2.4Errorestimation
2.4.1Relativeerrorsintotalenergyandangularmomentum
Accordingtooneofthebasicpropertiesofsymplecticintegrators，whichconservethephysicallyconservativequantitieswell(totalorbitalenergyandangularmomentum)，ourlong-termnumericalintegrationsseemtohavebeenperformedwithverysmallerrors.Theaveragedrelativeerrorsoftotalenergy(∼10−9)andoftotalangularmomentum(∼10−11)haveremainednearlyconstantthroughouttheintegrationperiod(Fig.1).Thespecialstartupprocedure，warmstart，wouldhavereducedtheaveragedrelativeerrorintotalenergybyaboutoneorderofmagnitudeormore.
RelativenumericalerrorofthetotalangularmomentumδAA0andthetotalenergyδEE0inournumericalintegrationsN±1，2，3，whereδEandδAaretheabsolutechangeofthetotalenergyandtotalangularmomentum，respectively，andE0andA0aretheirinitialvalues.ThehorizontalunitisGyr.
Notethatdifferentoperatingsystems，differentmathematicallibraries，anddifferenthardwarearchitecturesresultindifferentnumericalerrors，throughthevariationsinround-offerrorhandlingandnumericalalgorithms.IntheupperpanelofFig.1，wecanrecognizethissituationinthesecularnumericalerrorinthetotalangularmomentum，whichshouldberigorouslypreserveduptomachine-εprecision.
2.4.2Errorinplanetarylongitudes
SincethesymplecticmapspreservetotalenergyandtotalangularmomentumofN-bodydynamicalsystemsinherentlywell，thedegreeoftheirpreservationmaynotbeagoodmeasureoftheaccuracyofnumericalintegrations，especiallyasameasureofthepositionalerrorofplanets，i.e.theerrorinplanetarylongitudes.Toestimatethenumericalerrorintheplanetarylongitudes，weperformedthefollowingprocedures.Wecomparedtheresultofourmainlong-termintegrationswithsometestintegrations，whichspanmuchshorterperiodsbutwithmuchhigheraccuracythanthemainintegrations.Forthispurpose，weperformedamuchmoreaccurateintegrationwithastepsizeof0.125d(164ofthemainintegrations)spanning3×105yr，startingwiththesameinitialconditionsasintheN−1integration.Weconsiderthatthistestintegrationprovidesuswitha‘pseudo-true’solutionofplanetaryorbitalevolution.Next，wecomparethetestintegrationwiththemainintegration，N−1.Fortheperiodof3×105yr，weseeadifferenceinmeananomaliesoftheEarthbetweenthetwointegrationsof∼0.52°(inthecaseoftheN−1integration).Thisdifferencecanbeextrapolatedtothevalue∼8700°，about25rotationsofEarthafter5Gyr，sincetheerroroflongitudesincreaseslinearlywithtimeinthesymplecticmap.Similarly，thelongitudeerrorofPlutocanbeestimatedas∼12°.ThisvalueforPlutoismuchbetterthantheresultinKinoshita&&Nakai(1996)wherethedifferenceisestimatedas∼60°.
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