笔迷楼>穿越小说>死在火星上>对火星轨道变化问题的最后解释
  作者君在作品相关中已经解释过这个问题,并在此列出相关参考文献中的一篇开源论文。

  以下是文章内容:

  Long-termintegrationsandstabilityofplanetaryorbitsinourSolarsystem

  Abstract

  Wepresenttheresultsofverylong-termnumericalintegrationsofplanetaryorbitalmotionsover109-yrtime-spansincludingallnineplanets.Aquickinspectionofournumericaldatashowsthattheplanetarymotion,atleastinoursimpledynamicalmodel,seemstobequitestableevenoverthisverylongtime-span.Acloserlookatthelowest-frequencyoscillationsusingalow-passfiltershowsusthepotentiallydiffusivecharacterofterrestrialplanetarymotion,especiallythatofMercury.ThebehaviouroftheeccentricityofMercuryinourintegrationsisqualitativelysimilartotheresultsfromJacquesLaskar'ssecularperturbationtheory(e.g.emax∼0.35over∼±4Gyr).However,therearenoapparentsecularincreasesofeccentricityorinclinationinanyorbitalelementsoftheplanets,whichmayberevealedbystilllonger-termnumericalintegrations.Wehavealsoperformedacoupleoftrialintegrationsincludingmotionsoftheouterfiveplanetsoverthedurationof±5×1010yr.TheresultindicatesthatthethreemajorresonancesintheNeptune–Plutosystemhavebeenmaintainedoverthe1011-yrtime-span.

  1Introduction

  1.1Definitionoftheproblem

  ThequestionofthestabilityofourSolarsystemhasbeendebatedoverseveralhundredyears,sincetheeraofNewton.Theproblemhasattractedmanyfamousmathematiciansovertheyearsandhasplayedacentralroleinthedevelopmentofnon-lineardynamicsandchaostheory.However,wedonotyethaveadefiniteanswertothequestionofwhetherourSolarsystemisstableornot.Thisispartlyaresultofthefactthatthedefinitionoftheterm‘stability’isvaguewhenitisusedinrelationtotheproblemofplanetarymotionintheSolarsystem.Actuallyitisnoteasytogiveaclear,rigorousandphysicallymeaningfuldefinitionofthestabilityofourSolarsystem.

  Amongmanydefinitionsofstability,hereweadopttheHilldefinition(Gladman1993):actuallythisisnotadefinitionofstability,butofinstability.Wedefineasystemasbecomingunstablewhenacloseencounteroccurssomewhereinthesystem,startingfromacertaininitialconfiguration(Chambers,Wetherill&Boss1996;Ito&Tanikawa1999).AsystemisdefinedasexperiencingacloseencounterwhentwobodiesapproachoneanotherwithinanareaofthelargerHillradius.Otherwisethesystemisdefinedasbeingstable.HenceforwardwestatethatourplanetarysystemisdynamicallystableifnocloseencounterhappensduringtheageofourSolarsystem,about±5Gyr.Incidentally,thisdefinitionmaybereplacedbyoneinwhichanoccurrenceofanyorbitalcrossingbetweeneitherofapairofplanetstakesplace.Thisisbecauseweknowfromexperiencethatanorbitalcrossingisverylikelytoleadtoacloseencounterinplanetaryandprotoplanetarysystems(Yoshinaga,Kokubo&Makino1999).OfcoursethisstatementcannotbesimplyappliedtosystemswithstableorbitalresonancessuchastheNeptune–Plutosystem.

  1.2Previousstudiesandaimsofthisresearch

  Inadditiontothevaguenessoftheconceptofstability,theplanetsinourSolarsystemshowacharactertypicalofdynamicalchaos(Sussman&Wisdom1988,1992).Thecauseofthischaoticbehaviourisnowpartlyunderstoodasbeingaresultofresonanceoverlapping(Murray&Holman1999;Lecar,Franklin&Holman2001).However,itwouldrequireintegratingoveranensembleofplanetarysystemsincludingallnineplanetsforaperiodcoveringseveral10Gyrtothoroughlyunderstandthelong-termevolutionofplanetaryorbits,sincechaoticdynamicalsystemsarecharacterizedbytheirstrongdependenceoninitialconditions.

