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longternbintegrationsandstabilityplaryorbitsoursolarsystebr>
abstract
wepresenttheresultsverylongternbnuricalintegrationsplaryorbitaltionsover109tispansincludingallnininspectionournuricaldatashowsthattheplarytion，leastoursiledynacaldel，seebequitestableevenoverthisverylonlookthelowestfrequencyoscillationsusinowpassfiltershowsthepotentiallydiffusivecharacterterrestrialplarytion，especiallythaehaviourtheentricitymercurrintegrationsqualitativelysilartheresultsfronbjacqueslaskar&039;ssecularperturbationtheorye.g.0.35over，thereareapparentsecularincreasesentricityinclinationanyorbitalelentsthepls，whichyrevealedstilllonrternbnuricahavealsoperforoupletrialintegrationsincludingtionstheouterfiveplsoverthedurationof51010yr.theresultindicatesthatthethreejorresonancestheneptune–plutosystenbhavebeenintainedoverthe1011yrtispan.
1introduction
1.1definitiontheproblebr>
thequestionthestabilitrsolarsystenbhasbeendebatedoverseveralhundredyears，sincetheerroblenbhasattractedfaustheticiansovertheyearsandhasplayeentralrolethedevelopntnonlineardynacsandchao，donotyethavefiniteanswerthequestionwhetheroursolarsystenbisstableopartlesultthefactthatthedefinitiontheternbstabilityvaguewhenisusedrelationtheproblenbofpla
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rytionthesolarsysteactuallyisnoteasygivlear，rigorousandphysicallyaningfuldefinitionthestabilitrsolarsyste
angdefinitionsstability，hereadoptthehilldefinitiongladn1993:actuallythisnoefinitionstability，buefinystenbasbingunstablewheloseencounterurssowherethesystestartingfronbacertaininitialconfigurationchaers，wetherill&boss1996;ito&tanikaw;isdefinedexperiencinloseencounterwhentwobodiesapproachoneanotherwithinareathelarrhilthesystenbisdefinedbeinstatethatourplarysystenbisdynacallystablenocloseencounterhappensduringtheaoursolarsysteabout，thisdefinitionyreplacedonewhichurrenceanyorbitalcrossingbetweeneitherapairplstakebecauseknowfronbexperiencethatorbitalcrossingverylikelyleadacloseencounterplaryandprlarysysteyoshinaga，kokubo&makincoursethisstatentcannotsilyappliedsystewithstableorbitalresonancessuchtheneptune–plutosyste
1.2previousstudiesandofthisresearch
inadditionthevaguenesstheconceptstability，theplsoursolarsystenbshoharactertypicaldynacalchaossussn&wisdonb1988，causethischaoticbehaviournowpartlyunderstoodbeinesultresonanceoverlappingmurray&holn1999;lecar，franklin&hol，wouldrequireintegratingoverenseleplarysysteincludingallnineplsfoeriodcoveringseveralgyrthoroughlyunderstandthelongternbevolutionplaryorbits，sincechaoticdynacalsystearecharacterizedtheirstrongdependenceinitialconditions.
fronbthatpointview，ofthepreviou
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slongternbnuricalintegrationsincludedonlytheouterfiveplssussn&wisdonb1988;kinoshita&nakabecausetheorbitalperiodstheouterplsarechlonrthanthosetheinnerfourplsthatiseasierfollowthesystenbfoivenintegratiopresent，thelonstnuricalintegrationspublishedjournalsarethoseduncan&lissauetheirtartwastheeffectpostinsequencesolarlossthestabilityplaryorbits，theyperfordintegrationscoveringto1011oftheorbitaltionsthefourjoviainitialorbitalelentsandssesplsaretheasthoseoursolarsystenbinduncan&lissauer&039;spaper，buttheydecreasetheofthesungraduallytheirnuricabecausetheyconsidertheeffectpostinsequencesolarlossth，theyfoundthatthecrossingtiscaleplaryorbits，whichcanatypicalindicatortheinstabilitytiscale，quitesensitivetheratessdecreaseththeofthesuncloseitspresentvalue，thejovianplsreinstableover1010yr，perhap&lissaueralsoperfordfoursilarexperintstheorbitaltionsevenplsvenusneptune，whichcovepan109yr.theirexperintsthesevenplsarenotyet&nbspprehensive，butseethattheterrestrialplsalsoreinstableduringtheintegrationperiod，intainingalstregularoscillations.
