对火星轨道变化问题的最后解释(第2/3页)
accordingtooneofthebasicpropertiesofsymplecticintegrators，whichconservethephysicallyconservativequantitieswell(totalorbitalenergyandangularmomentum)，ourlong-termnumericalintegrationsseemtohavebeenperformedwithverysmallerrors.theaveragedrelativeerrorsoftotalenergy(～10?9)andoftotalangularmomentum(～10?11)haveremainednearlyconstantthroughouttheintegrationperiod(fig.1).thespecialstartupprocedure，warmstart，wouldhavereducedtheaveragedrelativeerrorintotalenergybyaboutoneorderofmagnitudeormore.
relativenumericalerrorofthetotalangularmomentumδa/a0andthetotalenergyδe/e0inournumericalintegrationsn±1，2，3，whereδeandδaaretheabsolutechangeofthetotalenergyandtotalangularmomentum，respectively，ande0anda0aretheirinitialvalues.thehorizontalunitisgyr.
notethatdifferentoperatingsystems，differentmathematicallibraries，anddifferenthardwarearchitecturesresultindifferentnumericalerrors，throughthevariationsinround-offerrorhandlingandnumericalalgorithms.intheupperpaneloffig.1，wecanrecognizethissituationinthesecularnumericalerrorinthetotalangularmomentum，whichshouldberigorouslypreserveduptomachine-eprecision.
2.4.2errorinplanetarylongitudes
sincethesymplecticmapspreservetotalenergyandtotalangularmomentumofn-bodydynamicalsystemsinherentlywell，thedegreeoftheirpreservationmaynotbeagoodmeasureoftheaccuracyofnumericalintegrations，especiallyasameasureofthepositionalerrorofplanets，i.e.theerrorinplanetarylongitudes.toestimatethenumericalerrorintheplanetarylongitudes，weperformedthefollowingprocedures.wparedtheresultofourmainlong-termintegrationswithsometestintegrations，whichspanmuchshorterperiodsbutwithmuchhigheraccuracythanthemainintegrations.forthispurpose，weperformedamuchmoreaccurateintegrationwithastepsizeof0.125d(1/64ofthemainintegrations)spanning3x105yr，startingwiththesameinitialconditionsasinthen?1integration.weconsiderthatthistestintegrationprovidesuswitha‘pseudo-true’solutionofplanetaryorbitalevolution.next，wparethetestintegrationwiththemainintegration，n?1.fortheperiodof3x105yr，weseeadifferenceinmeananomaliesoftheearthbetweenthetwointegrationsof～0.52°(inthecaseofthen?1integration).thisdifferencecanbeextrapolatedtothevalue～8700°，about25rotationsofearthafter5gyr，sincetheerroroflongitudesincreaseslinearlywithtimeinthesymplecticmap.similarly，thelongitudeerrorofplutocanbeestimatedas～12°.thisvalueforplutoismuchbetterthantheresultinkinoshita&nakai(1996)wherethedifferenceisestimatedas～60°.
3numericalresults–i.glanceattherawdata
inthissectionwebrieflyreviewthelong-termstabilityofplanetaryorbitalmotionthroughsomesnapshotsofrawnumericaldata.theorbitalmotionofplanetsindicateslong-termstabilityinallofournumericalintegrations:noorbitalcrossingsnorcloseencountersbetweenanypairofplanetstookplace.
3.1generaldescriptionofthestabilityofplanetaryorbits
first，webrieflylookatthegeneralcharacterofthelong-termstabilityofplanetaryorbits.ourinterestherefocusesparticularlyontheinnerfourterrestrialplanetsforwhichtheorbitaltime-scalesaremuchshorterthanthoseoftheouterfiveplanets.aswecanseeclearlyfromtheplanarorbitalconfigurationsshowninfigs2and3，orbitalpositionsoftheterrestrialplanetsdifferlittlebetweentheinitialandfinalpartofeachnumericalintegration，whichspansseveralgyr.thesolidlinesdenotingthepresentorbitsoftheplanetsliealmostwithintheswarmofdotseveninthefinalpartofintegrations(b)and(d).thisindicatesthatthroughouttheentireintegrationperiodthealmostregularvariationsofplanetaryorbitalmotionremainnearlythesameastheyareatpresent.
