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对火星轨道变化问题的最后解释 (第6/14页)
A/A0andthetotalenergyδE/E0inournumericalintegrationsN±1,2,3,whereδEandδAaretheabsolutechangeofthetotalenergyandtotalangularmomentum,respectively,andE0andA0aretheirinitialvalues.ThehorizontalunitisGyr. Notethatdifferentoperatingsystems,differentmathematicallibraries,anddifferenthardwarearchitecturesresultindifferentnumericalerrors,throughthevariationsinround-offerrorhandlingandnumericalalgorithms.IntheupperpanelofFig.1,wecanrecognizethissituationinthesecularnumericalerrorinthetotalangularmomentum,whichshouldberigorouslypreserveduptomachine-εprecision. 2.4.2Errorinplanetarylongitudes SincethesymplecticmapspreservetotalenergyandtotalangularmomentumofN-bodydynamicalsystemsinherentlywell,thedegreeoftheirpreservationmaynotbeagoodmeasureoftheaccuracyofnumericalintegrations,especiallyasameasureofthepositionalerrorofplanets,i.e.theerrorinplanetarylongitudes.Toestimatethenumericalerrorintheplanetarylongitudes,weperformedthefollowingprocedures.Wecomparedtheresultofourmainlong-termintegrationswithsometestintegrations,whichspanmuchshorterperiodsbutwithmuchhigheraccuracythanthemainintegrations.Forthispurpose,weperformedamuchmoreaccurateintegrationwithastepsizeof0.125d(1/64ofthemainintegrations)spanning3×105yr,startingwiththesameinitialconditionsasintheN?1integration.Weconsiderthatthistestintegrationprovidesuswitha‘pseudo-true’solutionofplanetaryorbitalevolution.Next,wecomparethetestintegrationwiththemainintegration,N?1.Fortheperiodof3×105yr,weseeadifferenceinmeananomaliesoftheEarthbetweenthetwointegrationsof~0.52°(inthecaseoftheN?1integration).Thisdifferencecanbeextrapolatedtothevalue~8700°,about25rotationsofEarthafter5Gyr,sincetheerroroflongitudesincreaseslinearlywithtimeinthesymplecticmap.Similarly,thelongitudeerrorofPlutocanbeestimatedas~12°.ThisvalueforPlutoismuchbetterthantheresultinKinoshita&Nakai(1996)wherethedifferenceisestimatedas~60°. 3Numericalresult