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：

lng-terintegratinsandstabilityfplanetaryrbitsinurslarsyste

abstrat

epresenttheresultsfverylng-ternuerialintegratinsfplanetaryrbitaltinsver109-yrtie-spansinludingallnineplanets.aquikinspetinfurnuerialdatashsthattheplanetarytin，atleastinursipledynaialdel，seestbequitestableevenverthisverylngtie-span.alserlkatthelest-frequenysillatinsusingal-passfiltershsustheptentiallydiffusiveharaterfterrestrialplanetarytin，espeiallythatferury.thebehaviurftheeentriityferuryinurintegratinsisqualitativelysiilarttheresultsfrjaqueslaskar'sseularperturbatinthery(e.g.eax～0.35ver～±4gyr).hever，therearenapparentseularinreasesfeentriityrinlinatininanyrbitaleleentsftheplanets，hihayberevealedbystilllnger-ternuerialintegratins.ehavealsperfredaupleftrialintegratinsinludingtinsftheuterfiveplanetsvertheduratinf±5x1010yr.theresultindiatesthatthethreeajrresnanesintheneptune–plutsystehavebeenaintainedverthe1011-yrtie-span.

1intrdutin

1.1definitinftheprble

thequestinfthestabilityfurslarsystehasbeendebatedverseveralhundredyears，sinetheerafnetn.theprblehasattratedanyfausatheatiiansvertheyearsandhasplayedaentralrleinthedevelpentfnn-lineardynaisandhasthery.hever，edntyethaveadefiniteansertthequestinfhetherurslarsysteisstablernt.thisispartlyaresultfthefatthatthedefinitinftheter‘stability’isvaguehenitisusedinrelatinttheprblefplanetarytinintheslarsyste.atuallyitisnteasytgivealear，rigrusandphysiallyeaningfuldefinitinfthestabilityfurslarsyste.

anganydefinitinsfstability，hereeadptthehilldefinitin(gladan1993):atuallythisisntadefinitinfstability，butfinstability.edefineasysteasbeingunstablehenalseenunterurssehereinthesyste，startingfraertaininitialnfiguratin(habers，etherillittanikaa1999).asysteisdefinedasexperieningalseenunterhentbdiesapprahneantherithinanareafthelargerhillradius.therisethesysteisdefinedasbeingstable.henefrardestatethaturplanetarysysteisdynaiallystableifnlseenunterhappensduringtheagefurslarsyste，abut±5gyr.inidentally，thisdefinitinaybereplaedbyneinhihanurrenefanyrbitalrssingbeteeneitherfapairfplanetstakesplae.thisisbeauseeknfrexperienethatanrbitalrssingisverylikelytleadtalseenunterinplanetaryandprtplanetarysystes(yshinaga，kkubakin1999).fursethisstateentanntbesiplyappliedtsystesithstablerbitalresnanessuhastheneptune–plutsyste.

1.2previusstudiesandaisfthisresearh

inadditintthevaguenessftheneptfstability，theplanetsinurslarsysteshaharatertypialfdynaialhas(sussanisd1988，1992).theausefthishatibehaviurisnpartlyunderstdasbeingaresultfresnaneverlapping(urraylear，franklinhlan2001).hever，ituldrequireintegratingveranenseblefplanetarysystesinludingallnineplanetsfraperidveringseveral10gyrtthrughlyunderstandthelng-terevlutinfplanetaryrbits，sinehatidynaialsystesareharaterizedbytheirstrngdependeneninitialnditins.

frthatpintfvie，anyfthepreviuslng-ternuerialintegratinsinludednlytheuterfiveplanets(sussankinshitanakai1996).thisisbeausetherbitalperidsftheuterplanetsaresuhlngerthanthseftheinnerfurplanetsthatitisuheasiertfllthesystefragivenintegratinperid.atpresent，thelngestnuerialintegratinspublishedinjurnalsarethsefdunanlissauer(1998).althughtheiraintargetastheeffetfpst-ain-sequeneslarasslssnthestabilityfplanetaryrbits，theyperfredanyintegratinsveringupt～1011yrftherbitaltinsfthefurjvianplanets.theinitialrbitaleleentsandassesfplanetsarethesaeasthsefurslarsysteindunanlissauer'spaper，buttheydereasetheassfthesungraduallyintheirnuerialexperients.thisisbeausetheynsidertheeffetfpst-ain-sequeneslarasslssinthepaper.nsequently，theyfundthattherssingtie-salefplanetaryrbits，hihanbeatypialindiatrftheinstabilitytie-sale，isquitesensitivettheratefassdereasefthesun.hentheassfthesunislsetitspresentvalue，thejvianplanetsreainstablever1010yr，rperhapslnger.dunanlissaueralsperfredfursiilarexperientsntherbitaltinfsevenplanets(venustneptune)，hihveraspanf～109yr.theirexperientsnthesevenplanetsarentyetprehensive，butitseesthattheterrestrialplanetsalsreainstableduringtheintegratinperid，aintainingalstregularsillatins.

