtotheresultsfrojacquesskar'ssecurperturbationtheory(egeax～035over～±4gyr)however，therearenoapparentsecurcreasesofeentricityorclationanyorbitalelentsoftheps，&nbshichayberevealedbystilllonger-ternuricaltegrations&nbsehavealsoperfordaupleoftrialtegrationscludgotionsoftheouterfivepsoverthedurationof±5x1010yrtheresultdicatesthatthethreeajorresonancestheneptune–plutosystehavebeenataedoverthe1011-yrti-span
1troduction
11defitionoftheproble
thequestionofthestabilityofoursorsystehasbeendebatedoverseveralhundredyears，scetheeraofnewtontheproblehasattractedanyfaoatheaticiansovertheyearsandhaspyedacentralrolethedevelopntofnon-leardynaicsandchaostheoryhowever，&nbsedonotyethaveadefiteanswertothequestionof&nbshetheroursorsysteisstableornotthisispartlyaresultofthefactthatthedefitionoftheter‘stability’isvague&nbshenitisedretiontotheprobleofparyotionthesorsysteactuallyitisnoteasytogiveaclear，rigoroandphysicallyangfuldefitionofthestabilityofoursorsyste
aonganydefitionsofstability，here&nbseadoptthehilldefition(gdan1993):actuallythisisnotadefitionofstability，butofstability&nbsedefeasysteasbegunstable&nbshenacloseenunterourssowherethesyste，startgfroacertaitialnfiguration(chabers，&nbsetherill&apapboss1996ito&apaptanikawa1999)asysteisdefedasexperiencgacloseenunter&nbshenobodiesapproachoneanother&nbsithanareaofthergerhillradiotherwisethesysteisdefedasbegstablehenceforward&nbsestatethatourparysysteisdynaicallystableifnocloseenunterhappensdurgtheageofoursorsyste，about±5gyrcidentally，thisdefitionayberepcedbyone&nbshichanourrenceofanyorbitalcrossgbeeeneitherofapairofpstakespcethisisbecae&nbseknowfroexperiencethatanorbitalcrossgisverylikelytoleadtoacloseenunterparyandproarysystes(yoshaga，kokubo&apapako1999)ofursethisstatentcannotbesiplyappliedtosystes&nbsithstableorbitalresonancessuchastheneptune–plutosyste
12previostudiesandaisofthisresearch
additiontothevaguenessofthenceptofstability，thepsoursorsysteshowacharactertypicalofdynaicalchaos(ssan&apap&nbsisdo1988，1992)thecaeofthischaoticbehaviourisnowpartlyunderstoodasbegaresultofresonanceoverppg(urray&apapholan1999lecar，frankl&apapholan2001)however，it&nbsouldrequiretegratgoveranensebleofparysystescludgallnepsforaperiodvergseveral10gyrtothoroughlyunderstandthelong-terevolutionofparyorbits，scechaoticdynaicalsystesarecharacterizedbytheirstrongdependenceonitialnditions
frothatpotofview，anyofthepreviolong-ternuricaltegrationscludedonlytheouterfiveps(ssan&apap&nbsisdo1988koshita&apapnakai1996)thisisbecaetheorbitalperiodsoftheouterpsaresouchlongerthanthoseofthenerfourpsthatitisucheasiertofollowthesysteforagiventegrationperiodatpresent，thelongestnuricaltegrationspublishedjournalsarethoseofduncan&apaplissauer(1998)althoughtheiratarget&nbsastheeffectofpost-a-sequencesorasslossonthestabilityofparyorbits，theyperfordanytegrationsvergupto～1011yroftheorbitalotionsofthefourjovianpstheitialorbitalelentsandassesofpsarethesaasthoseofoursorsysteduncan&apaplissauer'spaper，buttheydecreasetheassofthesungraduallytheirnuricalexperintsthisisbecaetheynsidertheeffectofpost-a-sequencesorasslossthepapernsequently，theyfoundthatthecrossgti-scaleofparyorbits，&nbshichcanbeatypicaldicatorofthestabilityti-scale，isquitesensitivetotherateofassdecreaseofthesun&nbshentheassofthesunisclosetoitspresentvalue，thejovianpsreastableover1010yr，orperhapslongerduncan&apaplissaueralsoperfordfoursiirexperintsontheorbitalotionofsevenps(ventoneptune)，&nbshichveraspanof～109yrtheirexperintsonthesevenpsarenotyetprehensive，butitseesthattheterrestrialpsalsoreastabledurgthetegrationperiod，atagalostregurosciltions
