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零点看书 > 死在火星上 > 对火星轨道变化问题的最后解释

对火星轨道变化问题的最后解释

对火星轨道变化问题的最后解释 (第1/2页)
  
  作者君在作品相关中其实已经解释过这个问题。
  
  不过仍然有人质疑——“你说得太含糊了”,“火星轨道的变化比你想象要大得多!”
  
  那好吧,既然作者君的简单解释不够有力,那咱们就看看严肃的东西,反正这本书写到现在,嚷嚷着本书BUG一大堆,用初高中物理在书中挑刺的人也不少。
  
  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.
  
  
  
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