  Fromthatpointofview,manyofthepreviouslong-termnumericalintegrationsincludedonlytheouterfiveplanets(Sussman&Wisdom1988;Kinoshita&Nakai1996).Thisisbecausetheorbitalperiodsoftheouterplanetsaresomuchlongerthanthoseoftheinnerfourplanetsthatitismucheasiertofollowthesystemforagivenintegrationperiod.Atpresent,thelongestnumericalintegrationspublishedinjournalsarethoseofDuncan&Lissauer(1998).Althoughtheirmaintargetwastheeffectofpost-main-sequencesolarmasslossonthestabilityofplanetaryorbits,theyperformedmanyintegrationscoveringupto∼1011yroftheorbitalmotionsofthefourjovianplanets.TheinitialorbitalelementsandmassesofplanetsarethesameasthoseofourSolarsysteminDuncan&Lissauer'spaper,buttheydecreasethemassoftheSungraduallyintheirnumericalexperiments.Thisisbecausetheyconsidertheeffectofpost-main-sequencesolarmasslossinthepaper.Consequently,theyfoundthatthecrossingtime-scaleofplanetaryorbits,whichcanbeatypicalindicatoroftheinstabilitytime-scale,isquitesensitivetotherateofmassdecreaseoftheSun.WhenthemassoftheSunisclosetoitspresentvalue,thejovianplanetsremainstableover1010yr,orperhapslonger.Duncan&Lissaueralsoperformedfoursimilarexperimentsontheorbitalmotionofsevenplanets(VenustoNeptune),whichcoveraspanof∼109yr.Theirexperimentsonthesevenplanetsarenotyetcomprehensive,butitseemsthattheterrestrialplanetsalsoremainstableduringtheintegrationperiod,maintainingalmostregularoscillations.

  Ontheotherhand,inhisaccuratesemi-analyticalsecularperturbationtheory(Laskar1988),Laskarfindsthatlargeandirregularvariationscanappearintheeccentricitiesandinclinationsoftheterrestrialplanets,especiallyofMercuryandMarsonatime-scaleofseveral109yr(Laskar1996).TheresultsofLaskar'ssecularperturbationtheoryshouldbeconfirmedandinvestigatedbyfullynumericalintegrations.

  Inthispaperwepresentpreliminaryresultsofsixlong-termnumericalintegrationsonallnineplanetaryorbits,coveringaspanofseveral109yr,andoftwootherintegrationscoveringaspanof±5×1010yr.Thetotalelapsedtimeforallintegrationsismorethan5yr,usingseveraldedicatedPCsandworkstations.Oneofthefundamentalconclusionsofourlong-termintegrationsisthatSolarsystemplanetarymotionseemstobestableintermsoftheHillstabilitymentionedabove,atleastoveratime-spanof±4Gyr.Actually,inournumericalintegrationsthesystemwasfarmorestablethanwhatisdefinedbytheHillstabilitycriterion:notonlydidnocloseencounterhappenduringtheintegrationperiod,butalsoalltheplanetaryorbitalelementshavebeenconfinedinanarrowregionbothintimeandfrequencydomain,thoughplanetarymotionsarestochastic.Sincethepurposeofthispaperistoexhibitandoverviewtheresultsofourlong-termnumericalintegrations,weshowtypicalexamplefiguresasevidenceoftheverylong-termstabilityofSolarsystemplanetarymotion.Forreaderswhohavemorespecificanddeeperinterestsinournumericalresults,wehavepreparedawebpage(access),whereweshowraworbitalelements,theirlow-passfilteredresults,variationofDelaunayelementsandangularmomentumdeficit,andresultsofoursimpletime–frequencyanalysisonallofourintegrations.

  InSection2webrieflyexplainourdynamicalmodel,numericalmethodandinitialconditionsusedinourintegrations.Section3isdevotedtoadescriptionofthequickresultsofthenumericalintegrations.Verylong-termstabilityofSolarsystemplanetarymotionisapparentbothinplanetarypositionsandorbitalelements.Aroughestimationofnumericalerrorsisalsogiven.Section4goesontoadiscussionofthelongest-termvariationofplanetaryorbitsusingalow-passfilterandincludesadiscussionofangularmomentumdeficit.InSection5,wepresentasetofnumericalintegrationsfortheouterfiveplanetsthatspans±5×1010yr.InSection6wealsodiscussthelong-termstabilityoftheplanetarymotionanditspossiblecause.