ontheotherhand，hisauratesei-analyticalsecurperturbationtheory(skar1988)，skarfdsthatrgeandirregurvariationscanappeartheeentricitiesandclationsoftheterrestrialps，especiallyofrcuryandarsonati-scaleofseveral109yr(skar1996)theresultsofskar'ssecurperturbationtheoryshouldbenfirdandvestigatedbyfullynuricaltegrations
thispaper&nbsepresentpreliaryresultsofsixlong-ternuricaltegrationsonallneparyorbits，vergaspanofseveral109yr，andofoothertegrationsvergaspanof±5x1010yrthetotalepsedtiforalltegrationsisorethan5yr，gseveraldedicatedpcsand&nbsorkstationsoneofthefundantalnclionsofourlong-tertegrationsisthatsorsysteparyotionseestobestabletersofthehillstabilityntionedabove，atleastoverati-spanof±4gyractually，ournuricaltegrationsthesyste&nbsasfarorestablethan&nbshatisdefedbythehillstabilitycriterion:notonlydidnocloseenunterhappendurgthetegrationperiod，butalsoalltheparyorbitalelentshavebeennfedanarrowregionbothtiandfrequencydoa，thoughparyotionsarestochasticscethepurposeofthispaperistoexhibitandoverviewtheresultsofourlong-ternuricaltegrations，&nbseshowtypicalexaplefiguresasevidenceoftheverylong-terstabilityofsorsysteparyotionforreaders&nbshohaveorespecificanddeeperterestsournuricalresults，&nbsehaveprepareda&nbsebpage(aess)，&nbshere&nbseshowraworbitalelents，theirlow-passfilteredresults，variationofdeunayelentsandangurontudeficit，andresultsofoursipleti–frequencyanalysisonallofourtegrations
section2&nbsebrieflyexpourdynaicalodel，nuricalthodanditialnditionsedourtegrationssection3isdevotedtoadescriptionofthequickresultsofthenuricaltegrationsverylong-terstabilityofsorsysteparyotionisapparentbothparypositionsandorbitalelentsaroughestiationofnuricalerrorsisalsogivensection4goesontoadiscsionofthelongest-tervariationofparyorbitsgalow-passfilterandcludesadiscsionofangurontudeficitsection5，&nbsepresentasetofnuricaltegrationsfortheouterfivepsthatspans±5x1010yrsection6&nbsealsodiscsthelong-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
&nbseutilizeasend-order&nbsisdo–holansyplecticapasourategrationthod(wisdo&apapholan1991koshita，yoshida&apapnakai1991)&nbsithaspecialstart-upproceduretoreducethetruncationerrorofanglevariables，‘warstart’(saha&apaptreae1992，1994)
thestepsizeforthenuricaltegrationsis8dthroughoutalltegrationsoftheneps(n±1，2，3)，&nbshichisabout1/11oftheorbitalperiodofthenerostp(rcury)asforthedeterationofstepsize，&nbsepartlyfollowtheprevionuricaltegrationofallnepsssan&apap&nbsisdo(1988，72d)andsaha&apaptreae(1994，225/32d)&nbseroundedthedecialpartofthetheirstepsizesto8toakethestepsizeaultipleof2ordertoreducetheauutionofround-offerrortheputationprocessesretiontothis，&nbsisdo&apapholan(1991)perfordnuricaltegrationsoftheouterfiveparyorbitsgthesyplecticap&nbsithastepsizeof400d，1/1083oftheorbitalperiodofjupitertheirresultseestobeaurateenough，&nbshichpartlyjtifiesourthodofdetergthestepsizehowever，scetheeentricityofjupiter(～005)isuchsallerthanthatofrcury(～02)，&nbseneedsocare&nbshen&nbseparethesetegrationssiplytersofstepsizes
thetegrationoftheouterfiveps(f±)，&nbsefixedthestepsizeat400d
&nbseadoptgas'fandgfunctionsthesyplecticaptogether&nbsiththethird-orderhalleythod(danby1992)asasolverforkeplerequationsthenuberofaxiuiterations&nbsesethalley'sthodis15，buttheyneverreachedtheaxiuanyofourtegrations
thetervalofthedataoutputis200000d(～547yr)forthecalcutionsofallneps(n±1，2，3)，andabout8000000d(～21903yr)forthetegrationoftheouterfiveps(f±)
