（本章未完，请点击下一页继续阅读）

Dividethelow-lengthofeachdatasegmentshouldbeamultipleof2inordertoapplytheFFT.

Eachfragmentofthedatahasalargeoverlappingpart:forexample，whentheithdatabeginsfromt=tiandendsatt=ti+T，thenextdatasegmentrangesfromti+δT≤ti+δT+T，whereδontinuethisdivisionuntilwereachacertainnumberNbywhichtn+Treachesthetotalintegrationlength.

WeapplyanFFTtoeachofthedatafragments，andobtainnfrequencydiagrams.

2.4Errorestimation

2.4.1Relativeerrorsintotalenergyandangularmomentum

Accordingtooneofthebasicpropertiesofsymplecticintegrators，whichconservethephysicallyconservativequantitieswell(totalorbitalenergyandangularmomentum)，ourlong-termaveragedrelativeerrorsoftotalenergy(?10?9)andoftotalangularmomentum(?10?11)haveremainednearlyconstantthroughouttheintegrationperiod(Fig.1).Thespecialstartupprocedure，warmstart，wouldhavereducedtheaveragedrelativeerrorintotalenergybyaboutoneorderofmagnitudeormore.

Verticalviewofthefourinnerplanetaryorbits(fromthez-axisdirection)attheinitialandfinalpartsoftheintegrationsN±xy-planeissettotheinvariantplaneofSolarsystemtotalangularmomentum.(a)TheinitialpartofN+1(t=).(b)ThefinalpartofN+1(t=4.).(c)TheinitialpartofN?1(t=0to?0.0547x109yr).(d)ThefinalpartofN?1(t=?3.9180x109to?3.9727x109yr).Ineachpanel，planets(takenfromDE245).

ThevariationofeccentricitiesandorbitalinclinationsfortheinnerfourplanetsintheinitialandfinalpartoftheintegrationN+，thecharacterofthevariationofplanetaryorbitalelementsdoesnotdiffersignificantlybetweentheinitialandfinalpartofeachintegration，atleastforVenus，elementsofMercury，especiallyitseccentricity，sispartlybecausetheorbitaltime-scaleoftheplanetistheshortestofalltheplanets，whichleadstoamorerapidorbitalesresultappearstobeinsomeagreementwithLaskar‘s(1994，1996)expectationsthatlargeandirregularvariationsappearintheeccentricitiesandinclinationsofMercuryonatime-ever，theeffectofthepossibleinstabilityoftheorbitofMercurymaynotfatallyaffecttheglobaillmentionbrieflythelong-termorbitalevolutionofMercurylaterinSection4usinglow-passfilteredorbitalelements.

Theorbitalmotionoftheouterfiveplanetsseemsrigorouslystableandquiteregularoverthistime-span(seealsoSection5).

RelativenumericalerrorofthetotalangularmomentumδA/A0andthetotalenergyδE/E0inournumericalintegrationsN±1，2，3，whereδEandδAaretheabsolutechangeofthetotalenergyandtotalangularmomentum，respectively，horizontalunitisGyr.

Notethatdifferentoperatingsystems，differentmathematicallibraries，anddifferenthardwarearchitecturesresultindifferentnumericalerrors，，wecanrecognizethissituationinthesecularnumericalerrorinthetotalangularmomentum，whichshouldberigorouslypreserveduptomachine-eprecision.

2.4.2Errorinplanetarylongitudes

SincethesymplecticmapspreservetotalenergyandtotalangularmomentumofN-bodydynamicalsystemsinherentlywell，thedegreeoftheirpreservationmaynotbeagoodmeasureoftheaccuracyofnumericalintegrations，especiallyasameasureofthepositionalerrorofplanets，i.stimatethenumericalerrorintheplanetarylongitudes，omparedtheresultofourmainlong-termintegrationswithsometestintegrations，whichspanmucthispurpose，(1/64ofthemainintegrations)spanning3x105yr，startingwiththesameinitialconditionsasintheN?onsiderthatthistestintegrationprovidesuswitha‘pseudo-true’t，weparethetestintegrationwiththemainintegration，N?theperiodof3x105yr，weseeadifferenceinmeananomaliesoftheEarthbetweenthetwointegrationsof?0.52°(inthecaseoftheN?1integration).Thisdifferencecanbeextrapolatedtothevalue?8700°，about25rotationsofEarthafter5Gyr，sinceilarly，thelongitudeerrorofPlutocanbeestimatedas?12°.ThisvalueforPlutoismuchbetterthantheresultinKinoshita&Nakai(1996)wherethedifferenceisestimatedas?60°.