  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(Wisdom&Holman1991;Kinoshita,Yoshida&Nakai1991)withaspecialstart-upproceduretoreducethetruncationerrorofanglevariables,‘warmstart’(Saha&Tremaine1992,1994).

  Thestepsizeforthenumericalintegrationsis8dthroughoutallintegrationsofthenineplanets(N±1,2,3),whichisabout1/11oftheorbitalperiodoftheinnermostplanet(Mercury).Asforthedeterminationofstepsize,wepartlyfollowthepreviousnumericalintegrationofallnineplanetsinSussman&Wisdom(1988,7.2d)andSaha&Tremaine(1994,225/32d).Weroundedthedecimalpartofthetheirstepsizesto8tomakethestepsizeamultipleof2inordertoreducetheaccumulationofround-offerrorinthecomputationprocesses.Inrelationtothis,Wisdom&Holman(1991)performednumericalintegrationsoftheouterfiveplanetaryorbitsusingthesymplecticmapwithastepsizeof400d,1/10.83oftheorbitalperiodofJupiter.Theirresultseemstobeaccurateenough,whichpartlyjustifiesourmethodofdeterminingthestepsize.However,sincetheeccentricityofJupiter(∼0.05)ismuchsmallerthanthatofMercury(∼0.2),weneedsomecarewhenwecomparetheseintegrationssimplyintermsofstepsizes.

  Intheintegrationoftheouterfiveplanets(F±),wefixedthestepsizeat400d.

  WeadoptGauss'fandgfunctionsinthesymplecticmaptogetherwiththethird-orderHalleymethod(Danby1992)asasolverforKeplerequations.ThenumberofmaximumiterationswesetinHalley'smethodis15,buttheyneverreachedthemaximuminanyofourintegrations.

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

  Althoughnooutputfilteringwasdonewhenthenumericalintegrationswereinprocess,weappliedalow-passfiltertotheraworbitaldataafterwehadcompletedallthecalculations.SeeSection4.1formoredetail.

  2.4Errorestimation

  2.4.1Relativeerrorsintotalenergyandangularmomentum

  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-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°.

  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=0to0.0547×109yr).(b)ThefinalpartofN+1(t=4.9339×108to4.9886×109yr).(c)TheinitialpartofN−1(t=0to−0.0547×109yr).(d)ThefinalpartofN−1(t=−3.9180×109to−3.9727×109yr).Ineachpanel,atotalof23684pointsareplottedwithanintervalofabout2190yrover5.47×107yr.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,sincetheamountofnumericaldatatobedecomposedintofrequencycomponentsisterriblyhuge(severaltensofGbytes).

  Atypicalexampleofthetime–frequencymapcreatedbytheaboveproceduresisshowninagrey-scalediagramasFig.5,whichshowsthevariationofperiodicityintheeccentricityandinclinationofEarthinN+2integration.InFig.5,thedarkareashowsthatatthetimeindicatedbythevalueontheabscissa,theperiodicityindicatedbytheordinateisstrongerthaninthelighterareaaroundit.WecanrecognizefromthismapthattheperiodicityoftheeccentricityandinclinationofEarthonlychangesslightlyovertheentireperiodcoveredbytheN+2integration.Thisnearlyregulartrendisqualitativelythesameinotherintegrationsandforotherplanets,althoughtypicalfrequenciesdifferplanetbyplanetandelementbyelement.

  4.2Long-termexchangeoforbitalenergyandangularmomentum

  Wecalculateverylong-periodicvariationandexchangeofplanetaryorbitalenergyandangularmomentumusingfilteredDelaunayelementsL,G,H.GandHareequivalenttotheplanetaryorbitalangularmomentumanditsverticalcomponentperunitmass.LisrelatedtotheplanetaryorbitalenergyEperunitmassasE=−μ2/2L2.Ifthesystemiscompletelylinear,theorbitalenergyandtheangularmomentumineachfrequencybinmustbeconstant.Non-linearityintheplanetarysystemcancauseanexchangeofenergyandangularmomentuminthefrequencydomain.Theamplitudeofthelowest-frequencyoscillationshouldincreaseifthesystemisunstableandbreaksdowngradually.However,suchasymptomofinstabilityisnotprominentinourlong-termintegrations.

