对火星轨道变化问题的最后解释（2/4）

（本部分涉及比较复杂的积分计算，作者君就不贴上来了，贴上来了起点也不一定能成功显示。）

weutilizeasecond-orderwisdom–holmansymplecticmapasourmainintegrationmethod(wisdom&holman1991;kinoshita，yoshida&nakai1991)withaspecialstart-upproceduretoreducethetruncationerrorofanglevariables，‘warmstart’(saha&tremaine1992，1994).

thestepsizeforthenumericalintegrationsis8dthroughoutallintegrationsofthenineplanets(n±1，2，3)，whichisabout1/11oftheorbitalperiodoftheinnermostplanet(mercury).asforthedeterminationofstepsize，wepartlyfollowthepreviousnumericalintegrationofallnineplanetsinsussman&wisdom(1988，)andsaha&tremaine(1994，225/32d).weroundedthedecimalpartofthetheirstepsizesto8tomakethestepsizeamultipleof2inordertoreducetheaccumulationofround-，wisdom&holman(1991)performednumericalintegrationsoftheouterfiveplanetaryorbitsusingthesymplecticmapwithastepsizeof400d，1/，，sincetheeccentricityofjupiter(∼)ismuchsmallerthanthatofmercury(∼)，

intheintegrationoftheouterfiveplanets(f±)，

weadoptgauss'fandgfunctionsinthesymplecticmaptogetherwiththethird-orderhalleymethod(danby1992)'smethodis15，

theintervalofthedataoutputis200000d(∼547yr)forthecalculationsofallnineplanets(n±1，2，3)，andabout8000000d(∼21903yr)fortheintegrationoftheouterfiveplanets(f±).

althoughnooutputfilteringwasdonewhenthenumericalintegrationswereinprocess，weappliedalow-

accordingtooneofthebasicpropertiesofsymplecticintegrators，whichconservethephysicallyconservativequantitieswell(totalorbitalenergyandangularmomentum)，ourlong-(∼10−9)andoftotalangularmomentum(∼10−11)haveremainednearlyconstantthroughouttheintegrationperiod().thespecialstartupprocedure，warmstart，

relativenumericalerrorofthetotalangularmomentumδa/a0andthetotalenergyδe/e0inournumericalintegrationsn±1，2，3，whereδeandδaaretheabsolutechangeofthetotalenergyandtotalangularmomentum，respectively，

notethatdifferentoperatingsystems，differentmathematicallibraries，anddifferenthardwarearchitecturesresultindifferentnumericalerrors，throughthevariationsinround-，wecanrecognizethissituationinthesecularnumericalerrorinthetotalangularmomentum，whichshouldberigorouslypreserveduptomachine-

sincethesymplecticmapspreservetotalenergyandtotalangularmomentumofn-bodydynamicalsystemsinherentlywell，thedegreeoftheirpreservationmaynotbeagoodmeasureoftheaccuracyofnumericalintegrations，especiallyasameasureofthepositionalerrorofplanets，，-termintegrationswithsometestintegrations，，(1/64ofthemainintegrations)spanning3x105yr，startingwiththesameinitialconditionsasinthen−‘pseudo-true’，wecomparethetestintegrationwiththemainintegration，n−，weseeadifferenceinmeananomaliesoftheearthbetweenthetwointegrationsof∼°(inthecaseofthen−1integration).thisdifferencecanbeextrapolatedtothevalue∼8700°，about25rotationsofearthafter5gyr，，thelongitudeerrorofplutocanbeestimatedas∼12°.thisvalueforplutoismuchbetterthantheresultinkinoshita&nakai(1996)wherethedifferenceisestimatedas∼60°.

3numericalresults–

inthissectionwebrieflyreviewthelong--termstabilityinallofournumericalintegrations:

first，webrieflylookatthegeneralcharacterofthelong--，orbitalpositionsoftheterrestrialplanetsdifferlittlebetweentheinitialandfinalpartofeachnumericalintegration，(b)and(d).