对火星轨道变化问题的最后解释[1/2页]

天才一秒记住本站地址：[58小说]https://m.58books.com最快更新！无广告！

    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：
    LongtermintegrationsandstabilityofplanetaryorbitsinourSolarsystem
    Abstract
    Wepresenttheresultsofverylongtermnumericalintegrationsofplanetaryorbitalmotionsover109yrtimespansincludingallnineplanets.Aquickinspectionofournumericaldatashowsthattheplanetarymotion，atleastinoursimpledynamicalmodel，seemstobequitestableevenoverthisverylongtimespan.Acloserlookatthelowestfrequencyoscillationsusingalowpassfiltershowsusthepotentiallydiffusivecharacterofterrestrialplanetarymotion，especiallythatofMercury.ThebehaviouroftheeccentricityofMercuryinourintegrationsisqualitativelysimilartotheresultsfromJacquesLaskar'ssecularperturbationtheory(e.g.emax～0.35over～±4Gyr).However，therearenoapparentsecularincreasesofeccentricityorinclinationinanyorbitalelementsoftheplanets，whichmayberevealedbystilllongertermnumericalintegrations.Wehavealsoperformedacoupleoftrialintegrationsincludingmotionsoftheouterfiveplanetsoverthedurationof±5×1010yr.TheresultindicatesthatthethreemajorresonancesintheNeptune?Plutosystemhavebeenmaintainedoverthe1011yrtimespan.
    1Introduction
    1.1Definitionoftheproblem
    ThequestionofthestabilityofourSolarsystemhasbeendebatedoverseveralhundredyears，sincetheeraofNewton.Theproblemhasattractedmanyfamousmathematiciansovertheyearsandhasplayedacentralroleinthedevelopmentofnonlineardynamicsandchaostheory.However，wedonotyethaveadefiniteanswertothequestionofwhetherourSolarsystemisstableornot.Thisispartlyaresultofthefactthatthedefinitionoftheterm‘stabilityisvaguewhenitisusedinrelationtotheproblemofplanetarymotionintheSolarsystem.Actuallyitisnoteasytogiveaclear，rigorousandphysicallymeaningfuldefinitionofthestabilityofourSolarsystem.
    Amongmanydefinitionsofstability，hereweadopttheHilldefinition(Gladman1993):actuallythisisnotadefinitionofstability，butofinstability.Wedefineasystemasbecomingunstablewhenacloseencounteroccurssomewhereinthesystem，startingfromacertaininitialconfiguration(Chambers，Wetherillamp;amp;Boss1996;Itoamp;amp;Tanikawa1999).AsystemisdefinedasexperiencingacloseencounterwhentwobodiesapproachoneanotherwithinanareaofthelargerHillradius.Otherwisethesystemisdefinedasbeingstable.HenceforwardwestatethatourplanetarysystemisdynamicallystableifnocloseencounterhappensduringtheageofourSolarsystem，about±5Gyr.Incidentally，thisdefinitionmaybereplacedbyoneinwhichanoccurrenceofanyorbitalcrossingbetweeneitherofapairofplanetstakesplace.Thisisbecauseweknowfromexperiencethatanorbitalcrossingisverylikelytoleadtoacloseencounterinplanetaryandprotoplanetarysystems(Yoshinaga，Kokuboamp;amp;Makino1999).OfcoursethisstatementcannotbesimplyappliedtosystemswithstableorbitalresonancessuchastheNeptune?Plutosystem.
    1.2Previousstudiesandaimsofthisresearch
    Inadditiontothevaguenessoftheconceptofstability，theplanetsinourSolarsystemshowacharactertypicalofdynamicalchaos(Sussmanamp;amp;Wisdom1988，1992).Thecauseofthischaoticbehaviourisnowpartlyunderstoodasbeingaresultofresonanceoverlapping(Murrayamp;amp;Holman1999;Lecar，Franklinamp;amp;Holman2001).However，itwouldrequireintegratingoveranensembleofplanetarysystemsincludingallnineplanetsforaperiodcoveringseveral10Gyrtothoroughlyunderstandthelongtermevolutionofplanetaryorbits，sincechaoticdynamicalsystemsarecharacterizedbytheirstrongdependenceoninitialconditions.
