笔趣阁 - 科幻小说 - 死在火星上在线阅读 - 第76章 对火星轨道变化问题的最后解释

第76章 对火星轨道变化问题的最后解释

    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-termiionsandstabilityofplaaryorbitsinoursolarsystem

    abstract

    wepresenttheresultsofverylong-termnumeritegrationsofplaaryorbitalmotionsover109-yrtime-spansincludingallnineplas.aquispeofournumericaldatashowsthattheplaarymotion,atleastinoursimpledynamicalmodel,seemstobequitestableevehisverylongtime-span.acloserlookatthelowest-frequencyoscillationsusingalow-passfiltershowsusthepotentiallydiffusivecharacterofterrestrialplaarymotion,especiallythatofmercury.thebehaviouroftheetriercuryinouriionsisqualitativelysimilartotheresultsfromjacqueslaskar'ssecularperturbationtheory(e.g.emax~0.35yr).however,therearenoapparentsecularincreasesofetricityorinationinanyorbitalelementsoftheplas,whichmayberevealedbystillloermnumeritegrations.wehavealsoperformedacoupleoftrialiionsincludingmotionsoftheouterfiveplasoverthedurationof±5x1010yr.theresultindicatesthatthethreemajorresoheune–plutosystemhavebeenmaintainedoverthe1011-yrtime-span.

    1introdu

    1.1definitionoftheproblem

    thequestionofthestabilityofoursolarsystemhasbeeedoverseveralhundredyears,siheeraofon.theproblemhasattractedmanyfamousmathematisovertheyearsandhasplayedatralroleinthedevelopmentofnon-lineardynamidchaostheory.however,wedohaveadefiniteahequestionofwhetheroursolarsystemisstableornot.thisispartlyaresultofthefactthatthedefinitionoftheterm‘stability’isvaguewhenitisusediiontotheproblemofplaarymotioninthesolarsystem.actuallyitisogiveaclear,rigorousandphysicallymeaningfuldefinitionofthestabilityofoursolarsystem.

    amongmanydefinitionsofstability,heretthehilldefinition(gladman1993):actuallythisisnotadefinitionofstability,butofinstability.wedefineasystemasbeingunstablewhenacloseenteroccurssomewhereiem,startingfromacertaininitialfiguration(chambers,wetherillitotanikawa1999).asystemisdefinedasexperiengacloseenterwhentwobodiesapproaeahinahelargerhillradius.otherwisethesystemisdefinedasbeingstable.henceforwardwestatethatourplaarysystemisdynamicallystableifnocloseenterhappensduringtheageofoursolarsystem,about±5gyr.ially,thisdefinitionmaybereplacedbyoneinwhioccurrenceofanyorbitalcrossiweeherofapairofplaakesplace.thisisbecauseweknowfromexperieanorbitalcrossingisverylikelytoleadtoacloseenterinplaaryandprotoplaarysystems(yoshinaga,kokubomakino1999).ofcoursethisstatementotbesimplyappliedtosystemswithstableorbitalresonancessuchastheune–plutosystem.

    1.2previousstudiesandaimsofthisresearch

    inadditiontothevaguenessoftheceptofstability,theplasinoursolarsystemshowacharactertypicalofdynamicalchaos(sussmanwisdom1988,1992).thecauseofthischaoticbehaviourisnowpartlyuoodasbeiofresonanceoverlapping(murraylecar,franklinholman2001).however,itwouldrequireiingoveranensembleofplaarysystemsincludingallnineplasforaperiodcseveral10gyrthlyuandthelong-termevolutionofplaaryorbits,sincechaotiamicalsystemsarecharacterizedbytheirstrongdependeninitialditions.

    fromthatpointofview,manyofthepreviouslong-termnumeritegrationsincludedonlytheouterfiveplas(sussmankinoshitanakai1996).thisisbecausetheorbitalperiodsoftheouterplasaresomugerthanthoseoftheinnerfourplahatitismucheasiertofollowthesystemfiveionperiod.atpresent,thelonumeritegrationspublishedinjournalsarethoseofdunlissauer(1998).althoughtheirmaintargetwastheeffectofpost-main-sequenasslossoabilityofplaaryorbits,theyperformedmanyiionscupto~1011yroftheorbitalmotionsofthefourjovianplaheinitialorbitalelementsandmassesofplasarethesameasthoseofoursolarsystemindunlissauer'spaper,buttheydecreasethemassofthesungraduallyintheirnumericalexperiments.thisisbecausetheysidertheeffectofpost-main-sequenasslossinthepaper.sequently,theyfoundthatthecrossingtime-scaleofplaaryorbits,whibeatypidicatoroftheinstabilitytime-scale,isquitesensitivetotherateofmassdecreaseofthesuhemassofthesunisclosetoitspresentvalue,thejovianplasremainstableover1010yr,orperhapslonger.dunlissaueralsoperformedfoursimilarexperimentsontheorbitalmotionofsevenplas(venustoune),whichcoveraspanof~109yr.theirexperimentsonthesevenplasarepreheitseemsthattheterrestrialplasalsoremainstableduriegrationperiod,maintainingalmularoscillations.