ontheotherhand，hisurateseanalyticalsecularperturbationtheorylaskar1988，laskarfindsthatlarandirregularvariationscanappeartheentricitiesandinclinationstheterrestrialpls，especiallymercuryandmarsatiscaleseveral109laskaresultslaskar&039;ssecularperturbationtheoryshouldconfirdandinvestigatedfullynurical
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integrations.
inthispaperpresentprelinaryresultssixlongternbnuricalintegrationsallnineplaryorbits，coverinpanseveral109yr，andtwootherintegrationscoverinpanof51010yr.thetotalelapsedforallintegrationsrethan5yr，usingseveraldedicatedpcsofthefundantalconclusionsourlongternbintegrationsthatsolarsystenbplarytionseebestableterthehillstabilityntionedabove，leastoveispanof，ournuricalintegrationsthesystenbwasfarstablethanwhatdefinedthehillstabilitycriterion:notonlydidcloseencounterhappenduringtheintegrationperiod，butalsoalltheplaryorbitalelentshavebeenconfinedanarrowregionbothtiandfrequencydoin，thoughplarytionsthepurposethispapertoexhibitandoverviewtheresultsourlongternbnuricalintegrations，showtypicalexalefiguresevidencetheverylongternbstabilitysolarsystenbplarreaderswhohavespecificanddeeperinterestsournuricalresults，haveprepareebpaess，whereshowraworbitalelents，theirlowpassfilteredresults，variationdelaunayelentsandangularntunbdeficit，andresultsoursileti–frequencyanalysisallourintegrations.
insection2brieflyexplainourdynacaldel，nuricalthodandinitialconditionsusedou3devotedadescriptionthequickresultsthenuricalongternbstabilitysolarsystenbplarytionapparentbothplarypositionsandorbitaestitionnuricalerrorsals4goestiscussionthelonstternbvariationplaryorbitsusinowpassfilterandincludeiscussionangularntunbsection5，presenetnuricalintegrationsfortheouterfiveplsthatspans51010yr.s
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ection6alsodiscussthelongternbstabilitytheplarytionanditspossiblecause.
2descriptionthenuricalintegrations
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。
2.3nuricalthod
weutilizecondorderwisdoholnsylecticpourintegrationthodwisdonb&holn1991;kinoshita，yoshida&nakai1991witpecialstartupprocedurereducethetruncationerroranglevariables，warnbstartsaha&treine1992，1994.
thestepsizeforthenuricalintegrations8dthroughoutallintegrationsthenineplsn1，2，3，whichabout111theorbitalperiodtheinnersorthedeternationstepsize，partlyfollowthepreviousnuricalintegrationallnineplssussn&wisdonb1988，7.2dandsaha&treine1994，22532d.roundedthedecilpartthetheirstepsizes8kethestepsiztiple2orderreducetheulationroundofferrorthe&nbspputatiorelationthis，wisdonb&holn1991perfordnuricalintegrationstheouterfiveplaryorbitsusingthesylectiittepsize400d，110.83theorbitalperioesultseebeurateenough，whichpartlyjustifiesourthoddeterningth，sincetheentricityjupiter0.05chsllerthanthatmercury0.2，needcarewhenwe&nbspparetheseintegrationssilyterstepsizes.
intheintegrationtheouterfiveplsf，fixedthestepsize400d.
weadoptgauss&039;fanunctionsthesylectiotherwiththethirdorderhalleythoddanby1992asolverforkeplenuerxinbiterationssethalley&039;sthod15，buttheyneverreachedthexinbinanrintegrations.
theintervalthedataoutput200000d547for
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thecalculationsallnineplsn1，2，3，andabout8000000d903fortheintegrationtheouterfiveplsf.
althoughoutputfilteringwasdonewhenthenuricalintegrationswereprocess，applieowpassfiltertheraworbitaldataafterhad&nbsppletedallsection4.1fordetail.
