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第76章 对火星轨道变化问题的最后解释（第1页）

&esp;&esp;作者君在作品相关中其实已经解释过这个问题。

&esp;&esp;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！”

&esp;&esp;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。

&esp;&esp;以下是文章内容：

&esp;&esp;long-tertegrationsandstabilityofparyorbitsourrsyste

&esp;&esp;abstract

&esp;&esp;wepresenttheresultfverylong-ternuricaltegrationfparyorbitalotionver109-yrti-spanscdgallnepsaickspectionofournuricaldatashowsthattheparyotion，atleastoursipledynaicalodel，seestobeitestableevenoverthisverylongti-spanacloserlookatthelowest-freencyosciltionsgalow-passfiltershowsthepotentiallydiffivecharacterofterrestrialparyotion，especiallythatofrcurythebehaviouroftheeentricityofrcuryourtegrationsisalitativelysiirtotheresultsfrojacesskar&039;ssecurperturbationtheory（egeax～035over～±4gyr）however，therearenoapparentsecurcreasefeentricityorclationanyorbitalelentftheps，whichayberevealedbystilllonr-ternuricaltegrationswehavealperfordaupleoftrialtegrationscdgotionftheouterfivepverthedurationof±5x1010yrtheresultdicatesthatthethreeajorrenancestheneptune–ptosystehavebeenataedoverthe1011-yrti-span

&esp;&esp;1troduction

&esp;&esp;11defitionoftheproble

&esp;&esp;theestionofthestabilityofourrsystehasbeendebatedoverseveralhundredyears，scetheeraofnewtontheproblehasattractedanyfaoatheaticianvertheyearsandhaspyedacentralrolethedevelopntofnon-leardynaicsandchaostheoryhowever，wedonotyethaveadefiteanswertotheestionofwhetherourrsysteisstableornotthisispartlyaresultofthefactthatthedefitionoftheter‘stability’isvaguewhenitisedretiontotheprobleofparyotionthersysteactuallyitisnoteasytogiveaclear，rigoroandphysicallyangfuldefitionofthestabilityofourrsyste

&esp;&esp;aonganydefitionfstability，hereweadoptthehilldefition（gdan1993）:actuallythisisnotadefitionofstability，butofstabilitywedefeasysteasbegunstablewhenacloseenunterourswherethesyste，startgfroacertaitialnfiguration（chabers，wetherillitotanikawa1999）asysteisdefedasexperiencgacloseenunterwhenobodiesapproachoneanotherwithanareaoftherrhillradiotherwisethesysteisdefedasbegstablehenceforwardwestatethatourparysysteisdynaicallystableifnocloseenunterhappensdurgtheaofourrsyste，about±5gyrcidentally，thisdefitionayberepcedbyonewhichanourrenceofanyorbitalcrossgbeeeneitherofapairofpstakespcethisisbecaeweknowfroxperiencethatanorbitalcrossgisverylikelytoleadtoacloseenunterparyandproarysystes（yoshaga，kokuboako1999）ofursethisstatentcannotbesiplyappliedtosysteswithstableorbitalrenancessuchastheneptune–ptosyste

&esp;&esp;12previostudiesandaifthisresearch

&esp;&esp;additiontothevaguenesfthenceptofstability，thepsourrsysteshowacharactertypicalofdynaicalchaos（ssanwisdo1988，1992）thecaeofthischaoticbehaviourisnowpartlderstoodasbegaresultofrenanceoverppg（urraylecar，franklholan2001）however，iouldreiretegratgoveranensebleofparysystescdgallnepsforaperiodvergseveral10gyrtothoroughlderstandthelong-tervotionofparyorbits，scechaoticdynaicalsystesarecharacterizedbytheirstrongdependenceonitialnditions

&esp;&esp;frothatpotofview，anyofthepreviolong-ternuricaltegrationscdedonlytheouterfiveps（ssankoshitanakai1996）thisisbecaetheorbitalperiodftheouterpsareuchlonrthanthoseofthenerfourpsthatitisucheasiertofollowthesysteforagiventegrationperiodatpresent，thelonstnuricaltegrationspublishedjournalsarethoseofduncanlissauer（1998）althoughtheiratarastheeffectofpost-a-seencerasslosnthestabilityofparyorbits，theyperfordanytegrationsvergupto～1011yroftheorbitalotionfthefourjovianpstheitialorbitalelentsandassefpsarethesaasthoseofourrsysteduncanlissauer&039;spaper，buttheydecreasetheasfthesungraduallytheirnuricalexperintsthisisbecaetheynsidertheeffectofpost-a-seencerasslossthepapernseently，theyfoundthatthecrossgti-scaleofparyorbits，whichcanbeatypicaldicatorofthestabilityti-scale，isitesensitivetotherateofassdecreaseofthesunwhentheasfthesunisclosetoitspresentvae，thejovianpsreastableover1010yr，orperhapslonrduncanlissaueralperfordfoursiirexperintntheorbitalotionofsevenps（ventoneptune），whichveraspanof～109yrtheirexperintnthesevenpsarenotyetprehensive，butitseesthattheterrestrialpsalreastabledurgthetegrationperiod，atagalostregurosciltions

