第76章 对火星轨道变化问题的最后解释(6/14)
chine-eprecision.
2.4.2errorinplanetarylongitudes
sincethesymplecticmapspreservetotalenergyandtotalangularmomentumofn-bodydynamicalsystemsinherentlywell，thedegreeoftheirpreservationmaynotbeagoodmeasureoftheaccuracyofnumericalintegrations，especiallyasameasureofthepositionalerrorofplanets，i.e.theerrorinplanetarylongitudes.toestimatethenumericalerrorintheplanetarylongitudes，weperformedthefollowingprocedures.wecomparedtheresultofourmainlong-termintegrationswithsometestintegrations，whichspanmuchshorterperiodsbutwithmuchhigheraccuracythanthemainintegrations.forthispurpose，weperformedamuchmoreaccurateintegrationwithastepsizeof0.125d(164ofthemainintegrations)spanning3x105yr，startingwiththesameinitialconditionsasinthen?1integration.weconsiderthatthistestintegrationprovidesuswitha‘pseudo-true’solutionofplanetaryorbitalevolution.next，wecomparethetestintegrationwiththemainintegration，n?1.fortheperiodof3x105yr，weseeadifferenceinmeananomaliesoftheearthbetweenthetwointegrationsof～0.52°(inthecaseofthen?1integration).thisdifferencecanbeextrapolatedtothevalue～8700°，about25rotationsofearthafter5gyr，sincetheerroroflongitudesincreaseslinearlywithtimeinthesymplecticmap.similarly，thelongitudeerrorofplutocanbeestimatedas～12°.thisvalueforplutoismuchbetterthantheresultinkinoshitanakai(1996)wherethedifferenceisestimatedas～60°.
3numericalresults–i.glanceattherawdata
inthissectionwebrieflyreviewthelong-termstabilityofplanetaryorbitalmotionthroughsomesnapshotsofrawnumericaldata.theorbitalmotionofplanetsindicateslong-termstabilityinallofournumericalintegrations:noorbitalcrossingsnorcloseencountersbetweenanypairofplanetstookplace.
3.1generaldescriptionofthestabilityofplanetaryorbits
first，webrieflylookatthegeneralcharacterofthelong-termstabilityofplanetaryorbits.ourinterestherefocusesparticularlyontheinnerfourterrestrialplanetsforwhichtheorbitaltime-scalesaremuchshorterthanthoseoftheouterfiveplanets.aswecanseeclearlyfromtheplanarorbitalconfigurationsshowninfigs2and3，o
2.4.2errorinplanetarylongitudes
sincethesymplecticmapspreservetotalenergyandtotalangularmomentumofn-bodydynamicalsystemsinherentlywell，thedegreeoftheirpreservationmaynotbeagoodmeasureoftheaccuracyofnumericalintegrations，especiallyasameasureofthepositionalerrorofplanets，i.e.theerrorinplanetarylongitudes.toestimatethenumericalerrorintheplanetarylongitudes，weperformedthefollowingprocedures.wecomparedtheresultofourmainlong-termintegrationswithsometestintegrations，whichspanmuchshorterperiodsbutwithmuchhigheraccuracythanthemainintegrations.forthispurpose，weperformedamuchmoreaccurateintegrationwithastepsizeof0.125d(164ofthemainintegrations)spanning3x105yr，startingwiththesameinitialconditionsasinthen?1integration.weconsiderthatthistestintegrationprovidesuswitha‘pseudo-true’solutionofplanetaryorbitalevolution.next，wecomparethetestintegrationwiththemainintegration，n?1.fortheperiodof3x105yr，weseeadifferenceinmeananomaliesoftheearthbetweenthetwointegrationsof～0.52°(inthecaseofthen?1integration).thisdifferencecanbeextrapolatedtothevalue～8700°，about25rotationsofearthafter5gyr，sincetheerroroflongitudesincreaseslinearlywithtimeinthesymplecticmap.similarly，thelongitudeerrorofplutocanbeestimatedas～12°.thisvalueforplutoismuchbetterthantheresultinkinoshitanakai(1996)wherethedifferenceisestimatedas～60°.
3numericalresults–i.glanceattherawdata
inthissectionwebrieflyreviewthelong-termstabilityofplanetaryorbitalmotionthroughsomesnapshotsofrawnumericaldata.theorbitalmotionofplanetsindicateslong-termstabilityinallofournumericalintegrations:noorbitalcrossingsnorcloseencountersbetweenanypairofplanetstookplace.
3.1generaldescriptionofthestabilityofplanetaryorbits
first，webrieflylookatthegeneralcharacterofthelong-termstabilityofplanetaryorbits.ourinterestherefocusesparticularlyontheinnerfourterrestrialplanetsforwhichtheorbitaltime-scalesaremuchshorterthanthoseoftheouterfiveplanets.aswecanseeclearlyfromtheplanarorbitalconfigurationsshowninfigs2and3，o
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