verticalviewofthefourinnerplanetaryorbits(fromthez-axisdirection)attheinitialandfinalpartsoftheintegrationsn±1.theaxesunitsareau.thexy-planeissettotheinvariantplaneofsolarsystemtotalangularmomentum.(a)theinitialpartofn+1(t=0to0.0547x109yr).(b)thefinalpartofn+1(t=4.9339x108to4.9886x109yr).(c)theinitialpartofn?1(t=0to?0.0547x109yr).(d)thefinalpartofn?1(t=?3.9180x109to?3.9727x109yr).ineachpanel，atotalof23684pointsareplottedwithanintervalofabout2190yrover5.47x107yr.solidlinesineachpaneldenotethepresentorbitsofthefourterrestrialplanets(takenfromde245).
thevariationofeccentricitiesandorbitalinclinationsfortheinnerfourplanetsintheinitialandfinalpartoftheintegrationn+1isshowninfig.4.asexpected，thecharacterofthevariationofplanetaryorbitalelementsdoesnotdiffersignificantlybetweentheinitialandfinalpartofeachintegration，atleastforvenus，earthandmars.theelementsofmercury，especiallyitseccentricity，seemtochangetoasignificantextent.thisispartlybecausetheorbitaltime-scaleoftheplanetistheshortestofalltheplanets，whichleadstoamorerapidorbitalevolutionthanotherplanets;theinnermostplanetmaybenearesttoinstability.thisresultappearstobeinsomeagreementwithlaskar's(1994，1996)expectationsthatlargeandirregularvariationsappearintheeccentricitiesandinclinationsofmercuryonatime-scaleofseveral109yr.however，theeffectofthepossibleinstabilityoftheorbitofmercurymaynotfatallyaffecttheglobalstabilityofthewholeplanetarysystemowingtothesmallmassofmercury.wewillmentionbrieflythelong-termorbitalevolutionofmercurylaterinsection4usinglow-passfilteredorbitalelements.
theorbitalmotionoftheouterfiveplanetsseemsrigorouslystableandquiteregularoverthistime-span(seealsosection5).
3.2time–frequencymaps
althoughtheplanetarymotionexhibitsverylong-termstabilitydefinedasthenon-existenceofcloseencounterevents，thechaoticnatureofplanetarydynamicscanchangetheoscillatoryperiodandamplitudeofplanetaryorbitalmotiongraduallyoversuchlongtime-spans.evensuchslightfluctuationsoforbitalvariationinthefrequencydomain，particularlyinthecaseofearth，canpotentiallyhaveasignificanteffectonitssurfaceclimatesystemthroughsolarinsolationvariation(cf.berger1988).
togiveanoverviewofthelong-termchangeinperiodicityinplanetaryorbitalmotion，weperformedmanyfastfouriertransformations(ffts)alongthetimeaxis，andsuperposedtheresultingperiodgramstodrawtwo-dimensionaltime–frequencymaps.thespecificapproachtodrawingthesetime–frequencymapsinthispaperisverysimple–muchsimplerthanthewaveletanalysisorlaskar's(1990，1993)frequencyanalysis.
dividethelow-passfilteredorbitaldataintomanyfragmentsofthesamelength.thelengthofeachdatasegmentshouldbeamultipleof2inordertoapplythefft.
eachfragmentofthedatahasalargeoverlappingpart:forexample，whentheithdatabeginsfromt=tiandendsatt=ti+t，thenextdatasegmentrangesfromti+δt≤ti+δt+t，whereδt?t.wecontinuethisdivisionuntilwereachacertainnumbernbywhichtn+treachesthetotalintegrationlength.
weapplyanffttoeachofthedatafragments，andobtainnfrequencydiagrams.
ineachfrequencydiagramobtainedabove，thestrengthofperiodicitycanbereplacedbyagrey-scale(orcolour)chart.
weperformthereplacement，andconnectallthegrey-scale(orcolour)chartsintoonegraphforeachintegration.thehorizontalaxisofthesenewgraphsshouldbethetime，i.e.thestartingtimesofeachfragmentofdata(ti，wherei=1，…，n).theverticalaxisrepresentstheperiod(orfrequency)oftheoscillationoforbitalelements.
wehaveadoptedanfftbecauseofitsoverwhelmingspeed，sincetheamountofnumericaldatatobedposedintofrequencponentsisterriblyhuge(severaltensofgbytes).
atypicalexampleofthetime–frequencymapcreatedbytheaboveproceduresisshowninagrey-scalediagramasfig.5，whichshowsthevariationofperiodicityintheeccentricityandinclinationofearthinn+2integration.infig.5，thedarkareashowsthatatthetimeindicatedbythevalueontheabscissa，theperiodicityindicatedbytheordinateisstrongerthaninthelighterareaaroundit.wecanrecognizefromthismapthattheperiodicityoftheeccentricityandinclinationofearthonlychangesslightlyovertheentireperiodcoveredbythen+2integration.thisnearlyregulartrendisqualitativelythesameinotherintegrationsandforotherplanets，althoughtypicalfrequenciesdifferplanetbyplanetandelementbyelement.