nthetherhand，inhisauratesei-analytialseularperturbatinthery(laskar1988)，laskarfindsthatlargeandirregularvariatinsanappearintheeentriitiesandinlinatinsftheterrestrialplanets，espeiallyferuryandarsnatie-salefseveral109yr(laskar1996).theresultsflaskar'sseularperturbatintheryshuldbenfiredandinvestigatedbyfullynuerialintegratins.

inthispaperepresentpreliinaryresultsfsixlng-ternuerialintegratinsnallnineplanetaryrbits，veringaspanfseveral109yr，andfttherintegratinsveringaspanf±5x1010yr.thettalelapsedtiefrallintegratinsisrethan5yr，usingseveraldediatedpsandrkstatins.nefthefundaentalnlusinsfurlng-terintegratinsisthatslarsysteplanetarytinseestbestableintersfthehillstabilityentinedabve，atleastveratie-spanf±4gyr.atually，inurnuerialintegratinsthesysteasfarrestablethanhatisdefinedbythehillstabilityriterin:ntnlydidnlseenunterhappenduringtheintegratinperid，butalsalltheplanetaryrbitaleleentshavebeennfinedinanarrreginbthintieandfrequenydain，thughplanetarytinsaresthasti.sinethepurpsefthispaperistexhibitandvervietheresultsfurlng-ternuerialintegratins，eshtypialexaplefiguresasevideneftheverylng-terstabilityfslarsysteplanetarytin.frreadershhaverespeifianddeeperinterestsinurnuerialresults，ehavepreparedaebpage(aess)，hereeshrarbitaleleents，theirl-passfilteredresults，variatinfdelaunayeleentsandangularentudefiit，andresultsfursipletie–frequenyanalysisnallfurintegratins.

insetin2ebrieflyexplainurdynaialdel，nuerialethdandinitialnditinsusedinurintegratins.setin3isdevtedtadesriptinfthequikresultsfthenuerialintegratins.verylng-terstabilityfslarsysteplanetarytinisapparentbthinplanetarypsitinsandrbitaleleents.arughestiatinfnuerialerrrsisalsgiven.setin4gesntadisussinfthelngest-tervariatinfplanetaryrbitsusingal-passfilterandinludesadisussinfangularentudefiit.insetin5，epresentasetfnuerialintegratinsfrtheuterfiveplanetsthatspans±5x1010yr.insetin6ealsdisussthelng-terstabilityftheplanetarytinanditspssibleause.

2desriptinfthenuerialintegratins

（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。）

2.3nuerialethd

eutilizeasend-rderisd–hlansypletiapasurainintegratinethd(isdkinshita，yshidanakai1991)ithaspeialstart-uppreduretreduethetrunatinerrrfanglevariables，‘arstart’(sahatreaine1992，1994).

thestepsizefrthenuerialintegratinsis8dthrughutallintegratinsfthenineplanets(n±1，2，3)，hihisabut111ftherbitalperidftheinnerstplanet(erury).asfrthedeterinatinfstepsize，epartlyfllthepreviusnuerialintegratinfallnineplanetsinsussanisd(1988，7.2d)andsahatreaine(1994，22532d).erundedthedeialpartfthetheirstepsizest8takethestepsizeaultiplef2inrdertreduetheauulatinfrund-fferrrintheputatinpresses.inrelatintthis，isdhlan(1991)perfrednuerialintegratinsftheuterfiveplanetaryrbitsusingthesypletiapithastepsizef400d，110.83ftherbitalperidfjupiter.theirresultseestbeaurateenugh，hihpartlyjustifiesurethdfdeteriningthestepsize.hever，sinetheeentriityfjupiter(～0.05)isuhsallerthanthatferury(～0.2)，eneedseareheneparetheseintegratinssiplyintersfstepsizes.

intheintegratinftheuterfiveplanets(f±)，efixedthestepsizeat400d.

eadptgauss'fandgfuntinsinthesypletiaptgetheriththethird-rderhalleyethd(danby1992)asaslverfrkeplerequatins.thenuberfaxiuiteratinsesetinhalley'sethdis15，buttheyneverreahedtheaxiuinanyfurintegratins.

theintervalfthedatautputis200000d(～547yr)frthealulatinsfallnineplanets(n±1，2，3)，andabut8000000d(～21903yr)frtheintegratinftheuterfiveplanets(f±).

althughnutputfilteringasdnehenthenuerialintegratinsereinpress，eappliedal-passfilterttherarbitaldataafterehadpletedallthealulatins.seesetin4.1frredetail.

