第76章 对火星轨道变化问题的最后解释 (第5/12页)

otalenergyandangularmomentum

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

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

    differeingsystems，differentmathematicallibraries，anddifferenthardwarearchitecturesresultindifferentnumericalerrors，throughthevariationsinround-offerrorhandlingandnumericalalgorithms.intheupperpaneloffig.1，wereizethissituationinthesecularnumericalerrorialangularmomentum，whichshouldberigorouslypreserveduptomae-eprecision.

    2.4.2errorinplaarylongitudes

    sihesymplecticmapspreservetotalenergyandtotalangularmomentumofn-bodydynamicalsystemsilywell，thedegreeoftheirpreservationmaynotbeagoodmeasureoftheaccuraeritegrations，especiallyasameasureofthepositionalerrorofplas，i.e.theerrorinplaarylongitudes.toestimatethenumericalerrorintheplaarylongitudes，weperformedthefollowingprocedures.weparedtheresultofourmainlong-termiionswithsometestiions，whichspanmuchshorterperiodsbutwithmuchhigheraccuracythanthemaiions.forthispurpose，weperformedamuchmoreaccurateiionsizeof0.125d(164ofthemaiions)spanning3x105yr，startingwiththesameinitialditionsasinthen?1iion.wesiderthatthistestiionprovidesusseudo-true’solutionofplaaryorbitalevolutio，weparethetestiionwiththemaiion，n?1.fortheperiodof3x105yr，weseeadifferenmeananomaliesoftheearthbetweewoiionsof～0.52°(inthecaseofthen?1iion).thisdifferenbeextrapolatedtothevalue～8700°，about25rotatiohafter5gyr