  InFig.7,thetotalorbitalenergyandangularmomentumofthefourinnerplanetsandallnineplanetsareshownforintegrationN+2.Theupperthreepanelsshowthelong-periodicvariationoftotalenergy(denotedasE-E0),totalangularmomentum(G-G0),andtheverticalcomponent(H-H0)oftheinnerfourplanetscalculatedfromthelow-passfilteredDelaunayelements.E0,G0,H0denotetheinitialvaluesofeachquantity.Theabsolutedifferencefromtheinitialvaluesisplottedinthepanels.ThelowerthreepanelsineachfigureshowE-E0,G-G0andH-H0ofthetotalofnineplanets.Thefluctuationshowninthelowerpanelsisvirtuallyentirelyaresultofthemassivejovianplanets.

  Comparingthevariationsofenergyandangularmomentumoftheinnerfourplanetsandallnineplanets,itisapparentthattheamplitudesofthoseoftheinnerplanetsaremuchsmallerthanthoseofallnineplanets:theamplitudesoftheouterfiveplanetsaremuchlargerthanthoseoftheinnerplanets.Thisdoesnotmeanthattheinnerterrestrialplanetarysubsystemismorestablethantheouterone:thisissimplyaresultoftherelativesmallnessofthemassesofthefourterrestrialplanetscomparedwiththoseoftheouterjovianplanets.Anotherthingwenoticeisthattheinnerplanetarysubsystemmaybecomeunstablemorerapidlythantheouteronebecauseofitsshorterorbitaltime-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.

  ItisnotclearatthemomentwhytheVenus–Earthpairexhibitsanegativecorrelationinenergyexchangeandapositivecorrelationinangularmomentumexchange.Wemaypossiblyexplainthisthroughobservingthegeneralfactthattherearenoseculartermsinplanetarysemimajoraxesuptosecond-orderperturbationtheories(cf.Brouwer&Clemence1961;Boccaletti&Pucacco1998).Thismeansthattheplanetaryorbitalenergy(whichisdirectlyrelatedtothesemimajoraxisa)mightbemuchlessaffectedbyperturbingplanetsthanistheangularmomentumexchange(whichrelatestoe).Hence,theeccentricitiesofVenusandEarthcanbedisturbedeasilybyJupiterandSaturn,whichresultsinapositivecorrelationintheangularmomentumexchange.Ontheotherhand,thesemimajoraxesofVenusandEartharelesslikelytobedisturbedbythejovianplanets.ThustheenergyexchangemaybelimitedonlywithintheVenus–Earthpair,whichresultsinanegativecorrelationintheexchangeoforbitalenergyinthepair.

  Asfortheouterjovianplanetarysubsystem,Jupiter–SaturnandUranus–Neptuneseemtomakedynamicalpairs.However,thestrengthoftheircouplingisnotasstrongcomparedwiththatoftheVenus–Earthpair.

  5±5×1010-yrintegrationsofouterplanetaryorbits

  Sincethejovianplanetarymassesaremuchlargerthantheterrestrialplanetarymasses,wetreatthejovianplanetarysystemasanindependentplanetarysystemintermsofthestudyofitsdynamicalstability.Hence,weaddedacoupleoftrialintegrationsthatspan±5×1010yr,includingonlytheouterfiveplanets(thefourjovianplanetsplusPluto).Theresultsexhibittherigorousstabilityoftheouterplanetarysystemoverthislongtime-span.Orbitalconfigurations(Fig.12),andvariationofeccentricitiesandinclinations(Fig.13)showthisverylong-termstabilityoftheouterfiveplanetsinboththetimeandthefrequencydomains.Althoughwedonotshowmapshere,thetypicalfrequencyoftheorbitaloscillationofPlutoandtheotherouterplanetsisalmostconstantduringtheseverylong-termintegrationperiods,whichisdemonstratedinthetime–frequencymapsonourwebpage.

