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对火星轨道变化问题的最后解释(第1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。

那么作者君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。ThebehaviouroftheeccentricityofMercuryinourintegrationsisqualitativelysimilartotheresultsfromJacquesLaskarssecularperturbationtheory(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&Lissauerspaper,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)。TheresultsofLaskarssecularperturbationtheoryshouldbeconfirmedandinvestigatedbyfullynumericalintegrations。

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),whichisabout111oftheorbitalperiodoftheinnermostplanet(Mercury)。Asforthedeterminationofstepsize,wepartlyfollowthepreviousnumericalintegrationofallnineplanetsinSussman&Wisdom(1988,7。2d)andSaha&Tremaine(1994,22532d)。Weroundedthedecimalpartofthetheirstepsizesto8tomakethestepsizeamultipleof2inordertoreducetheaccumulationofround-offerrorinthecomputationprocesses。Inrelationtothis,Wisdom&Holman(1991)performednumericalintegrationsoftheouterfiveplanetaryorbitsusingthesymplecticmapwithastepsizeof400d,110。83oftheorbitalperiodofJupiter。Theirresultseemstobeaccurateenough,whichpartlyjustifiesourmethodofdeterminingthestepsize。However,sincetheeccentricityofJupiter(~0。05)ismuchsmallerthanthatofMercury(~0。2),weneedsomecarewhenwecomparetheseintegrationssimplyintermsofstepsizes。

Intheintegrationoftheouterfiveplanets(F±),wefixedthestepsizeat400d。

WeadoptGaussfandgfunctionsinthesymplecticmaptogetherwiththethird-orderHalleymethod(Danby1992)asasolverforKeplerequations。ThenumberofmaximumiterationswesetinHalleysmethodis15,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。

RelativenumericalerrorofthetotalangularmomentumδAA0andthetotalenergyδEE0inournumericalintegrationsN±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(164ofthemainintegrations)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)。

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