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：

    longternbsp;integrations and stabilityplary orbitsour solar systebr>

    abstract

    we present the resultsvery longternbsp;nurical integrationsplary orbital tions over 109tispans including all nin inspectionour nurical data shows that the plary tion，leastour sile dynacal del， seebe quite stable even over this very lon lookthe lowestfrequency oscillations usinowpass filter showsthe potentially diffusive characterterrestrial plary tion， especially thaehaviourthe entricitymercurr integrationsqualitatively silarthe results fronbsp;jacques laskar&039;s secular perturbation theory e.g.0.35 over， there areapparent secular increasesentricityinclinationany orbital elentsthe pls， which yrevealedstill lonrternbsp;nurica have also perforoupletrial integrations including tionsthe outer five pls over the duration of51010 yr. the result indicates that the three jor resonancesthe neptune–pluto systenbsp;have been intained over the 1011yr tispan.

    1 introduction

    1.1definitionthe problebr>

    the questionthe stabilitr solar systenbsp;has been debated over several hundred years， since the erroblenbsp;has attractedfaus theticians over the years and has playeentral rolethe developntnonlinear dynacs and chao，do not yet havefinite answerthe questionwhether our solar systenbsp;is stable opartlesultthe fact that the definitionthe ternbsp;stabilityvague whenis usedrelationthe problenbsp;of plary tionthe solar syste actuallyis not easygivlear， rigorous and physically aningful definitionthe stabilitr solar syste

    angdefinitionsstability， hereadopt the hill definition gladn 1993: actually thisnoefinitionstability， buefinystenbsp;as bing unstable whelose encounter urs sowherethe syste starting fronbsp;a certain initial configuration chaers， wetherill & boss 1996; ito & tanikaw;is definedexperiencinlose encounter when two bodies approach one another withinareathe larr hil the systenbsp;is definedbeinstate that our plary systenbsp;is dynacally stableno close encounter happens during the aour solar syste about ， this definition yreplacedonewhichurrenceany orbital crossing between eithera pairpls takebecauseknow fronbsp;experience thatorbital crossingvery likelyleada close encounterplary and prlary syste yoshinaga， kokubo & makin course this statent cannotsily appliedsyste with stable orbital resonances suchthe neptune–pluto syste

    1.2previous studies andof this research

    in additionthe vaguenessthe conceptstability， the plsour solar systenbsp;shoharacter typicaldynacal chaos sussn & wisdonbsp;1988，  causethis chaotic behaviournow partly understoodbeinesultresonance overlapping murray & holn 1999; lecar， franklin & hol，would require integrating overenseleplary syste including all nine pls foeriod covering severalgyrthoroughly understand the longternbsp;evolutionplary orbits， since chaotic dynacal syste are characterizedtheir strong dependenceinitial conditions.

    fronbsp;that pointview，of the previous longternbsp;nurical integrations included only the outer five pls sussn & wisdonbsp;1988; kinoshita & nakabecause the orbital periodsthe outer pls arech lonr than thosethe inner four pls thatiseasierfollow the systenbsp;foiven integratio present， the lonst nurical integrations publishedjournals are thoseduncan & lissaue theirtart was the effectpostinsequence solarlossthe stabilityplary orbits， they perfordintegrations coveringto 1011of the orbital tionsthe four jovia initial orbital elents and ssespls are theas thoseour solar systenbsp;in duncan & lissauer&039;s paper， but they decrease theof the sun graduallytheir nuricabecause they consider the effectpostinsequence solarlossth， they found that the crossing tiscaleplary orbits， which cana typical indicatorthe instability tiscale，quite sensitivethe ratess decreaseth theof the suncloseits present value， the jovian pls rein stable over 1010 yr，perhap & lissauer also perford four silar experintsthe orbital tionseven pls venusneptune， which covepan109 yr. their experintsthe seven pls are not yet&nbspprehensive， butsee that the terrestrial pls also rein stable during the integration period， intaining alst regular oscillations.

    on the other hand，his urate seanalytical secular perturbation theory laskar 1988， laskar finds that lar and irregular variations can appearthe entricities and inclinationsthe terrestrial pls， especiallymercury and marsa tiscaleseveral 109laska resultslaskar&039;s secular perturbation theory shouldconfird and investigatedfully nurical integrations.