&esp;&esp;ontheotherhand，hisauratesei-analyticalsecurperturbationtheory（skar1988），skarfdsthatrandirregurvariationscanappeartheeentricitiesandclationftheterrestrialps，especiallyofrcuryandarnati-scaleofseveral109yr（skar1996）theresultfskar&039;ssecurperturbationtheoryshouldbenfirdandvestigatedbyfullynuricaltegrations

&esp;&esp;thispaperwepresentpreliaryresultfsixlong-ternuricaltegrationnallneparyorbits，vergaspanofseveral109yr，andofoothertegrationsvergaspanof±5x1010yrthetotalepsedtiforalltegrationsisorethan5yr，gseveraldedicatedpcsandworkstationsoneofthefundantalncsionfourlong-tertegrationsisthatrsysteparyotionseestobestableterfthehillstabilityntionedabove，atleastoverati-spanof±4gyractually，ournuricaltegrationsthesystewasfarorestablethanwhatisdefedbythehillstabilitycriterion:notonlydidnocloseenunterhappendurgthetegrationperiod，butalalltheparyorbitalelentshavebeennfedanarrowregionbothtiandfreencydoa，thoughparyotionsarestochasticscethepurposeofthispaperistoexhibitandoverviewtheresultfourlong-ternuricaltegrations，weshowtypicalexaplefiguresasevidenceoftheverylong-terstabilityofrsysteparyotionforreaderswhohaveorespecificanddeeperterestsournuricalresults，wehavepreparedawebpa（aess），whereweshowraworbitalelents，theirlow-passfilteredresults，variationofdeunayelentsandangurontudeficit，andresultfoursipleti–freencyanalysinallofourtegrations

&esp;&esp;section2webrieflyexpourdynaicalodel，nuricalthodanditialnditionsedourtegrationssection3isdevotedtoadescriptionoftheickresultfthenuricaltegrationsverylong-terstabilityofrsysteparyotionisapparentbothparypositionsandorbitalelentsaroughestiationofnuricalerrorsisalgivensection4goentoadiscsionofthelonst-tervariationofparyorbitsgalow-passfilterandcdesadiscsionofangurontudeficitsection5，wepresentasetofnuricaltegrationsfortheouterfivepsthatspans±5x1010yrsection6wealdiscsthelong-terstabilityoftheparyotionanditspossiblecae

&esp;&esp;2descriptionofthenuricaltegrations

&esp;&esp;（本部分涉及比较复杂的积分计算，作者君就不贴上来了，贴上来了也不一定能成功显示。）

&esp;&esp;23nuricalthod

&esp;&esp;weutilizeasend-orderwisdo–holansyplectiapaurategrationthod（wisdokoshita，yoshidanakai1991）withaspecialstart-upproceduretoreducethetruncationerrorofanglevariables，‘warstart’（sahatreae1992，1994）

&esp;&esp;thestepsizeforthenuricaltegrationsis8dthroughoutalltegrationftheneps（n±1，2，3），whichisabout111oftheorbitalperiodofthenerostp（rcury）asforthedeterationofstepsize，wepartlyfollowtheprevionuricaltegrationofallnepsssanwisdo（1988，72d）andsahatreae（1994，22532d）weroundedthedecialpartofthetheirstepsizesto8toakethestepsizeaultipleof2ordertoreducetheauutionofround-offerrortheputationprocessesretiontothis，wisdoholan（1991）perfordnuricaltegrationftheouterfiveparyorbitsgthesyplectiaithastepsizeof400d，11083oftheorbitalperiodofjupitertheirresultseestobeaurateenough，whichpartlyjtifieurthodofdetergthestepsizehowever，scetheeentricityofjupiter（～005）isuchsallerthanthatofrcury（～02），weneedcarewhenweparethesetegrationssiplyterfstepsizes

