对火星轨道变化问题的最后解释(第1/3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’isvaguewhenitisusedinrelationtotheproblemofplanetarymotioninthesolarsystem.actuallyitisnoteasytogiveaclear，rigorousandphysicallymeaningfuldefinitionofthestabilityofoursolarsystem.
amongmanydefinitionsofstability，hereweadoptthehilldefinition(gladman1993):actuallythisisnotadefinitionofstability，butofinstability.wedefineasystemasbingunstablewhenacloseencounteroccurssomewhereinthesystem，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.theirexperimentsonthesevenplanetsarenotyeprehensive，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&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-offerrorinthputationprocesses.inrelationtothis，wisdom&holman(1991)performednumericalintegrationsoftheouterfiveplanetaryorbitsusingthesymplecticmapwithastepsizeof400d，1/10.83oftheorbitalperiodofjupiter.theirresultseemstobeaccurateenough，whichpartlyjustifiesourmethodofdeterminingthestepsize.however，sincetheeccentricityofjupiter(～0.05)ismuchsmallerthanthatofmercury(～0.2)，weneedsomecarewhenwparetheseintegrationssimplyintermsofstepsizes.
intheintegrationoftheouterfiveplanets(f±)，wefixedthestepsizeat400d.
weadoptgauss'fandgfunctionsinthesymplecticmaptogetherwiththethird-orderhalleymethod(danby1992)asasolverforkeplerequations.thenumberofmaximumiterationswesetinhalley'smethodis15，buttheyneverreachedthemaximuminanyofourintegrations.
theintervalofthedataoutputis200000d(～547yr)forthecalculationsofallnineplanets(n±1，2，3)，andabout8000000d(～21903yr)fortheintegrationoftheouterfiveplanets(f±).
althoughnooutputfilteringwasdonewhenthenumericalintegrationswereinprocess，weappliedalow-passfiltertotheraworbitaldataafterwehapletedallthecalculations.seesection4.1formoredetail.
2.4errorestimation
2.4.1relativeerrorsintotalenergyandangularmomentum
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