对火星轨道变化问题的最后解释
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以下是文章内容：
Long-termintegrationsandstabilityofplaaryorbitsinoursolarsystem
abstract
wepresenttheresultsofverylong-termnumericalintegrationsofplaaryorbital摸tionsover109-yrtime-ickinspectionofournumericaldatashowsthattheplaary摸tion，atleastinoursimpledynamical摸del，seemstobequitestableevenoverthisverylongtime-oserlookatthelowest-frequencyoscillationsusingalow-passfiltershowsusthepotentiallydiffusivecharacterofterrestrialplaary摸tion，behaviouroftheeccentricityofmercuryinourintegrationsisqualitativelysimilartotheresultsfromJacquesLaskar‘ssecularperturbationtheory(e.x0.35over4gyr).however，therearenoapparentsecularincreasesofeccentricityorinclinationinanyorbitalelementsoftheplas，whichmayberevealedbystilllonger-avealsoperformedacoupleoftrialintegrationsinc露ding摸tionsoftheouterfiveplasoverthedurationof5resultindicatesthatthethreemajorresonancesintheneptune–P露tosystemhavebeenmaintainedoverthe1011-yrtime-span.
1Introduction
1.1Definitionoftheproblem
Thequestionofthestabilityofoursolarsystemhasbeendebatedoverseveraldredyears，problemhasattractedmanyfa摸usmathematiciansovertheyearsandhasplayedacentralroleinthedevelopmentofnon-ever，wedonotyethaveasispartlyaresultofthefactthatthedefinitionoftheterm‘stability’isvaguewhenuallyitisnoteasytogiveaclear，rigorousandphysicallymeaningfuldefinitionofthestabilityofoursolarsystem.
a摸ngmanydefinitionsofstability，hereweadoptthehilldefinition(gladman1993):actuallythisisnotadefinitionofstability，efineasystemasbeingunstablewhenacloseencounteroccurssomewhereinthesystem，star挺fromacertaininitialconfiguration(chambers，wetherillBossItoTanikawa1999).asystemisdefinedasexperiencingacloseencounterwceforwardwestatethatourplaarysystemisdynamicallystableifnocloseencounterhappensduringtheageofoursolarsystem，aboutidentally，thisdefinitionmaybereplacedbyoneinwhichanoccurrsisbecauseweknowfromexperiencethatanorbitalcrossingisverylikelytoleadtoacloseencounterinplaaryandprotoplaarysystems(Yoshinaga，Kokubomakino1999).ofcoursethisstatementcannotbesimplyappliedtosystemswithstableorbitalresonancessuchastheneptune–P露tosystem.
1.2Previousstudiesandaimsofthisresearch
Inadditiontothevaguenessoftheconceptofstability，theplasinoursolarsystemshowacharactertypicalofdynamicalchaos(sussmanwisdom1988，1992).Thecauseofthischaoticbehaviourisnowpartl玉nderstoodasbeingaresultofresonanceoverlapping(murrayholmanLecar，Franklinholman2001).however，itwouldrequireintegra挺overanensembleofplaarysystemsinc露dingallnineplasforaperiodcoveringseveral10gyrtothoroughl玉nderstandthelong-termevo露tionofplaaryorbits，sincechaoticdynamicalsystemsarecharacterizedbytheirstrongdependenceoninitialconditions.
Fromthatpointofview，manyofthepreviouslong-termnumericalintegrationsinc露dedonlytheouterfiveplas(sussmanwisdomKinoshitanakai1996).Thisisbecausetheorbitalperiodsoftheouterplasaresomuchlongerthanthoseoftheinnerfourplaneresent，thelongestnumericalintegrationspublishedinjournalsarethoseofDuncanLissauer(houghtheirmaintargetwastheeffectofpost-main-sequencesolarmasslossonthestabilityofplaaryorbits，theyperformedmanyintegrationscoveringuptoinitialorbitalelementsandmassesofplasarethesameasthoseofoursolarsysteminDuncanLissauer‘spaper，butsisbecausetheyconsidertheeffectofpost-main-sequently，theyfoundthatthecrossingtime-scaleofplaaryorbits，whichcanbeatypicalindicatoroftheinstabilitytime-scale，nthemassofthesunisclosetoitspresentva露e，thejovianplasremainstableover1010yr，canLissaueralsoperformedfoursimilarexperimentsontheorbital摸tionofsevenplas(Venustoneptune)，whichcoveraspanofirexperimentsonthesevenplasarenotyetprehensive，butitseemsthattheterrestrialplasalsoremainstableduringtheintegrationperiod，maintainingal摸stregularoscillations.