    Fromthatpointofview，manyofthepreviouslongtermnumericalintegrationsincludedonlytheouterfiveplanets(Sussmanamp;amp;Wisdom1988;Kinoshitaamp;amp;Nakai1996).Thisisbecausetheorbitalperiodsoftheouterplanetsaresomuchlongerthanthoseoftheinnerfourplanetsthatitismucheasiertofollowthesystemforagivenintegrationperiod.Atpresent，thelongestnumericalintegrationspublishedinjournalsarethoseofDuncanamp;amp;Lissauer(1998).Althoughtheirmaintargetwastheeffectofpostmainsequencesolarmasslossonthestabilityofplanetaryorbits，theyperformedmanyintegrationscoveringupto～1011yroftheorbitalmotionsofthefourjovianplanets.TheinitialorbitalelementsandmassesofplanetsarethesameasthoseofourSolarsysteminDuncanamp;amp;Lissauer'spaper，buttheydecreasethemassoftheSungraduallyintheirnumericalexperiments.Thisisbecausetheyconsidertheeffectofpostmainsequencesolarmasslossinthepaper.Consequently，theyfoundthatthecrossingtimescaleofplanetaryorbits，whichcanbeatypicalindicatoroftheinstabilitytimescale，isquitesensitivetotherateofmassdecreaseoftheSun.WhenthemassoftheSunisclosetoitspresentvalue，thejovianplanetsremainstableover1010yr，orperhapslonger.Duncanamp;amp;Lissaueralsoperformedfoursimilarexperimentsontheorbitalmotionofsevenplanets(VenustoNeptune)，whichcoveraspanof～109yr.Theirexperimentsonthesevenplanetsarenotyetcomprehensive，butitseemsthattheterrestrialplanetsalsoremainstableduringtheintegrationperiod，maintainingalmostregularoscillations.
    Ontheotherhand，inhisaccuratesemianalyticalsecularperturbationtheory(Laskar1988)，Laskarfindsthatlargeandirregularvariationscanappearintheeccentricitiesandinclinationsoftheterrestrialplanets，especiallyofMercuryandMarsonatimescaleofseveral109yr(Laskar1996).TheresultsofLaskar'ssecularperturbationtheoryshouldbeconfirmedandinvestigatedbyfullynumericalintegrations.
    Inthispaperwepresentpreliminaryresultsofsixlongtermnumericalintegrationsonallnineplanetaryorbits，coveringaspanofseveral109yr，andoftwootherintegrationscoveringaspanof±5×1010yr.Thetotalelapsedtimeforallintegrationsismorethan5yr，usingseveraldedicatedPCsandworkstations.OneofthefundamentalconclusionsofourlongtermintegrationsisthatSolarsystemplanetarymotionseemstobestableintermsoftheHillstabilitymentionedabove，atleastoveratimespanof±4Gyr.Actually，inournumericalintegrationsthesystemwasfarmorestablethanwhatisdefinedbytheHillstabilitycriterion:notonlydidnocloseencounterhappenduringtheintegrationperiod，butalsoalltheplanetaryorbitalelementshavebeenconfinedinanarrowregionbothintimeandfrequencydomain，thoughplanetarymotionsarestochastic.Sincethepurposeofthispaperistoexhibitandoverviewtheresultsofourlongtermnumericalintegrations，weshowtypicalexamplefiguresasevidenceoftheverylongtermstabilityofSolarsystemplanetarymotion.Forreaderswhohavemorespecificanddeeperinterestsinournumericalresults，wehavepreparedawebpage(access)，whereweshowraworbitalelements，theirlowpassfilteredresults，variationofDelaunayelementsandangularmomentumdeficit，andresultsofoursimpletime?frequencyanalysisonallofourintegrations.
    InSection2webrieflyexplainourdynamicalmodel，numericalmethodandinitialconditionsusedinourintegrations.Section3isdevotedtoadescriptionofthequickresultsofthenumericalintegrations.VerylongtermstabilityofSolarsystemplanetarymotionisapparentbothinplanetarypositionsandorbitalelements.Aroughestimationofnumericalerrorsisalsogiven.Section4goesontoadiscussionofthelongesttermvariationofplanetaryorbitsusingalowpassfilterandincludesadiscussionofangularmomentumdeficit.InSection5，wepresentasetofnumericalintegrationsfortheouterfiveplanetsthatspans±5×1010yr.InSection6wealsodiscussthelongtermstabilityoftheplanetarymotionanditspossiblecause.
    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
    WeutilizeasecondorderWisdom?Holmansymplecticmapasourmainintegrationmethod(Wisdomamp;amp;Holman1991;Kinoshita，Yoshidaamp;amp;Nakai1991)withaspecialstartupproceduretoreducethetruncationerrorofanglevariables，‘warmstart(Sahaamp;amp;Tremaine1992，1994).
    Thestepsizeforthenumericalintegrationsis8dthroughoutallintegrationsofthenineplanets(N±1，2，3)，whichisabout1\/11oftheorbitalperiodoftheinnermostplanet(Mercury).Asforthedeterminationofstepsize，wepartlyfollowthepreviousnumericalintegrationofallnineplanetsinSussmanamp;amp;Wisdom(1988，7.2d)andSahaamp;amp;Tremaine(1994，225\/32d).Weroundedthedecimalpartofthetheirstepsizesto8tomakethestepsizeamultipleof2inordertoreducetheaccumulationofroundofferrorinthecomputationprocesses.Inrelationtothis，Wisdomamp;amp;Holman(1991)performednumericalintegrationsoftheouterfiveplanetaryorbitsusingthesymplecticmapwithastepsizeof400d，1\/10.83oftheorbitalperiodofJupiter.Theirresultseemstobeaccurateenough，whichpartlyjustifiesourmethodofdeterminingthestepsize.However，sincetheeccentricityofJupiter(～0.05)ismuchsmallerthanthatofMercury(～0.2)，weneedsomecarewhenwecomparetheseintegrationssimplyintermsofstepsizes.
