对火星轨道变化问题的最后解释 (第5/14页)

Sussman&Wisdom(1988，7.2d)andSaha&Tremaine(1994，225/32d).Weroundedthedecimalpartofthetheirstepsizesto8tomakethestepsizeamultipleof2inordertoreducetheaccumulationofround-offerrorinthecomputationprocesses.Inrelationtothis，Wisdom&Holman(1991)performednumericalintegrationsoftheouterfiveplanetaryorbitsusingthesymplecticmapwithastepsizeof400d，1/10.83oftheorbitalperiodofJupiter.Theirresultseemstobeaccurateenough，whichpartlyjustifiesourmethodofdeterminingthestepsize.However，sincetheeccentricityofJupiter(～0.05)ismuchsmallerthanthatofMercury(～0.2)，weneedsomecarewhenwecomparetheseintegrationssimplyintermsofstepsizes.

    Intheintegrationoftheouterfiveplanets(F±)，wefixedthestepsizeat400d.

    WeadoptGauss'fandgfunctionsinthesymplecticmaptogetherwiththethird-orderHalleymethod(Danby1992)asasolverforKeplerequations.ThenumberofmaximumiterationswesetinHalley'smethodis15，buttheyneverreachedthemaximuminanyofourintegrations.

    Theintervalofthedataoutputis200000d(～547yr)forthecalculationsofallnineplanets(N±1，2，3)，andabout8000000d(～21903yr)fortheintegrationoftheouterfiveplanets(F±).

    Althoughnooutputfilteringwasdonewhenthenumericalintegrationswereinprocess，weappliedalow-passfiltertotheraworbitaldataafterwehadcompletedallthecalculations.SeeSection4.1formoredetail.

    2.4Errorestimation

    2.4.1Relativeerrorsintotalenergyandangularmomentum

    Accordingtooneofthebasicpropertiesofsymplecticintegrators，whichconservethephysicallyconservativequantitieswell(totalorbitalenergyandangularmomentum)，ourlong-termnumericalintegrationsseemtohavebeenperformedwithverysmallerrors.Theaveragedrelativeerrorsoftotalenergy(～10?9)andoftotalangularmomentum(～10?11)haveremainednearlyconstantthroughouttheintegrationperiod(Fig.1).Thespecialstartupprocedure，warmstart，wouldhavereducedtheaveragedrelativeerrorintotalenergybyaboutoneorderofmagnitudeormore.

    Relativenumericalerrorofthetotalangularmomentumδ