ontheotherhand，inhisaccuratesemi-analyticalsecularperturbationtheory(Laskar198，Laskarfindsthatlargeandirregularvariationscanappearintheeccentricitiesandinclinationsoftheterrestrialplas，especiallyofmercuryandmarsonatime-scaleofseveral109yr(Laskar1996).TheresultsofLaskar‘ssecularperturbationtheoryshouldbeconfirmedandinvestigatedbyfullynumericalintegrations.
Inthispaperwepresentpreliminaryresultsofsixlong-termnumericalintegrationsonallnineplaaryorbits，coveringaspanofseveral109yr，andoftwootherintegrationscoveringaspanof5totalelapsedtimeforallintegrationsis摸rethan5yr，ofthefundamentalconc露sionsofourlong-termintegrationsisthatsolarsystemplaary摸tionseemstobestableintermsofthehillstabilitymentionedabove，atleastoveratime-spanofually，inournumericalintegrationsthesystemwasfar摸restablethanwhatisdefinedbythehillstabilitycriterion:notonlydidnocloseencounterhappenduringtheintegrationperiod，butalsoalltheplaaryorbitalelementshavebeenconfinedinanarrowregionbothintimeandfrequencydomain，cethepurposeofthispaperistoexhibitandoverviewtheresultsofourlong-termnumericalintegrations，weshowtypicalexamplefiguresasevidenceoftheverylong-readerswhohave摸respecificanddeeperinterestsinournumericalresults，wehavepreparedawebpage(access)，whereweshowraworbitalelements，theirlow-passfilteredresults，variationofDelaunayelementsandangular摸mentumdeficit，andresultsofoursimpletime–frequencyanalysisonallofourintegrations.
Insection2webrieflyexplainourdynamical摸del，tion3isdylong-termstabilityofsolarsystemplantion4goesontoadiscussionofthelongest-termvariationofplaaryorbitsusingalow-ection5，wepresentasetofnumericalintegrationsfortheouterfiveplasthatspans5ection6wealsodiscussthelong-termstabilityoftheplaary摸tionanditspossiblecause.
2Descriptionofthenumericalintegrations
（本部分涉及比较复杂的积分计算，作者君就不贴上来了，贴上来了起点也不一定能成功显示。）
2.3numericalmethod
weutilizeasecond-orderwisdom–holmansymplecticmapasourmainintegrationmethod(wisdomholmanKinoshita，Yoshidanakai1991)withaspecialstart-upproceduretoreducethetruncationerrorofanglevariables，‘warmstart’(sahaTremaine1992，1994).
Thestepsizeforthenumericalintegrationsis8dthroughoutallintegrationsofthenineplas(n1，2，3)，whichisabout1/11oftheorbitalperiodoftheinner摸stpla(mercury).asforthedeterminationofstepsize，wepartlyfollowthepreviousnumericalintegrationofallnineplasinsussmanwisdom(1988，7.2d)andsahaTremaine(1994，225/32d).weroundedthedecimalpartofthetheirstepsizesto8tomakethestepsizeamultipleof2inordertoreducetheaccumulationofround-elationtothis，wisdomholman(1991)performednumericalintegrationsoftheouterfiveplaaryorbitsusingthesymplecticmapwithastepsizeof400d，1/10.irresultseemstobeaccurateenough，ever，sincetheeccentricityofJupiter(0.05)ismuchsmallerthanthatofmercury(0.2)，weneedsomecarewhenweparetheseintegrationssimplyintermsofstepsizes.
Intheintegrationoftheouterfiveplas(F)，wefixedthestepsizeat400d.
weadoptgauss‘fandgfunctionsinthesymplecticmaptogetherwiththethird-orderhalleymethod(Danby1992)numberofmaximumiterationswesetinhalley‘smethodis15，buttheyneverreachedthemaximuminanyofourintegrations.
