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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)