ionsofallninelas(±1，2，3)，andabout8000000d(～21903yr)fortheintegrationoftheouterfivelas(±).
lthoughnooututfilteringasdonehenthenumericalintegrationsereinrocess，ealiedalo-assfiltertotheraorbitaldataafterehadletedallthecalculations.eeection4.1formoredetail.
2.4rrorestimation
2.4.1elativeerrorsintotalenergyandangularmomentum
ordingtooneofthebasicroertiesofsymlecticintegrators，hichconservethehysicallyconservativequantitiesell(totalorbitalenergyandangularmomentum)，ourlong-termnumericalintegrationsseemtohavebeenerformedithverysmallerrors.heaveragedrelativeerrorsoftotalenergy(～10?9)andoftotalangularmomentum(～10?11)haveremainednearlyconstantthroughouttheintegrationeriod(ig.1).hesecialstarturocedure，armstart，ouldhavereducedtheaveragedrelativeerrorintotalenergybyaboutoneorderofmagnitudeormore.
elativenumericalerrorofthetotalangularmomentumδ0andthetotalenergyδ0inournumericalintegrations±1，2，3，hereδandδaretheabsolutechangeofthetotalenergyandtotalangularmomentum，resectively，and0and0aretheirinitialvalues.hehorizontalunitisyr.
otethatdifferentoeratingsystems，differentmathematicallibraries，anddifferenthardarearchitecturesresultindifferentnumericalerrors，throughthevariationsinround-offerrorhandlingandnumericalalgorithms.ntheueranelofig.1，ecanrecognizethissituationinthesecularnumericalerrorinthetotalangularmomentum，hichshouldberigorouslyreservedutomachine-recision.
2.4.2rrorinlaarylongitudes
incethesymlecticmasreservetotalenergyandtotalangularmomentumof-bodydynamicalsystemsinherentlyell，thedegreeoftheirreservationm
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