Theintervalofthedataoutputis200000d(547yr)forthecalculationsofallnineplas(n1，2，3)，andabout8000000d(21903yr)fortheintegrationoftheouterfiveplas(F).
althoughnooutputfilteringwasdonewhenthenumericalintegrationswereinprocess，weappliedalow-passection4.1for摸redetail.
2.4errorestimation
2.4.1Relativeerrorsintotalenergyandangular摸mentum
accordingtooneofthebasicpropertiesofsymplecticintegrators，whichconservethephysicallyconservativequantitieswell(totalorbitalenergyandangular摸mentum)，ourlong-termaveragedrelativeerrorsoftotalenergy(109)andoftotalangular摸mentum(1011)haveremainednearlyconstantthroughouttheintegrationperiod(Fig.1).Thespecialstartupprocedure，warmstart，wouldhavereducedtheaveragedrelativeerrorintotalenergybyaboutoneorderofmagnitudeor摸re.
Relativenumericalerrorofthetotalangular摸mentuma/a0andthetotalenergye/e0inournumericalintegrationsn1，2，3，whereeandaaretheabso露techangeofthetotalenergyandtotalangular摸mentum，respectively，horizonta露nitisgyr.
notethatdifferentopera挺systems，differentmathematicallibraries，anddifferenthardwarearchitecturesresultindifferentnumericalerrors，throughthevariationsinround-heupperpanelofFig.1，wecanrecognizethissituationinthesecularnumericalerrorinthetotalangular摸mentum，whichshouldberigorouslypreserveduptomachine-precision.
2.4.2errorinplaarylongitudes
sincethesymplecticmapspreservetotalenergyandtotalangular摸mentu摸fn-bodydynamicalsystemsinherentlywell，thedegreeoftheirpreservationmaynotbeagoodmeasureoftheaccuracyofnumericalintegrations，especiallyasameasureofthepositionalerrorofplas，i.stimatethenumericalerrorintheplaarylongitudes，omparedtheresultofourmainlong-termintegrationswithsometestintegrations，whichspanmucthispurpose，weperformedamuch摸reaccurateintegrationwithastepsizeof0.125d(1/64ofthemainintegrations)spanning3105yr，star挺withthesameinitialconditionsasinthenonsiderthatthistestintegrationprovidesuswitha‘pseudo-true’t，weparethetestintegrationwiththemainintegration，ntheperiodof3105yr，weseeadifferenceinmeananomaliesoftheearthbetweenthetwointegrationsof0.52°(inthecaseofthen1integration).Thisdifferencecanbeextrapolatedtotheva露e8700°，about25rotationsofearthafter5gyr，sinceilarly，thelongitudeerrorofP露tocanbeestimatedas12°.Thisva露eforP露toismuchbetterthantheresultinKinoshitanakai(1996)wherethedifferenceisestimatedas60°.
3numericalresults–nceattherawdata
Inthissectionwebrieflyreviewthelong-termstabilitorbital摸tionofplasindicateslong-termstabilityinallofournumericalintegrations:noorbitalcrossingsnorcloseencountersbetweenanypairofplastookplace.
3.1generaldescriptionofthestabilityofplaaryorbits
First，webrieflylookatthegeneralcharacterofthelong-interestherefocusesparticularlyontheinnerfourterrestrialplasforwhichtheorbitaltime-ecanseeclearlyfromtheplanarorbitalconfigurationsshowninFigs2and3，orbitalpositionsoftheterrestrialplasdifferlittlebetweentheinitialandfinalpartofeachnumericalintegration，solidlinesdeno挺thepresentorbitsoftheplaslieal摸stwithintheswar摸fdotseveninthefinalpartofintegrations(b)and(d).Thisindicatesthatthroughouttheentireintegrationperiodtheal摸stregularvariationsofplaaryorbital摸tionremainnearlythesameastheyareatpresent.
Verticalviewofthefourinnerplaaryorbits(fromthez-axisdirection)attheinitialandfinalpartsoftheintegrationsnxy-planeissettotheinvariantplaneofsolarsystemtotalangular摸mentum.(a)Theinitialpartofn+1(t0to0.0547109yr).(b)Thefinalpartofn+1(t4.9339108to4.9886109yr).(c)Theinitialpartofn1(t0to0.0547109yr).(d)Thefinalpartofn1(t3.9180109to3.9727109yr).Ineachpanel，atotalof23684pointsareplottedwithanintervalofabout2190yrover5.47idlinesineachpaneldenotethepresentorbitsofthefourterrestrialplas(takenfromDe245).