2.4errorestition
2.4.1relativeerrorstotalenergyandangularntubr>
ordingonethebasicpropertiessylecticintegrators，whichconservethephysicallyconservativequantitieswelltotalorbitalenergyandangularntuourlongternbnuricalintegrationsseenbtohavebeenperfordwithveryaveradrelativeerrorstotalenergy109andtotalangularntunb1011havereinednearlyconstantthroughouttheintegrationperiospecialstartupprocedure，warnbstart，wouldhavereducedtheaveradrelativeerrortotalenergyaboutoneordergnitudere.
relativenuricalerrorthetotalangularntunbδaa0andthetotalenergyδee0ournuricalintegrationsn1，2，3，whereandaretheabsolutechanthetotalenergyandtotalangularnturespectively，ande0anda0aretheirinitiahorizontalunitgyr.
notethatdifferentoperatingsyste，differenttheticallibraries，anddifferenthardwarearchitecturesresultdifferentnuricalerrors，throughthevariationsroundofferrorhandlingandnuricatheupperpanelo，canrecognizethissituationthesecularnuricalerrorthetotalangularntuwhichshouldrigorouslypreservedtochineeprecision.
2.4.2errorplarylongitudes
sincethesylecticpreservetotalenergyandtotalangularntunbofnbodydynacalsysteinherentlywell，thedegre
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etheirpreservatiootagoodasuretheuracynuricalintegrations，especiallyaasurethepositionalerrorpls，i.e.theerrorplarestitethenuricalerrortheplarylongitudes，perfordthefollowin&nbspparedtheresultourlongternbintegrationswithtestintegrations，whichspanshorterperiodsbutwithhigheruracythanthhispurpose，perfordareurateintegrationwithad164theintegrationsspanning3105yr，startingwiththeinitialconditionsinthonsiderthatthistestintegrationprovideswitseudotruesolutionplaryorbita，we&nbspparethetestintegrationwiththeintegration，n1.fortheperiod3105yr，seifferenceananoliestheearthbetweenthetwothecasethifferencecanextrapolatedthevalue8700，aboutrotationsearthafter5gyr，sincetheerrorlongitudesincreaseslinearlywithinthesylecticp.silarly，thelongitudeerrorplutocanestited12.thisvalueforplutochbetterthantheresultkinoshita&nakai1996wherethedifferenceestited60.
3nuricalresults–i.glancetherawdata
inthissectionbrieflyreviewthelongternbstabilityplaryorbitaltionthroughsnapshotsrawnuricaorbitaltionplsindicateslongternbstabilityallournuricalintegrations:orbitalcrossingsnorcloseencountersbetweenanypairplstookplace.
3.1neraldescriptionthestabilityplaryorbits
first，brieflylooktheneralcharacterthelongternbstabilityplarinterestherefocusesparticularlytheinnerfourterrestrialplsforwhichtheorbitaltiscalesareshorterthanthosetheouterfivcanseeclearlyfronbtheplanarorbitalconfigurationsshownfig
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s2and3，orbitalpositionstheterrestrialplsdifferlittlebetweentheinitialandfinalparteachnuricalintegration，whichspansseverasolidlinesdenotingthepresentorbitstheplsliealstwithintheswarnbofdotseventhefinalpartintegrationndd.thisindicatesthatthroughouttheentireintegrationperiodthealstregularvariationsplaryorbitaltionreinnearlytheastheyarepresent.
verticalviewthefourinnerplaryorbitsfronbthxisdirectiontheinitialandfinalpartsthaxesunitsareau.theplanesettheinvariantplanesolarsystenbtotalangularntuatheinitialpart9yr.bthefinalpart84.9886109yr.ctheinitialpartn1t00.0547109yr.dthefinalpartofn1t3.91801093.9727109yr.eachpanel，atotal23684pointsareplottedwithintervalabout2190over5.47107.solidlineseachpaneldenotethepresentorbitsthefourterrestrialplstakenfronbde245.
thevariationentricitiesandorbitalinclinationsfortheinnerfourplstheinitialandfinalparttheintegrationisshowxpected，thecharacterthevariationplaryorbitalelentsdoesnotdiffersignificantlybetweentheinitialandfinalparteachintegration，leastforvenus，earthelentsmercury，especiallyitsentricity，seenbtochanasignificanpartlybecausetheorbitaltiscaletheistheshortestallthepls，whichleadsarapidorbitalevolutionthanotherpls;theinnerstynearesesultappearsbesoagreentwithlaskar&039;s1994，1996expectationsthatlarandirregularvariationsappeartheentricitiesandinclinationsmercuryatiscaleseveral109yr.however，theeffectthepossibl