  Inthesetwointegrations,therelativenumericalerrorinthetotalenergywas∼10−6andthatofthetotalangularmomentumwas∼10−10.

  5.1ResonancesintheNeptune–Plutosystem

  Kinoshita&Nakai(1996)integratedtheouterfiveplanetaryorbitsover±5.5×109yr.TheyfoundthatfourmajorresonancesbetweenNeptuneandPlutoaremaintainedduringthewholeintegrationperiod,andthattheresonancesmaybethemaincausesofthestabilityoftheorbitofPluto.Themajorfourresonancesfoundinpreviousresearchareasfollows.Inthefollowingdescription,λdenotesthemeanlongitude,Ωisthelongitudeoftheascendingnodeandϖisthelongitudeofperihelion.SubscriptsPandNdenotePlutoandNeptune.

  MeanmotionresonancebetweenNeptuneandPluto(3:2).Thecriticalargumentθ1=3λP−2λN−ϖPlibratesaround180°withanamplitudeofabout80°andalibrationperiodofabout2×104yr.

  TheargumentofperihelionofPlutoωP=θ2=ϖP−ΩPlibratesaround90°withaperiodofabout3.8×106yr.ThedominantperiodicvariationsoftheeccentricityandinclinationofPlutoaresynchronizedwiththelibrationofitsargumentofperihelion.ThisisanticipatedinthesecularperturbationtheoryconstructedbyKozai(1962).

  ThelongitudeofthenodeofPlutoreferredtothelongitudeofthenodeofNeptune,θ3=ΩP−ΩN,circulatesandtheperiodofthiscirculationisequaltotheperiodofθ2libration.Whenθ3becomeszero,i.e.thelongitudesofascendingnodesofNeptuneandPlutooverlap,theinclinationofPlutobecomesmaximum,theeccentricitybecomesminimumandtheargumentofperihelionbecomes90°.Whenθ3becomes180°,theinclinationofPlutobecomesminimum,theeccentricitybecomesmaximumandtheargumentofperihelionbecomes90°again.Williams&Benson(1971)anticipatedthistypeofresonance,laterconfirmedbyMilani,Nobili&Carpino(1989).

  Anargumentθ4=ϖP−ϖN+3(ΩP−ΩN)libratesaround180°withalongperiod,∼5.7×108yr.

  Inournumericalintegrations,theresonances(i)–(iii)arewellmaintained,andvariationofthecriticalargumentsθ1,θ2,θ3remainsimilarduringthewholeintegrationperiod(Figs14–16).However,thefourthresonance(iv)appearstobedifferent:thecriticalargumentθ4alternateslibrationandcirculationovera1010-yrtime-scale(Fig.17).ThisisaninterestingfactthatKinoshita&Nakai's(1995,1996)shorterintegrationswerenotabletodisclose.

  6Discussion

  Whatkindofdynamicalmechanismmaintainsthislong-termstabilityoftheplanetarysystem?Wecanimmediatelythinkoftwomajorfeaturesthatmayberesponsibleforthelong-termstability.First,thereseemtobenosignificantlower-orderresonances(meanmotionandsecular)betweenanypairamongthenineplanets.JupiterandSaturnareclosetoa5:2meanmotionresonance(thefamous‘greatinequality’),butnotjustintheresonancezone.Higher-orderresonancesmaycausethechaoticnatureoftheplanetarydynamicalmotion,buttheyarenotsostrongastodestroythestableplanetarymotionwithinthelifetimeoftherealSolarsystem.Thesecondfeature,whichwethinkismoreimportantforthelong-termstabilityofourplanetarysystem,isthedifferenceindynamicaldistancebetweenterrestrialandjovianplanetarysubsystems(Ito&Tanikawa1999,2001).WhenwemeasureplanetaryseparationsbythemutualHillradii(R_),separationsamongterrestrialplanetsaregreaterthan26RH,whereasthoseamongjovianplanetsarelessthan14RH.Thisdifferenceisdirectlyrelatedtothedifferencebetweendynamicalfeaturesofterrestrialandjovianplanets.Terrestrialplanetshavesmallermasses,shorterorbitalperiodsandwiderdynamicalseparation.Theyarestronglyperturbedbyjovianplanetsthathavelargermasses,longerorbitalperiodsandnarrowerdynamicalseparation.Jovianplanetsarenotperturbedbyanyothermassivebodies.