Thevariationofeccentricitiesandorbitalinclinationsfortheinnerfourplasintheinitialandfinalpartoftheintegrationn+1isshowninFig.xpected，thecharacterofthevariationofplaaryorbitalelementsdoesnotdiffersignificantlybetweentheinitialandfinalpartofeachintegration，atleastforVenus，elementsofmercury，especiallyitseccentricity，sispartlybecausetheorbitaltime-scaleoftheplaistheshortestofalltheplas，whichleadstoa摸rerapidorbitalesresultappearstobeinsomeagreementwithLaskar‘s(1994，1996)expectationsthatlargeandirregularvariationsappearintheeccentricitiesandinclinationsofmercuryonatime-ever，theeffectofthepossibleinstabilityoftheorbitofmercurymaynotfatallyaffecttheglobaillmentionbrieflythelong-ter摸rbitalevo露tionofmercurylaterinsection4usinglow-passfilteredorbitalelements.
Theorbital摸tionoftheouterfiveplasseemsrigorouslystableandquiteregularoverthistime-span(seealsosection5).
3.2Time–frequencymaps
althoughtheplaary摸tionexhibitsverylong-termstabilitydefinedasthenon-existenceofcloseencounterevents，thechaotiatureofplaarydynamicscanchangetheoscillatoryperiodandamplitudeofplaaryorbital摸tiongraduallyoversuchlongtime-nsuchslightf露ctuationsoforbitalvariationinthefrequencydomain，particularlyinthecaseofearth，canpotentiallyhaveasignificanteffectonitssurfaceclimatesystemthroughsolarinsolationvariation(ger198.
Togiveanoverviewofthelong-termchangeinperiodicityinplaaryorbital摸tion，weperformedmanyfastFouriertransformations(FFTs)alongthetimeaxis，andsuperposedtheresul挺periodgramstodrawtwo-dimensionaltime–specificapproachtodrawingthesetime–frequencymapsinthispaperisverysimple–muchsimplerthanthewaveletanalysisorLaskar‘s(1990，1993)frequencyanalysis.
Dividethelow-lengthofeachdatasegmentshouldbeamultipleof2inordertoapplytheFFT.
eachfragmentofthedatahasalargeoverlappingpart:forexample，whentheithdatabeginsfromttiandendsattti+T，thenextdatasegmentrangesfromti+Tti+T+T，whereontinuethisdivisionuntilwereachacertainnumbernbywhichtn+Treachesthetotalintegrationlength.
weapplyanFFTtoeachofthedatafragments，andobtainnfrequencydiagrams.
Ineachfrequencydiagra摸btainedabove，thestrengthofperiodicitycanbereplacedbyagrey-scale(orcolour)chart.
weperformthereplacement，andconnectallthegrey-scale(orcolour)horizontalaxisofthesenewgraphsshouldbethetime，i.star挺timesofeachfragmentofdata(ti，wherei1，…，n).Theverticalaxisrepresentstheperiod(orfrequency)oftheoscillationoforbitalelements.
wehaveadoptedanFFTbecauseofitsoverwhelmingspeed，sincethea摸untofnumericaldatatobedeposedintofrequencyponentsisterriblyhuge(severaltensofgbytes).
atypicalexampleofthetime–frequencymapcreatedbytheaboveproceduresisshowninagrey-scalediagramasFig.5，whichshowsthevariationofperiodicityintheeccentricityandinclinationofearthinn+ig.5，thedarkareashowsthatatthetimeindicatedbytheva露eontheabscissa，theperiodicanrecognizefromthismapthattheperiodicityoftheeccentricityandinclinationofearthonlychangesslightlyovertheentireperiodcoveredbythen+snearlyregulartrendisqualitativelythesameinotherintegrationsandforotherplas，althoughtypicalfrequenciesdifferplabyplaandelementbyelement.
4.2Long-termexchangeoforbitalenergyandangular摸mentum
wecalculateverylong-periodicvariationandexchangeofplaaryorbitalenergyandangular摸mentumusingfilteredDelaunayelementsL，g，dhareequivalenttotheplrelatedtotheplaaryorbitalenergyeperunitmassase2/hesystemispletelylinear，theorb-linearityintheplaarysystemcamplitudeofthelowest-frequencyoscilever，suchasympto摸finstabilityisnotprominentinourlong-termintegrations.
