第76章 对火星轨道变化问题的最后解释(4/10)

ize.however，siheetricityofjupiter(～0.05)ismuchsmallerthanthatofmercury(～0.2)，weneedsomecarewhenweparetheseiionssimplyintermsofstepsizes.

iegrationoftheouterfiveplas(f±)，wefixedthestepsizeat400d.

tgauss'fandgfunsinthesymplecticmaptogetherwiththethird-orderhalleymethod(danby1992)asasolverforkeplerequations.thenumberofmaximumiteratioinhalley'smethodis15，buttheyneverreachedthemaximuminanyofouriions.

theintervalofthedataoutputis200000d(～547yr)forthecalculationsofallnineplas(n±1，2，3)，andabout8000000d(～21903yr)fortheiionoftheouterfiveplas(f±).

althoughnooutputfilteringwasdohenumeritegrationswereinprocess，liedalow-passfiltertotheraworbitaldataafterwehadpletedallthecalculations.seese4.1formoredetail.

2.4errorestimatio

2.4.1relativeerrorsintotalenergyandangularmomentum

acctoohebasicpropertiesofsymplectitegrators，whiservethephysicallyservativequantitieswell(totalorbitalenergyandangularmomentum)，-termnumeritegratioohavebeenperformedwithverysmallerrors.theaveragedrelativeerrorsoftotalenergy(～10?9)andoftotalangularmomentum(～10?11)haveremainednearlystantthroughouttheiionperiod(fig.1).thespecialstartupprocedure，warmstart，wouldhavereducedtheaveragedrelativeerrorintotalenergybyaboutoneorderofmagnitudeormore.

relativenumericalerrorofthetotalangularmomentumδaa0aalenergyδee0inournumeritegrationsn±1，2，3，whereδeandδaaretheabsolutegeofthetotalenergyandtotalangularmomentum，respectively，ande0anda0aretheirinitialvalues.thehorizontalunitisgyr.