althoughnooutputfilterg&nbsasdone&nbshenthenuricaltegrations&nbsereprocess，&nbseappliedalow-passfiltertotheraworbitaldataafter&nbsehadpletedallthecalcutionsseesection41fororedetail
24errorestiation
241retiveerrorstotalenergyandangurontu
aordgtooneofthebasicpropertiesofsyplectictegrators，&nbshichnservethephysicallynservativequantities&nbsell(totalorbitalenergyandangurontu)，ourlong-ternuricaltegrationsseetohavebeenperford&nbsithverysallerrorstheaveragedretiveerrorsoftotalenergy(～10?9)andoftotalangurontu(～10?11)havereaednearlynstantthroughoutthetegrationperiod(fig1)thespecialstartupprocedure，&nbsarstart，&nbsouldhavereducedtheaveragedretiveerrortotalenergybyaboutoneorderofagnitudeorore
retivenuricalerrorofthetotalangurontuδa/a0andthetotalenergyδe/e0ournuricaltegrationsn±1，2，3，&nbshereδeandδaaretheabsolutechangeofthetotalenergyandtotalangurontu，respectively，ande0anda0aretheiritialvaluesthehorizontalunitisgyr
notethatdifferentoperatgsystes，differentatheaticallibraries，anddifferenthardwarearchitecturesresultdifferentnuricalerrors，throughthevariationsround-offerrorhandlgandnuricalalgorithstheupperpaneloffig1，&nbsecanregnizethissituationthesecurnuricalerrorthetotalangurontu，&nbshichshouldberigorolypreserveduptoache-eprecision
242errorparylongitudes
scethesyplecticapspreservetotalenergyandtotalangurontuofn-bodydynaicalsystesherently&nbsell，thedegreeoftheirpreservationaynotbeagoodasureoftheauracyofnuricaltegrations，especiallyasaasureofthepositionalerrorofps，ietheerrorparylongitudestoestiatethenuricalerrortheparylongitudes，&nbseperfordthefollogprocedures&nbseparedtheresultofouralong-tertegrations&nbsithsotesttegrations，&nbshichspanuchshorterperiodsbut&nbsithuchhigherauracythantheategrationsforthispurpose，&nbseperfordauchoreauratetegration&nbsithastepsizeof0125d(1/64oftheategrations)spanng3x105yr，startg&nbsiththesaitialnditionsasthen?1tegration&nbsensiderthatthistesttegrationprovides&nbsitha‘pseudo-true’solutionofparyorbitalevolutionnext，&nbseparethetesttegration&nbsiththeategration，n?1fortheperiodof3x105yr，&nbseseeadifferenceananoaliesoftheearthbeeentheotegrationsof～052°(thecaseofthen?1tegration)thisdifferencecanbeextrapotedtothevalue～8700°，about25rotationsofearthafter5gyr，scetheerroroflongitudescreaseslearly&nbsithtithesyplecticapsiirly，thelongitudeerrorofplutocanbeestiatedas～12°thisvalueforplutoisuchbetterthantheresultkoshita&apapnakai(1996)&nbsherethedifferenceisestiatedas～60°
3nuricalresults–ignceattherawdata
thissection&nbsebrieflyreviewthelong-terstabilityofparyorbitalotionthroughsosnapshotsofrawnuricaldatatheorbitalotionofpsdicateslong-terstabilityallofournuricaltegrations:noorbitalcrossgsnorcloseenuntersbeeenanypairofpstookpce
31generaldescriptionofthestabilityofparyorbits
first，&nbsebrieflylookatthegeneralcharacterofthelong-terstabilityofparyorbitsourterestherefocesparticurlyonthenerfourterrestrialpsfor&nbshichtheorbitalti-scalesareuchshorterthanthoseoftheouterfivepsas&nbsecanseeclearlyfrothepnarorbitalnfigurationsshownfigs2and3，orbitalpositionsoftheterrestrialpsdifferlittlebeeentheitialandfalpartofeachnuricaltegration，&nbshichspansseveralgyrthesolidlesdenotgthepresentorbitsofthepsliealost&nbsiththeswarofdotseventhefalpartoftegrations(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，atotalof23684potsareplotted&nbsithantervalofabout2190yrover547x107yrsolidleseachpaneldenotethepresentorbitsofthefourterrestrialps(takenfrode245)
thevariationofeentricitiesandorbitalclationsforthenerfourpstheitialandfalpartofthetegrationn+1isshownfig4asex