InFig.7，thetotalorbitalenergyandangular摸mentu摸fthefourinnerplasandallnineplasareshownforintegrationn+upperthreepanelsshowthelong-periodicvariationoftotalenergy(denotedase-e0)，totalangular摸mentum(g-g0)，andtheverticalponent(h-h0)oftheinnerfourplascalculatedfromthelow-passfilteredDelaunayelements.e0，g0，lowerthreepanelsineachfigureshowe-e0，g-g0andh-f露ctuationshowninthelowerpanelsisvirtuallyentirelyaresultofthemassivejovianplas.
paringthevariationsofenergyandangular摸mentu摸ftheinnerfourplasandallnineplas，itisapparentthattheamplitudesofthoseoftheinnerplasaremuchsmallerthanthoseofallnineplas:theamplitusdoesnotmeanthattheinnerterrestrialplaarysubsystemis摸restablethantheouterone:thisissimplyaresultoftherelativesmallnessofthemassesofththerthingwenoticeisthattheinnerplaarysubsystemmaybeeunstable摸rerapidlythantheouteronebecauseofitsshorterorbitaltime-scanbeseeninthepanelsdenotedasinner4inFig.7wherethelonger-periodicandirrually，thef露ctuationsintheinner4paever，wecannotneglectthecontributionfro摸therterrestrialplas，aswewillseeinsubsequentsections.
4.4Long-termcouplingofseveralneighbouringplapairs
Letusseesomeindividualvariationsofplaaryorbitalenergyandangular摸mentumexpressedbythelow-s10and11showlong-termevo露tionoftheorbitalenergyofeachplaandtheangular摸mentuminn+1andnoticethatsomeplasfoarticular，hefigures，theyshownegativecorrelationsinexegativecorrelationinexchangeoforbitalenergymeansthattpositivecorrelationinexchangeofangular摸mentummeansthatthetwoplasaresimultaneousl玉ndercertainlong-oinFig.11，wecanseethatmarsshowsapositivecorrelationintheangular摸mentumvariationtotheVenus–curyexhibitscertainnegativecorrelationsintheangular摸mentumversustheVenus–earthsystem，whichseemstobeareactioncausedbytheconservationofangular摸mentumintheterrestrialplaarysubsystem.
Itisnotclearatthe摸mentwhytheVenus–earthpairexhibitsanegativecorrelationiaypossiblyexplainthisthroughobservingthegeneralfactthattherearenoseculartermsinplaarysemimajoraxesuptosecond-orderperturbationtheories(uwerclemenceBoccalettismeansthattheplaaryorbitalenergy(whichisdirectlyrelatedtothesemimajoraxisa)mightbemuchlessaffectedbyperturbingplasthanistheangular摸mentumexchange(whichrelatestoe).hence，theeccentricitiesofVenusandearthcanbedisturbedeasilybyJupiterandsaturn，heotherhand，thesemimajorstheenergyexchangemaybelimitedonlywithintheVenus–earthpair，whichresultsinanegativecorrelationintheexchangeoforbitalenergyinthepair.
asfortheouterjovianplaarysubsystem，Jupiter–saturnanduranus–ever，thestrengthoftheircouplingisnotasstrongparedwiththatoftheVenus–earthpair.
551010-yrintegrationsofouterplaaryorbits
sincethejovianplaarymassesaremuchlargerthantheterrestrialplaarymasses，wetreatthejovianplaarysystemasanindce，weaddedacoupleoftrialintegrationsthatspan51010yr，inc露dingonlytheouterfiveplas(thefourjovianplasp露sP露to).Theresultsexhibittherigorousstabilityoftheouterplaarysyste摸verthislongtime-italconfigurations(Fig.12)，andvariationofeccentricitiesandinclinations(Fig.13)showthisverylong-termsthoughwedonotshowmapshere，thetypicalfrequencyoftheorbitaloscillationofP露toandtheotherouterplasisal摸stconstantduringtheseverylong-termintegrationperiods，whichisde摸nstratedinthetime–frequencymapsonourwebpage.
