对火星轨道变化问题的最后解释（2/4）

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-ε

sincethesymplecticmapspreservetotalenergyandtotalangularmomentumofn-bodydynamicalsystemsinherentlywell，thedegreeoftheirpreservationmaynotbeagoodmeasureoftheaccuracyofnumericalintegrations，especiallyasameasureofthepositionalerrorofplanets，，-termintegrationswithsometestintegrations，，(1/64ofthemainintegrations)spanning3×105yr，startingwiththesameinitialconditionsasinthen?‘pseudo-true’，wecomparethetestintegrationwiththemainintegration，n?×105yr，weseeadifferenceinmeananomaliesoftheearthbetweenthetwointegrationsof～°(inthecaseofthen?1integration).thisdifferencecanbeextrapolatedtothevalue～8700°，about25rotationsofearthafter5gyr，，thelongitudeerrorofplutocanbeestimatedas～12°.thisvalueforplutoismuchbetterthantheresultinkinoshita&nakai(1996)wherethedifferenceisestimatedas～60°.

3numericalresults–

inthissectionwebrieflyreviewthelong--termstabilityinallofournumericalintegrations:

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

verticalviewofthefourinnerplanetaryorbits(fromthez-axisdirection)attheinitialandfinalpartsoftheintegrationsn±-(a)theinitialpartofn+1(t=×109yr).(b)thefinalpartofn+1(t=××109yr).(c)theinitialpartofn?1(t=0to?×109yr).(d)thefinalpartofn?1(t=?×109to?×109yr).ineachpanel，×(takenfromde245).