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第76章 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——“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±5x1010yr.theresultindicatesthatthethreemajorresonancesintheneptune–plutosystemhavebeenmaintainedoverthe1011-yrtime-span.

1introduction

1.1definitionoftheproblem

thequestionofthestabilityofoursolarsystemhasbeendebatedoverseveralhundredyears，sincetheeraofnewton.theproblemhasattractedmanyfamousmathematiciansovertheyearsandhasplayedacentralroleinthedevelopmentofnon-lineardynamicsandchaostheory.however，wedonotyethaveadefiniteanswertothequestionofwhetheroursolarsystemisstableornot.thisispartlyaresultofthefactthatthedefinitionoftheterm‘stability’isvaguewhenitisusedinrelationtotheproblemofplanetarymotioninthesolarsysteactuallyitisnoteasytogiveaclear，rigorousandphysicallymeaningfuldefinitionofthestabilityofoursolarsyste

amongmanydefinitionsofstability，hereweadoptthehilldefinition（gladman1993）:actuallythisisnotadefinitionofstability，butofinstability.wedefineasystemasbecomingunstablewhenacloseencounteroccurssomewhereinthesystem，startingfromacertaininitialconfiguration（chambers，wetherill&ito&&tanikawa1999）.asystemisdefinedasexperiencingacloseencounterwhentwobodiesapproachoneanotherwithinanareaofthelargerhillradius.otherwisethesystemisdefinedasbeingstable.henceforwardwestatethatourplanetarysystemisdynamicallystableifnocloseencounterhappensduringtheageofoursolarsystem，about±5gyr.incidentally，thisdefinitionmaybereplacedbyoneinwhichanoccurrenceofanyorbitalcrossingbetweeneitherofapairofplanetstakesplace.thisisbecauseweknowfromexperiencethatanorbitalcrossingisverylikelytoleadtoacloseencounterinplanetaryandprotoplanetarysystems（yoshinaga，kokubo&&makino1999）.ofcoursethisstatementcannotbesimplyappliedtosystemswithstableorbitalresonancessuchastheneptune–plutosyste

1.2previousstudiesandaimsofthisresearch

inadditiontothevaguenessoftheconceptofstability，theplanetsinoursolarsystemshowacharactertypicalofdynamicalchaos（sussman&&wisdom1988，1992）.thecauseofthischaoticbehaviourisnowpartlyunderstoodasbeingaresultofresonanceoverlapping（murray&lecar，franklin&&holman2001）.however，itwouldrequireintegratingoveranensembleofplanetarysystemsincludingallnineplanetsforaperiodcoveringseveral10gyrtothoroughlyunderstandthelong-termevolutionofplanetaryorbits，sincechaoticdynamicalsystemsarecharacterizedbytheirstrongdependenceoninitialconditions.

fromthatpointofview，manyofthepreviouslong-termnumericalintegrationsincludedonlytheouterfiveplanets（sussman&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±5x1010yr.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±5x1010yr.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&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.

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.

relativenumericalerrorofthetotalangularmomentumδaa0andthetotalenergyδee0inournumericalintegrationsn±1，2，3，whereδeandδaaretheabsolutechangeofthetotalenergyandtotalangularmomentum，respectively，ande0anda0aretheirinitialvalues.thehorizontalunitisgyr.

notethatdifferentoperatingsystems，differentmathematicallibraries，anddifferenthardwarearchitecturesresultindifferentnumericalerrors，throughthevariationsinround-offerrorhandlingandnumericalalgorithms.intheupperpaneloffig.1，wecanrecognizethissituationinthesecularnumericalerrorinthetotalangularmomentum，whichshouldberigorouslypreserveduptomachine-eprecision.

2.4.2errorinplanetarylongitudes

sincethesymplecticmapspreservetotalenergyandtotalangularmomentumofn-bodydynamicalsystemsinherentlywell，thedegreeoftheirpreservationmaynotbeagoodmeasureoftheaccuracyofnumericalintegrations，especiallyasameasureofthepositionalerrorofplanets，i.e.theerrorinplanetarylongitudes.toestimatethenumericalerrorintheplanetarylongitudes，weperformedthefollowingprocedures.wecomparedtheresultofourmainlong-termintegrationswithsometestintegrations，whichspanmuchshorterperiodsbutwithmuchhigheraccuracythanthemainintegrations.forthispurpose，weperformedamuchmoreaccurateintegrationwithastepsizeof0.125d（164ofthemainintegrations）spanning3x105yr，startingwiththesameinitialconditionsasinthen?1integration.weconsiderthatthistestintegrationprovidesuswitha‘pseudo-true’solutionofplanetaryorbitalevolution.next，wecomparethetestintegrationwiththemainintegration，n?1.fortheperiodof3x105yr，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-planeissettotheinvariantplaneofsolarsystemtotalangularmomentu（a）theinitialpartofn+1（t=0to0.0547x109yr）.（b）thefinalpartofn+1（t=4.9339x108to4.9886x109yr）.（c）theinitialpartofn?1（t=0to?0.0547x109yr）.（d）thefinalpartofn?1（t=?3.9180x109to?3.9727x109yr）.ineachpanel，atotalof23684pointsareplottedwithanintervalofabout2190yrover5.47x107yr.solidlinesineachpaneldenotethepresentorbitsofthefourterrestrialplanets（takenfromde245）.

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