4.2long-termexchangeoforbitalenergyandangularmomentum
wecalculateverylong-periodicvariationandexchangeofplanetaryorbitalenergyandangularmomentumusingfiltereddelaunayelementsl，g，h.gandhareequivalenttotheplanetaryorbitalangularmomentumanditsverticaponentperunitmass.lisrelatedtotheplanetaryorbitalenergyeperunitmassase=?μ2/2l2.ifthesystemipletelylinear，theorbitalenergyandtheangularmomentumineachfrequencybinmustbeconstant.non-linearityintheplanetarysystemcancauseanexchangeofenergyandangularmomentuminthefrequencydomain.theamplitudeofthelowest-frequencyoscillationshouldincreaseifthesystemisunstableandbreaksdowngradually.however，suchasymptomofinstabilityisnotprominentinourlong-termintegrations.
infig.7，thetotalorbitalenergyandangularmomentumofthefourinnerplanetsandallnineplanetsareshownforintegrationn+2.theupperthreepanelsshowthelong-periodicvariationoftotalenergy(denotedase-e0)，totalangularmomentum(g-g0)，andtheverticaponent(h-h0)oftheinnerfourplanetscalculatedfromthelow-passfiltereddelaunayelements.e0，g0，h0denotetheinitialvaluesofeachquantity.theabsolutedifferencefromtheinitialvaluesisplottedinthepanels.thelowerthreepanelsineachfigureshowe-e0，g-g0andh-h0ofthetotalofnineplanets.thefluctuationshowninthelowerpanelsisvirtuallyentirelyaresultofthemassivejovianplanets.
comparingthevariationsofenergyandangularmomentumoftheinnerfourplanetsandallnineplanets，itisapparentthattheamplitudesofthoseoftheinnerplanetsaremuchsmallerthanthoseofallnineplanets:theamplitudesoftheouterfiveplanetsaremuchlargerthanthoseoftheinnerplanets.thisdoesnotmeanthattheinnerterrestrialplanetarysubsystemismorestablethantheouterone:thisissimplyaresultoftherelativesmallnessofthemassesofthefourterrestrialplanetparedwiththoseoftheouterjovianplanets.anotherthingwenoticeisthattheinnerplanetarysubsystemmaybeunstablemorerapidlythantheouteronebecauseofitsshorterorbitaltime-scales.thiscanbeseeninthepanelsdenotedasinner4infig.7wherethelonger-periodicandirregularoscillationsaremoreapparentthaninthepanelsdenotedastotal9.actually，thefluctuationsintheinner4panelsaretoalargeextentasaresultoftheorbitalvariationofthemercury.however，wecannotneglectthecontributionfromotherterrestrialplanets，aswewillseeinsubsequentsections.
4.4long-termcouplingofseveralneighbouringplanetpairs
letusseesomeindividualvariationsofplanetaryorbitalenergyandangularmomentumexpressedbythelow-passfiltereddelaunayelements.figs10and11showlong-termevolutionoftheorbitalenergyofeachplanetandtheangularmomentuminn+1andn?2integrations.wenoticethatsomeplanetsformapparentpairsintermsoforbitalenergyandangularmomentumexchange.inparticular，venusandearthmakeatypicalpair.inthefigures，theyshownegativecorrelationsinexchangeofenergyandpositivecorrelationsinexchangeofangularmomentum.thenegativecorrelationinexchangeoforbitalenergymeansthatthetwoplanetsformacloseddynamicalsystemintermsoftheorbitalenergy.thepositivecorrelationinexchangeofangularmomentummeansthatthetwoplanetsaresimultaneouslyundercertainlong-termperturbations.candidatesforperturbersarejupiterandsaturn.alsoinfig.11，wecanseethatmarsshowsapositivecorrelationintheangularmomentumvariationtothevenus–earthsystem.mercuryexhibitscertainnegativecorrelationsintheangularmomentumversusthevenus–earthsystem，whichseemstobeareactioncausedbytheconservationofangularmomentumintheterrestrialplanetarysubsystem.
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