thetegrationoftheouterfiveplas（f&psn;），wefixedthestepsizeat400d

weadoptgas&39;fandgfunctionsthesyplecticaptotherwiththethird-orderhalleythod（danby1992）asalverforkeplereationsthenuberofaxiuiterationswesethalley&39;sthodis15，buttheyneverreachedtheaxiuanyofourtegrations

thetervalofthedataoutputis200000d（&si;547yr）forthecalculationsofallneplas（n&psn;1，2，3），andabout8000000d（&si;21903yr）forthetegrationoftheouterfiveplas（f&psn;）

althoughnooutputfiltergwasdonewhenthenuricaltegrationswereprocess，weappliedalow-passfiltertotheraworbitaldataafterwehadpletedallthecalculationsseesection41fororedetail

24errorestiation

241relativeerrorstotalenergyandangularontu

aordgtooneofthebasicpropertiesofsyplectictegrators，whichnservethephysicallynservativeantitieswell（totalorbitalenergyandangularontu），ourlong-ternuricaltegrationsseetohavebeenperfordwithverysallerrorstheaveradrelativeerrorsoftotalenergy（&si;10&;9）andoftotalangularontu（&si;10&;11）havereaednearlynstantthroughoutthetegrationperiod（fig1）thespecialstartupprocedure，warstart，wouldhavereducedtheaveradrelativeerrortotalenergybyaboutoneorderofagnitudeorore

relativenuricalerrorofthetotalangularontuδaa0andthetotalenergyδee0ournuricaltegrationsn&psn;1，2，3，whereδeandδaaretheabte插nofthetotalenergyandtotalangularontu，respectively，ande0anda0aretheiritialvaesthehorizontalunitisgyr

notethatdifferentoperatgsystes，differentatheaticallibraries，anddifferenthardwarearchitecturesresultdifferentnuricalerrors，throughthevariationsround-offerrorhandlgandnuricalalgorithstheupperpaneloffig1，wecanregnizethissituationthesecularnuricalerrorthetotalangularontu，whichshouldberigorolypreserveduptoache-εprecision

242errorplaarylongitudes

scethesyplecticapspreservetotalenergyandtotalangularontuofn-bodydynaicalsystesherentlywell，thedegreeoftheirpreservationaynotbeagoodasureoftheauracyofnuricaltegrations，especiallyasaasureofthepositionalerrorofplas，ietheerrorplaarylongitudestoestiatethenuricalerrortheplaarylongitudes，weperfordthefollogproceduresweparedtheresultofouralong-tertegrationswithtesttegrations，whichspanuchshorterperiodsbutwithuchhigherauracythantheategrationsforthispurpose，weperfordauchoreauratetegrationwithastepsizeof0125d（164oftheategrations）spanng3&tis;105yr，startgwiththesaitialnditionsasthen&;1tegrationwensiderthatthistesttegrationprovideswitha&lso;pseudo-true&rso;tionofplaaryorbitalevotionnext，weparethetesttegrationwiththeategration，n&;1fortheperiodof3&tis;105yr，weseeadifferenceananoaliesoftheearthbeeentheotegrationsof&si;052°（thecaseofthen&;1tegration）thisdifferencecanbeextrapolatedtothevae&si;8700°，about25rotationsofearthafter5gyr，scetheerroroflongitudescreaseslearlywithtithesyplecticapsiilarly，thelongitudeerrorofptocanbeestiatedas&si;12°thisvaeforptoisuchbetterthantheresultkoshita≈ap;nakai（1996）wherethedifferenceisestiatedas&si;60°

3nuricalresults–iglanceattherawdata

thissectionwebrieflyreviewthelong-terstabilityofplaaryorbitalotionthroughsnapshotsofrawnuricaldatatheorbitalotionofplasdicateslong-terstabilityallofournuricaltegrations:noorbitalcrossgsnorcloseenuntersbeeenanypairofplastookplace

31neraldescriptionofthestabilityofplaaryorbits

first，webrieflylookattheneral插racterofthelong-terstabilityofplaaryorbitsourterestherefocesparticularlyonthenerfourterrestrialplasforwhichtheorbitalti-scalesareuchshorterthanthoseoftheouterfiveplasaswecanseeclearlyfrotheplanarorbitalnfigurationsshownfigs2and3，orbitalpositionsoftheterrestrialplasdifferlittlebeeentheitialandfalpartofeachnuricaltegration，whichspansseveralgyrthelidlesdenotgthepresentorbitsoftheplasliealostwiththeswarofdotseventhefalpartoftegrations（b）and（d）thisdicatesthatthroughouttheentiretegrationperiodthealostregularvariationsofplaaryorbitalotionreanearlythesaastheyareatpresent

verticalviewofthefournerplaaryorbits（frothez-axisdirection）attheitialandfalpartsofthetegrationsn&psn;1theaxesunitsareauthexy-planeissettothevariantplaneoflarsystetotalangularontu（a）theitialpartofn+1（t=0to00547&tis;109yr）（b）thefalpartofn+1（t=49339&tis;108to49886&tis;109yr）（c）theitialpartofn&;1（t=0to&;00547&tis;109yr）（d）thefalpartofn&;1（t=&;39180&tis;109to&;39727&tis;109yr）eachpanel，atotalof23684potsareplottedwithantervalofabout2190yrover547&tis;107yrlidleseachpaneldenotethepresentorbitsofthefourterrestrialplas（takenfrode245）