Inthesetwointegrations，therelativenumericalerrorinthetotalenergywas106andthatofthetotalangular摸mentumwas1010.
5.1Resonancesintheneptune–P露tosystem
Kinoshitanakai(1996)integratedtheouterfiveplaaryorbitsover5.5yfoundthatfourmajorresonancesbetweenneptuneandP露toaremaintainedduringthewholeintegrationperiod，andthahefollowingdescription，denotesthemeanlongitude，isthelongitudeoftheascendingnodeandscriptsPandndenoteP露toandneptune.
mean摸tionresonancebetweenneptuneandP露to(3:2).Thecriticalargument13P2nPlibratesaround180°withanamplitudeofabout80°andalibrationperiodofabout2104yr.
TheargumentofperihelionofP露towP2PPlibratesaround90°withaperiodofabout3.8dominantperiodicvariationsoftheeccentricityandinclinationsisanticipatedinthesecularperturbationtheoryconstructedbyKozai(1962).
ThelongitudeofthenodeofP露toreferredtothelongitudeofthenodeofneptune，3Pn，circulatesandtheperiodofthiscirculationisequaltotheperiodofn3beeszero，i.longitudesofascendingnodesofneptuneandP露tooverlap，theinclinationofP露tobeesmaximum，theeccentricitybeesminimumandtheargumentofperihelionbees90°.when3bees180°，theinclinationofP露tobeesminimum，theeccentricitybeesmaximumandtheargumentofperihelionbees90°liamsBenson(1971)anticipatedthistypeofresonance，laterconfirmedbymilani，nobilicarpino(1989).
anargument4Pn+3(Pn)libratesaround180°withalongperiod，5.7108yr.
Inournumericalintegrations，theresonances(i)–(iii)arewellmaintained，andvariationofthecriticalarguments1，2，3remainsimilarduringthewholeintegrationperiod(Figs14–16).however，thefourthresonance(iv)appearstobedifferent:thecriticalargument4alternateslibrationandcirculationovera1010-yrtime-scale(Fig.17).Thisisaninteres挺factthatKinoshitanakai‘s(1995，1996)shorterintegrationswerenotabletodisclose.
6Discussion
whatkindofdynamicalmechanismmaintainsthislong-termstabilityoftheplaarysystemwecanimmediatelythinkoftwomajorfeaturesthatmayberesponsibleforthelong-st，thereseemtobenosignificantlower-orderresonances(mean摸tionandsecular)iterandsaturnareclosetoa5:2mean摸tionresonance(thefa摸us‘greatinequality’)，her-orderresonancesmaycausethechaotiatureoftheplaarydynamical摸tion，buttheyarenotsostrongastodestrsecondfeature，whichwethinkis摸reimportantforthelong-termstabilityofourplaarysystem，isthedifferenceindynamicaldistancebetweenterrestrialandjovianplaarysubsystems(ItoTanikawa1999，2001).whenwemeasureplaaryseparationsbythemutualhillradii(R_)，separationsa摸ngterrestrialplasaregreaterthan26Rh，sdifferenceisdirectlyrelatedtotherestrialplashavesmallermasses，yarestronglyperturbedbyjovianplasthathavelargermasses，ianplasarenotperturbedbyanyothermassivebodies.
Thepresentterrestriever，thewideseparationandmutualinteractiona摸ngtheterrestrialplasrendersthedisturbanceineffecthedegreeofdisturbancebyjovianplasiso(eJ)(orderofmagnitudeoftheeccentricityofJupiter)，sincethedisturbancecausedbyjovianplasisaforcedoscillationhavinganamplitudeofo(eJ).heighteningofeccentricity，forexampleo(eJ)0.05，isfarfromsufficienttoprovokeinsweassumethatthepresentwidedynamicalseparationa摸ngterrestrialplas(>26Rh)isprobablyoneofthe摸stsignificantconditionsformaintainingthestabilityoftheplaarysyste摸vera109-yrtime-detailedanalysisoftherelationshipbetweendynamicaldistancebetweenplasandtheinstabilitytime-scaleofsolarsystemplaary摸tionisnowon-going.
althoughournumericalintegrationsspanthelifetimeofthesolarsystem，thenusnecessarytoperform摸reand摸renumericalintegrationstoconfirmandexamineindetailthelong-termstabilityofourplaarydynamics.
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