3Numericalresults–nceattherawdata

3.2Time–frequencymaps

Althoughtheplanetarymotionexhibitsverylong-termstabilitydefinedasthenon-existenceofcloseencounterevents，thechaoticnatureofplanetarydynamicscanchangetheoscillatoryperiodandamplitudeofplanetaryorbitalmotiongraduallyoversuchlongtime-nsuchslightfluctuationsoforbitalvariationinthefrequencydomain，particularlyinthecaseofEarth，canpotentiallyhaveasignificanteffectonitssurfaceclimatesystemthroughsolarinsolationvariation(ger198.

Togiveanoverviewofthelong-termchangeinperiodicityinplanetaryorbitalmotion，weperformedmanyfastFouriertransformations(FFTs)alongthetimeaxis，andsuperposedtheresultingperiodgramstodrawtwo-dimensionaltime–specificapproachtodrawingthesetime–frequencymapsinthispaperisverysimple–muchsimplerthanthewaveletanalysisorLaskar‘s(1990，1993)frequencyanalysis.

Atypicalexampleofthetime–frequenc，whichshowsthevariationofperiodicityintheeccentricityandinclinationofEarthinN+ig.5，thedarkareashowsthatatthetimeindicatedbythevalueontheabscissa，theperiodicanrecognizefromthismapthattheperiodicityoftheeccentricityandinclinationofEarthonlychangesslightlyovertheentireperiodcoveredbytheN+snearlyregulartrendisqualitativelythesameinotherintegrationsandforotherplanets，althoughtypicalfrequenciesdifferplanetbyplanetandelementbyelement.

4.2Long-termexchangeoforbitalenergyandangularmomentum

Wecalculateverylong-periodicvariationandexchangeofplanetaryorbitalenergyandangularmomentumusingfilteredDelaunayelementsL，G，dHareequivalenttothep

Ineachfrequencydiagramobtainedabove，thestrengthofperiodicitycanbereplacedbyagrey-scale(orcolour)chart.

Weperformthereplacement，andconnectallthegrey-scale(orcolour)horizontalaxisofthesenewgraphsshouldbethetime，i.startingtimesofeachfragmentofdata(ti，wherei=1，…，n).Theverticalaxisrepresentstheperiod(orfrequency)oftheoscillationoforbitalelements.

WehaveadoptedanFFTbecauseofitsoverwhelmingspeed，sincetheamountofnumericaldatatobedeposedintofrequencyponentsisterriblyhuge(severaltensofGbytes).

对火星轨道变化问题的最后解释 (第2/3页)

mplecticmaptogetherwiththethird-orderHalleymethod(Danby1992)numberofmaximumiterationswesetinHalley‘smethodis15，buttheyneverreachedthemaximuminanyofourintegrations.

Theintervalofthedataoutputis200000d(?547yr)forthecalculationsofallnineplanets(N±1，2，3)，andabout8000000d(?21903yr)fortheintegrationoftheouterfiveplanets(F±).

Althoughnooutputfilteringwasdonewhenthenumericalintegrationswereinprocess，

Inthissectionwebrieflyreviewthelong-termstabilitorbitalmotionofplanetsindicateslong-termstabilityinallofournumericalintegrations:noorbitalcrossingsnorcloseencountersbetweenanypairofplanetstookplace.

3.1Generaldescriptionofthestabilityofplanetaryorbits

First，webrieflylookatthegeneralcharacterofthelong-interestherefocusesparticularlyontheinnerfourterrestrialplanetsforwhichtheorbitaltime-ecanseeclearlyfromtheplanarorbitalconfigurationsshowninFigs2and3，orbitalpositionsoftheterrestrialplanetsdifferlittlebetweentheinitialandfinalpartofeachnumericalintegration，solidlinesdenotingthepresentorbitsoftheplanetsliealmostwithintheswarmofdotseveninthefinalpartofintegrations(b)and(d).Thisindicatesthatthroughouttheentireintegrationperiodthealmostregularvariationsofplanetaryorbitalmotionremainnearlythesameastheyareatpresent.

阅读死在火星上最新章节 请关注舞文小说网(www.wushuzw.info)