    Intheintegrationoftheouterfiveplanets(F±)，wefixedthestepsizeat400d.
    WeadoptGauss'fandgfunctionsinthesymplecticmaptogetherwiththethirdorderHalleymethod(Danby1992)asasolverforKeplerequations.ThenumberofmaximumiterationswesetinHalley'smethodis15，buttheyneverreachedthemaximuminanyofourintegrations.
    Theintervalofthedataoutputis200000d(～547yr)forthecalculationsofallnineplanets(N±1，2，3)，andabout8000000d(～21903yr)fortheintegrationoftheouterfiveplanets(F±).
    Althoughnooutputfilteringwasdonewhenthenumericalintegrationswereinprocess，weappliedalowpassfiltertotheraworbitaldataafterwehadcompletedallthecalculations.SeeSection4.1formoredetail.
    2.4Errorestimation
    2.4.1Relativeerrorsintotalenergyandangularmomentum
    Accordingtooneofthebasicpropertiesofsymplecticintegrators，whichconservethephysicallyconservativequantitieswell(totalorbitalenergyandangularmomentum)，ourlongtermnumericalintegrationsseemtohavebeenperformedwithverysmallerrors.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，throughthevariationsinroundofferrorhandlingandnumericalalgorithms.IntheupperpanelofFig.1，wecanrecognizethissituationinthesecularnumericalerrorinthetotalangularmomentum，whichshouldberigorouslypreserveduptomachineεprecision.
    2.4.2Errorinplanetarylongitudes
    SincethesymplecticmapspreservetotalenergyandtotalangularmomentumofNbodydynamicalsystemsinherentlywell，thedegreeoftheirpreservationmaynotbeagoodmeasureoftheaccuracyofnumericalintegrations，especiallyasameasureofthepositionalerrorofplanets，i.e.theerrorinplanetarylongitudes.Toestimatethenumericalerrorintheplanetarylongitudes，weperformedthefollowingprocedures.Wecomparedtheresultofourmainlongtermintegrationswithsometestintegrations，whichspanmuchshorterperiodsbutwithmuchhigheraccuracythanthemainintegrations.Forthispurpose，weperformedamuchmoreaccurateintegrationwithastepsizeof0.125d(1\/64ofthemainintegrations)spanning3×105yr，startingwiththesameinitialconditionsasintheN?1integration.Weconsiderthatthistestintegrationprovidesuswitha‘pseudotruesolutionofplanetaryorbitalevolution.Next，wecomparethetestintegrationwiththemainintegration，N?1.Fortheperiodof3×105yr，weseeadifferenceinmeananomaliesoftheEarthbetweenthetwointegrationsof～0.52°(inthecaseoftheN?1integration).Thisdifferencecanbeextrapolatedtothevalue～8700°，about25rotationsofEarthafter5Gyr，sincetheerroroflongitudesincreaseslinearlywithtimeinthesymplecticmap.Similarly，thelongitudeerrorofPlutocanbeestimatedas～12°.ThisvalueforPlutoismuchbetterthantheresultinKinoshitaamp;amp;Nakai(1996)wherethedifferenceisestimatedas～60°.
    3Numericalresults?I.Glanceattherawdata
    Inthissectionwebrieflyreviewthelongtermstabilityofplanetaryorbitalmotionthroughsomesnapshotsofrawnumericaldata.Theorbitalmotionofplanetsindicateslongtermstabilityinallofournumericalintegrations:noorbitalcrossingsnorcloseencountersbetweenanypairofplanetstookplace.
    3.1Generaldescriptionofthestabilityofplanetaryorbits
    First，webrieflylookatthegeneralcharacterofthelongtermstabilityofplanetaryorbits.Ourinterestherefocusesparticularlyontheinnerfourterrestrialplanetsforwhichtheorbitaltimescalesaremuchshorterthanthoseoftheouterfiveplanets.AswecanseeclearlyfromtheplanarorbitalconfigurationsshowninFigs2and3，orbitalpositionsoftheterrestrialplanetsdifferlittlebetweentheinitialandfinalpartofeachnumericalintegration，whichspansseveralGyr.Thesolidlinesdenotingthepresentorbitsoftheplanetsliealmostwithintheswarmofdotseveninthefinalpartofintegrations(b)and(d).Thisindicatesthatthroughouttheentireintegrationperiodthealmostregularvariationsofplanetaryorbitalmotionremainnearlythesameastheyareatpresent.
    Verticalviewofthefourinnerplanetaryorbits(fromthezaxisdirection)attheinitialandfinalpartsoftheintegrationsN±1.Theaxesunitsareau.ThexyplaneissettotheinvariantplaneofSolarsystemtotalangularmomentum.(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.The

对火星轨道变化问题的最后解释[1/2页]

『加入书签，方便阅读』