  Thepresentterrestrialplanetarysystemisstillbeingdisturbedbythemassivejovianplanets.However,thewideseparationandmutualinteractionamongtheterrestrialplanetsrendersthedisturbanceineffective;thedegreeofdisturbancebyjovianplanetsisO(eJ)(orderofmagnitudeoftheeccentricityofJupiter),sincethedisturbancecausedbyjovianplanetsisaforcedoscillationhavinganamplitudeofO(eJ).Heighteningofeccentricity,forexampleO(eJ)∼0.05,isfarfromsufficienttoprovokeinstabilityintheterrestrialplanetshavingsuchawideseparationas26RH.Thusweassumethatthepresentwidedynamicalseparationamongterrestrialplanets(>26RH)isprobablyoneofthemostsignificantconditionsformaintainingthestabilityoftheplanetarysystemovera109-yrtime-span.Ourdetailedanalysisoftherelationshipbetweendynamicaldistancebetweenplanetsandtheinstabilitytime-scaleofSolarsystemplanetarymotionisnowon-going.

  AlthoughournumericalintegrationsspanthelifetimeoftheSolarsystem,thenumberofintegrationsisfarfromsufficienttofilltheinitialphasespace.Itisnecessarytoperformmoreandmorenumericalintegrationstoconfirmandexamineindetailthelong-termstabilityofourplanetarydynamics.

  ——以上文段引自Ito,T.&Tanikawa,K.Long-termintegrationsandstabilityofplanetaryorbitsinourSolarSystem.Mon.Not.R.Astron.Soc.336,483–500(2002)

  这只是作者君参考的一篇文章,关于太阳系的稳定性。

  还有其他论文,不过也都是英文的,相关课题的中文文献很少,那些论文下载一篇要九美元(《Nature》真是暴利),作者君写这篇文章的时候已经回家,不在检测中心,所以没有数据库的使用权,下不起,就不贴上来了。 无尽的昏迷过后,时宇猛地从床上起身。想要看最新章节内容,请下载星星阅读app,无广告免费阅读最新章节内容。网站已经不更新最新章节内容,已经星星阅读小说APP更新最新章节内容。

  他大口的呼吸起新鲜的空气,胸口一颤一颤。

  迷茫、不解,各种情绪涌上心头。

  这是哪?

  随后,时宇下意识观察四周,然后更茫然了。

  一个单人宿舍?

  就算他成功得到救援,现在也应该在病房才对。

  还有自己的身体……怎么会一点伤也没有。

  带着疑惑,时宇的视线快速从房间扫过,最终目光停留在了床头的一面镜子上。

  镜子照出他现在的模样,大约十七八岁的年龄,外貌很帅。

  可问题是,这不是他!下载星星阅读app,阅读最新章节内容无广告免费

  之前的自己,是一位二十多岁气宇不凡的帅气青年,工作有段时间了。

  而现在,这相貌怎么看都只是高中生的年纪……

  这个变化,让时宇发愣很久。

  千万别告诉他,手术很成功……

  身体、面貌都变了,这根本不是手术不手术的问题了,而是仙术。

  他竟完全变成了另外一个人!

  难道……是自己穿越了?

  除了床头那摆放位置明显风水不好的镜子,时宇还在旁边发现了三本书。

  时宇拿起一看,书名瞬间让他沉默。

  《新手饲养员必备育兽手册》

  《宠兽产后的护理》

  《异种族兽耳娘评鉴指南》

  时宇:???

  前两本书的名字还算正常,最后一本你是怎么回事?

  “咳。”

  时宇目光一肃,伸出手来,不过很快手臂一僵。

  就在他想翻开第三本书,看看这究竟是个什么东西时,他的大脑猛地一阵刺痛,大量的记忆如潮水般涌现。

  冰原市。

  宠兽饲养基地。

  实习宠兽饲养员。网站即将关闭,下载星星阅读app为您提供大神天瑞说符的死在火星上

  御兽师?

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