    oherhand,inhisaccuratesemi-analyticalsecularperturbationtheory(laskar1988),laskarfindsthatlargeandirregularvariationsappearintheetricitiesandinationsoftheterrestrialplas,especiallyofmercuryandmarsonatime-scaleofseveral109yr(laskar1996).theresultsoflaskar'ssecularperturbationtheoryshouldbefirmedandiigatedbyfullynumeritegrations.

    inthispaperwepresentpreliminaryresultsofsixlong-termnumeritegrationsonallnineplaaryorbits,caspanofseveral109yr,andoftwootheriionscaspanof±5x1010yr.thetotalelapsedtimeforalliionsismorethan5yr,usingseveraldedicatedpdworkstations.ohefualclusions-termiionsisthatsolarsystemplaarymotioobestableintermsofthehillstabilitymentionedabove,atleastoveratime-spanof±4gyr.actually,inournumeritegratioemwasfarmorestablethanwhatisdefihehillstabilitycriterion:notonlydidnocloseenterhappenduriegrationperiod,butalsoalltheplaaryorbitalelementshavebeenfinedinanarrionbothintimeandfrequenain,thoughplaarymotioochasticethepurposeofthispaperistoexhibitandoverviewtheresults-termnumeritegrations,weshowtypicalexamplefiguresasevideheveryloabilityofsolarsystemplaarymotion.forreaderswhohavemorespecifiddeeperisinournumericalresults,reparedawebpage(access),whereweshowraworbitalelements,theirlow-passfilteredresults,variationofdelaunayelementsandangularmomentumdeficit,asofoursimpletime–frequenalysisonallofouriions.

    iion2webrieflyexplainourdynamicalmodel,numericalmethodandinitialditionsusedinouriioion3isdevotedtoadescriptionofthequickresultsofthenumeritegrations.veryloabilityofsolarsystemplaarymotionisapparentbothinplaarypositionsandorbitalelements.aroughestimationofnumericalerrorsisalsogiveiooadiscussionoftheloermvariationofplaaryorbitsusingalow-passfilterandincludesadiscussionofangularmomentumdeficit.iion5,wepreseofnumeritegrationsfortheouterfiveplahatspans±5x1010yr.iion6wealsodiscusstheloabilityoftheplaarymotionanditspossiblecause.

    2descriptionofthenumeritegrations

    (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

    weutilizeased-orderwisdom–holmansymplecticmapasourmaiiohod(wisdomkinoshita,yoshidanakai1991)ecialstart-upproceduretoreducethetruncationerrorofanglevariables,‘warmstart’(sahatremaine1992,1994).

    thestepsizeforthenumeritegrationsis8dthroughoutalliionsofthenineplas(n±1,2,3),whichisabout111oftheorbitalperiodoftheinnermostpla(mercury).asforthedeterminationofstepsize,wepartlyfollowthepreviousnumeritegrationofallnineplasinsussmanwisdom(1988,7.2d)andsahatremaine(1994,22532d).werouhedecimalpartofthetheirstepsizesto8tomakethestepsizeamultipleof2ioreducetheaccumulationofround-offerrorintheputationprocesses.iiontothis,wisdomholman(1991)performednumeritegrationsoftheouterfiveplaaryorbitsusingthesymplecticmapsizeof400d,110.83oftheorbitalperiodofjupiter.theirresultseemstobeaccurateenough,whichpartlyjustifiesourmethodofdeterminiepsize.however,siheetricityofjupiter(~0.05)ismuchsmallerthanthatofmercury(~0.2),weneedsomecarewhenweparetheseiionssimplyintermsofstepsizes.

    iegrationoftheouterfiveplas(f±),wefixedthestepsizeat400d.

    tgauss'fandgfunsinthesymplecticmaptogetherwiththethird-orderhalleymethod(danby1992)asasolverforkeplerequations.thenumberofmaximumiteratioinhalley'smethodis15,buttheyneverreachedthemaximuminanyofouriions.

    theintervalofthedataoutputis200000d(~547yr)forthecalculationsofallnineplas(n±1,2,3),andabout8000000d(~21903yr)fortheiionoftheouterfiveplas(f±).

    althoughnooutputfilteringwasdohenumeritegrationswereinprocess,liedalow-passfiltertotheraworbitaldataafterwehadpletedallthecalculations.seese4.1formoredetail.