2.4errrestiatin

2.4.1relativeerrrsinttalenergyandangularentu

ardingtnefthebasiprpertiesfsypletiintegratrs，hihnservethephysiallynservativequantitiesell(ttalrbitalenergyandangularentu)，urlng-ternuerialintegratinsseethavebeenperfredithverysallerrrs.theaveragedrelativeerrrsfttalenergy(～10?9)andfttalangularentu(～10?11)havereainednearlynstantthrughuttheintegratinperid(fig.1).thespeialstartuppredure，arstart，uldhavereduedtheaveragedrelativeerrrinttalenergybyabutnerderfagnituderre.

relativenuerialerrrfthettalangularentuδaa0andthettalenergyδee0inurnuerialintegratinsn±1，2，3，hereδeandδaaretheabslutehangefthettalenergyandttalangularentu，respetively，ande0anda0aretheirinitialvalues.thehrizntalunitisgyr.

ntethatdifferentperatingsystes，differentatheatiallibraries，anddifferenthardarearhiteturesresultindifferentnuerialerrrs，thrughthevariatinsinrund-fferrrhandlingandnuerialalgriths.intheupperpanelffig.1，eanregnizethissituatinintheseularnuerialerrrinthettalangularentu，hihshuldberigruslypreserveduptahine-epreisin.

2.4.2errrinplanetarylngitudes

sinethesypletiapspreservettalenergyandttalangularentufn-bdydynaialsystesinherentlyell，thedegreeftheirpreservatinayntbeagdeasureftheaurayfnuerialintegratins，espeiallyasaeasurefthepsitinalerrrfplanets，i.e.theerrrinplanetarylngitudes.testiatethenuerialerrrintheplanetarylngitudes，eperfredthefllingpredures.eparedtheresultfurainlng-terintegratinsithsetestintegratins，hihspanuhshrterperidsbutithuhhigherauraythantheainintegratins.frthispurpse，eperfredauhreaurateintegratinithastepsizef0.125d(164ftheainintegratins)spanning3x105yr，startingiththesaeinitialnditinsasinthen?1integratin.ensiderthatthistestintegratinprvidesusitha‘pseud-true’slutinfplanetaryrbitalevlutin.next，eparethetestintegratiniththeainintegratin，n?1.frtheperidf3x105yr，eseeadiffereneineananaliesftheearthbeteenthetintegratinsf～0.52°(intheasefthen?1integratin).thisdiffereneanbeextraplatedtthevalue～8700°，abut25rtatinsfearthafter5gyr，sinetheerrrflngitudesinreaseslinearlyithtieinthesypletiap.siilarly，thelngitudeerrrfplutanbeestiatedas～12°.thisvaluefrplutisuhbetterthantheresultinkinshitanakai(1996)herethediffereneisestiatedas～60°.

3nuerialresults–i.glaneattheradata

inthissetinebrieflyreviethelng-terstabilityfplanetaryrbitaltinthrughsesnapshtsfranuerialdata.therbitaltinfplanetsindiateslng-terstabilityinallfurnuerialintegratins:nrbitalrssingsnrlseenuntersbeteenanypairfplanetstkplae.

3.1generaldesriptinfthestabilityfplanetaryrbits

first，ebrieflylkatthegeneralharaterfthelng-terstabilityfplanetaryrbits.urinterestherefusespartiularlyntheinnerfurterrestrialplanetsfrhihtherbitaltie-salesareuhshrterthanthseftheuterfiveplanets.aseanseelearlyfrtheplanarrbitalnfiguratinsshninfigs2and3，rbitalpsitinsftheterrestrialplanetsdifferlittlebeteentheinitialandfinalpartfeahnuerialintegratin，hihspansseveralgyr.theslidlinesdentingthepresentrbitsftheplanetsliealstithinthesarfdtseveninthefinalpartfintegratins(b)and(d).thisindiatesthatthrughuttheentireintegratinperidthealstregularvariatinsfplanetaryrbitaltinreainnearlythesaeastheyareatpresent.

vertialviefthefurinnerplanetaryrbits(frthez-axisdiretin)attheinitialandfinalpartsftheintegratinsn±1.theaxesunitsareau.thexy-planeissetttheinvariantplanefslarsystettalangularentu.(a)theinitialpartfn+1(t=0t0.0547x109yr).(b)thefinalpartfn+1(t=4.9339x108t4.9886x109yr).()theinitialpartfn?1(t=0t?0.0547x109yr).(d)thefinalpartfn?1(t=?3.9180x109t?3.9727x109yr).ineahpanel，attalf23684pintsareplttedithanintervalfabut2190yrver5.47x107yr.slidlinesineahpaneldentethepresentrbitsfthefurterrestrialplanets(takenfrde245).