督抚小说

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).

请勿开启浏览器阅读模式,否则将导致章节内容缺失及无法阅读下一章。

网游:拾荒少年封神  剑道争锋  祁同伟:侯亮平下跪,我狂飙进步  民国道长  邪骨!  爱错难逃  重生之最强黑萌妻  无敌神锄  穿成短命女配,我靠养崽卷死他们  谢邀,人在长安,正准备造反  天工至圣  我的媳妇是阴山女帝  夜缕罗  外星系统进化  婚图不轨  四合院,开局一个太初仙境  校花之至尊高手  进化之眼  灾厄魔盒  千金医家  

热门小说推荐
惊悚狩猎计划[无限]

惊悚狩猎计划[无限]

预收文大佬们都混吃等死了无限v章防盗比例80,每晚21点更新,不要养肥温馨提示微恐怖,爆笑沙雕,热血爽文桀骜明艳通缉犯冷傲果决执行官狩猎计划重启的那一年,各个系统分区被选中的300万玩...

高武:我有一个合成栏

高武:我有一个合成栏

苏宇穿越,突然得到了一个合成栏,世间万物,均可合成!无论是什么东西,落在苏宇的手中,都能变成稀世珍宝!在苏宇手中,只要物品足够,菜刀都能变成屠龙宝刀。普通的白纸合成出了鸿蒙金卷。普通的衣服合成出了飘渺仙衣。普通的气血丹合成出了九转金丹。任何物品,在合成栏下,都能变成无数人趋之若鹜的宝贝!...

医路风云

医路风云

实习马上结束,对留院已经不抱任何希望的楚天羽有的只剩下对未来的迷茫,但就在这时候上帝跟他开了一个天大的玩笑,让他可以在末世与现实世界自由穿梭,一个崭新的大时代向楚天羽打开了一扇大门书友群11774886...

快穿归来当影后

快穿归来当影后

快穿世界中,无数任务后,白苏终于回归现实。蜗居一间,证件一张,伤腿一条。赤贫开局,做个龙套又如何这一次她只为自己活多年后名导惊喜直接进组她来还需试镜同行郁闷苏提名了看来又是陪跑影...

恭请陛下斩仙

恭请陛下斩仙

穿越者许墨辰成为古代大乾国的皇帝,为了在乱世中存活,不得已夜夜爆肝,导入一些先进的政策科技知识,富国强兵,最终结束乱世。统一天下之后,他决意建立特区发扬文治逐步全国推广,成就千古一帝。哪知道群臣力谏陛下要遵循祖制广纳后宫啊!于是放眼看去,这满朝文武六部九卿,皆是修仙者。有妖有魔,有精有灵。迟迟到来的后宫佳丽,个个也都飞天遁地。不过小场面不要慌,朕毕竟带着穿越神器来的。归心的,赐予仙缘丹,助其证飞升大道不服的,请天子剑斩杀!什么?准备对凡人万民下手???非常耐死,那就不要怪朕超时代发挥了!...

每日热搜小说推荐