    2.4errorestimation

    2.4.1relativeerrorsintotalenergyandangularmomentum

    acctoohebasicpropertiesofsymplectitegrators,whiservethephysicallyservativequantitieswell(totalorbitalenergyandangularmomentum),-termnumeritegratioohavebeenperformedwithverysmallerrors.theaveragedrelativeerrorsoftotalenergy(~10?9)andoftotalangularmomentum(~10?11)haveremainednearlystantthroughouttheiionperiod(fig.1).thespecialstartupprocedure,warmstart,wouldhavereducedtheaveragedrelativeerrorintotalenergybyaboutoneorderofmagnitudeormore.

    relativenumericalerrorofthetotalangularmomentumδaa0aalenergyδee0inournumeritegrationsn±1,2,3,whereδeandδaaretheabsolutegeofthetotalenergyandtotalangularmomentum,respectively,ande0anda0aretheirinitialvalues.thehorizontalunitisgyr.

    differeingsystems,differentmathematicallibraries,anddifferenthardwarearchitecturesresultindifferentnumericalerrors,throughthevariationsinround-offerrorhandlingandnumericalalgorithms.intheupperpaneloffig.1,wereizethissituationinthesecularnumericalerrorialangularmomentum,whichshouldberigorouslypreserveduptomae-eprecision.

    2.4.2errorinplaarylongitudes

    sihesymplecticmapspreservetotalenergyandtotalangularmomentumofn-bodydynamicalsystemsilywell,thedegreeoftheirpreservationmaynotbeagoodmeasureoftheaccuraeritegrations,especiallyasameasureofthepositionalerrorofplas,i.e.theerrorinplaarylongitudes.toestimatethenumericalerrorintheplaarylongitudes,weperformedthefollowingprocedures.weparedtheresultofourmainlong-termiionswithsometestiions,whichspanmuchshorterperiodsbutwithmuchhigheraccuracythanthemaiions.forthispurpose,weperformedamuchmoreaccurateiionsizeof0.125d(164ofthemaiions)spanning3x105yr,startingwiththesameinitialditionsasinthen?1iion.wesiderthatthistestiionprovidesusseudo-true’solutionofplaaryorbitalevolutio,weparethetestiionwiththemaiion,n?1.fortheperiodof3x105yr,weseeadifferenmeananomaliesoftheearthbetweewoiionsof~0.52°(inthecaseofthen?1iion).thisdifferenbeextrapolatedtothevalue~8700°,about25rotatiohafter5gyr,siheerroroflongitudesincreaseslinearlywithtimeinthesymplecticmap.similarly,thelongitudeerrorofplutobeestimatedas~12°.thisvalueforplutoismuchbetterthantheresultinkinoshitanakai(1996)wherethedifferenceisestimatedas~60°.

    3numericalresults–i.glaherawdata

    inthissewebrieflyreviewtheloabilityofplaaryorbitalmotihsomesnapshotsofrawnumericaldata.theorbitalmotionofplasindicatesloabilityinallofournumeritegrations:noorbitalcrossingsnorcloseentersbetairofplaookplace.

    3.1generaldescriptionofthestabilityofplaaryorbits

    first,webrieflylookatthegeneralcharacteroftheloabilityofplaaryorbits.ouriherefocusesparticularlyontheinnerfourterrestrialplasforwhichtheorbitaltime-scalesaremuchshorterthanthoseoftheouterfiveplas.asweseeclearlyfromtheplanarorbitalfigurationsshowninfigs2and3,orbitalpositionsoftheterrestrialplasdifferlittlebetweeialandfinalpartofeaumeritegration,whichspansseveralgyr.thesolidliingthepresentorbitsoftheplasliealmostwithintheswarmofdotseveninthefinalpartofiions(b)and(d).thisindicatesthatthroughouttheeegrationperiodthealmularvariationsofplaaryorbitalmotionremainnearlythesameastheyareatpresent.

    verticalviewofthefourinnerplaaryorbits(fromthez-axisdire)attheinitialandfinalpartsoftheiionsn±1.theaxesunitsareau.thexy-planeissettotheinvariantplaneofsolarsystemtotalangularmomentum.(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).ineael,atotalof23684pointsareplottedwithanintervalofabout2190yrover5.47x107yr.solidlinesineaeldehepresentorbitsofthefourterrestrialplaakenfromde245).

    thevariationofetricitiesandorbitalinationsfortheinnerfourplasiialandfinalpartoftheiionn 1isshowninfig.4.asexpected,thecharacterofthevariationofplaaryorbitalelementsdoesnotdiffersignifitlybetweeialandfinalpartofeategration,atleastforvehandmars.theelementsofmercury,especiallyitsetricity,seemtoget