&esp;&esp;thetegrationoftheouterfiveps（f±），wefixedthestepsizeat400d

&esp;&esp;weadoptgas&039;fandgfunctionsthesyplectiaptotherwiththethird-orderhalleythod（danby1992）asalverforkeplereationsthenuberofaxiuiterationswesethalley&039;sthodis15，buttheyneverreachedtheaxiuanyofourtegrations

&esp;&esp;thetervalofthedataoutputis200000d（～5yr）forthecalcutionfallneps（n±1，2，3），andabout8000000d（～21903yr）forthetegrationoftheouterfiveps（f±）

&esp;&esp;althoughnooutputfiltergwasdonewhenthenuricaltegrationswereprocess，weappliedalow-passfiltertotheraworbitaldataafterwehadpletedallthecalcutionsseesection41fororedetail

&esp;&esp;24errorestiation

&esp;&esp;241retiveerrorstotalenergyandangurontu

&esp;&esp;aordgtooneofthebasicpropertiefsyplectictegrators，whichnservethephysicallynservativeantitieswell（totalorbitalenergyandangurontu），ourlong-ternuricaltegrationsseetohavebeenperfordwithverysallerrorstheaveradretiveerrorftotalenergy（～10?9）andoftotangurontu（～10?11）havereaednearlynstantthroughoutthetegrationperiod（fig1）thespecialstartupprocedure，warstart，wouldhavereducedtheaveradretiveerrortotalenergybyaboutoneorderofagnitudeorore

&esp;&esp;retivenuricalerrorofthetotangurontuδaa0andthetotalenergyδee0ournuricaltegrationsn±1，2，3，whereδeandδaaretheabtechanofthetotalenergyandtotangurontu，respectively，ande0anda0aretheiritialvaesthehorizontanitisgyr

&esp;&esp;notethatdiffereneratgsystes，differentatheaticallibraries，anddifferenthardwarearchitecturesresultdifferentnuricalerrors，throughthevariationsround-offerrorhandlgandnuricalgorithstheupperpaneloffig1，wecanregnizethissituationthesecurnuricalerrorthetotangurontu，whichshouldberigorolypreserveduptoache-eprecision

&esp;&esp;242errorparylongitudes

&esp;&esp;scethesyplectiapspreservetotalenergyandtotangurontuofn-bodydynaicalsystesherentlywell，thedegreeoftheirpreservationaynotbeagoodasureoftheauracyofnuricaltegrations，especiallyasaasureofthepositionalerrorofps，ietheerrorparylongitudestoestiatethenuricalerrortheparylongitudes，weperfordthefollogproceduresweparedtheresultofouralong-tertegrationswithtesttegrations，whichspanuchshorterperiodsbuithuchhigherauracythantheategrationsforthispurpose，weperfordauchoreauratetegrationwithastepsizeof0125d（164oftheategrations）spanng3x105yr，startgwiththesaitialnditionsasthen?1tegrationwensiderthatthistesttegrationprovideswitha‘pseudo-true’tionofparyorbitalevotionnext，weparethetesttegrationwiththeategration，n?1fortheperiodof3x105yr，weseeadifferenceananoalieftheearthbeeentheotegrationf～052°（thecaseofthen?1tegration）thisdifferencecanbeextrapotedtothevae～8700°，about25rotationfearthafter5gyr，scetheerroroflongitudescreaseslearlywithtithesyplectiapsiirly，thelongitudeerrorofptocanbeestiatedas～12°thisvaeforptoisuchbetterthantheresultkoshitanakai（1996）wherethedifferenceisestiatedas～60°

&esp;&esp;3nuricalresults–ignceattherawdata

&esp;&esp;thissectionwebrieflyreviewthelong-terstabilityofparyorbitalotionthroughsnapshotfrawnuricaldatatheorbitalotionofpsdicateslong-terstabilityallofournuricaltegrations:noorbitalcrossgsnorcloseenuntersbeeenanypairofpstookpce

&esp;&esp;31neraldescriptionofthestabilityofparyorbits

&esp;&esp;first，webrieflylookattheneralcharacterofthelong-terstabilityofparyorbitsourterestherefocesparticurlyonthenerfourterrestrialpsforwhichtheorbitalti-scalesareuchshorterthanthoseoftheouterfivepsaswecanseeclearlyfrothepnarorbitalnfigurationsshownfigs2and3，orbitalpositionftheterrestrialpsdifferlittlebeeentheitiandfalpartofeachnuricaltegration，whichspansseveralgyrthelidlesdenotgthepresentorbitfthepsliealosiththeswarofdotseventhefalpartoftegrations（b）and（d）thisdicatesthatthroughouttheentiretegrationperiodthealostregurvariationfparyorbitalotionreanearlythesaastheyareatpresent

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