    in this paperpresent prelinary resultssix longternbsp;nurical integrationsall nine plary orbits， coverinpanseveral 109 yr， andtwo other integrations coverinpan of51010 yr. the total elapsedfor all integrationsre than 5 yr， using several dedicated pcsof the fundantal conclusionsour longternbsp;integrationsthat solar systenbsp;plary tion seebe stableterthe hill stability ntioned above，least oveispan of，our nurical integrations the systenbsp;was farstable than whatdefinedthe hill stability criterion: not only didclose encounter happen during the integration period， but also all the plary orbital elents have been confineda narrow region bothti and frequency doin， though plary tionsthe purposethis paperto exhibit and overview the resultsour longternbsp;nurical integrations，show typical exale figuresevidencethe very longternbsp;stabilitysolar systenbsp;plar readers who havespecific and deeper interestsour nurical results，have prepareebpa ess ， whereshow raw orbital elents， their lowpass filtered results， variationdelaunay elents and angular ntunbsp;deficit， and resultsour sile ti–frequency analysisallour integrations.

    in section 2briefly explain our dynacal del， nurical thod and initial conditions usedou 3devoteda descriptionthe quick resultsthe nurica longternbsp;stabilitysolar systenbsp;plary tionapparent bothplary positions and orbita estitionnurical errorsals 4 goestiscussionthe lonstternbsp;variationplary orbits usinowpass filter and includeiscussionangular ntunbsp; section 5，presenetnurical integrations for the outer five pls that spans51010 yr.section 6also discuss the longternbsp;stabilitythe plary tion and its possible cause.

    2 descriptionthe nurical integrations

    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。

    2.3 nurical thod

    we utilizecondorder wisdoholn sylectic pourintegration thod wisdonbsp;& holn 1991; kinoshita， yoshida & nakai 1991 witpecial startup procedurereduce the truncation errorangle variables，warnbsp;startsaha & treine 1992， 1994.

    the stepsize for the nurical integrations8 d throughout all integrationsthe nine pls n1，2，3， whichabout 111the orbital periodthe innersor the deternationstepsize，partly follow the previous nurical integrationall nine plssussn & wisdonbsp;1988， 7.2 d and saha & treine 1994， 22532 d.rounded the decil partthe their stepsizes8ke the stepsiztiple2orderreduce the ulationroundoff errorthe&nbspputatio relationthis， wisdonbsp;& holn 1991 perford nurical integrationsthe outer five plary orbits using the sylectiittepsize400 d， 110.83the orbital perioesult seebe urate enough， which partly justifies our thoddeterning th， since the entricityjupiter 0.05ch sller than thatmercury 0.2，needcare when we&nbsppare these integrations silyterstepsizes.

    in the integrationthe outer five pls f，fixed the stepsize400 d.

    we adopt gauss&039; f anunctionsthe sylectiother with the thirdorder halley thod danby 1992a solver for keple nuerxinbsp;iterationssethalley&039;s thod15， but they never reached the xinbsp;in anr integrations.

    the intervalthe data output200 000 d 547for the calculationsall nine pls n1，2，3， and about 8000 000 d903for the integrationthe outer five pls f.

    althoughoutput filtering was done when the nurical integrations wereprocess，applieowpass filterthe raw orbital data afterhad&nbsppleted allsection 4.1 fordetail.

    2.4 error estition

    2.4.1 relative errorstotal energy and angular ntubr>

    ordingonethe basic propertiessylectic integrators， which conserve the physically conservative quantities well total orbital energy and angular ntu our longternbsp;nurical integrations seenbsp;to have been perford with veryaverad relative errorstotal energy 109 andtotal angular ntunbsp;1011 have reined nearly constant throughout the integration perio special startup procedure， warnbsp;start， would have reduced the averad relative errortotal energyabout one ordergnitudere.

    relative nurical errorthe total angular ntunbsp;δaa0 and the total energy δee0our nurical integrationsn 1，2，3， whereandare the absolute chanthe total energy and total angular ntu respectively， ande0anda0are their initia horizontal unitgyr.

    note that different operating syste， different thetical libraries， and different hardware architectures resultdifferent nurical errors， through the variationsroundoff error handling and nurica the upper panel o，can recognize this situationthe secular nurical errorthe total angular ntu which shouldrigorously preservedto chinee precision.

    2.4.2 errorplary longitudes

    since the sylecticpreserve total energy and total angular ntunbsp;of nbody dynacal syste inherently well， the degreetheir preservatioota good asurethe uracynurical integrations， especiallya asurethe positional errorpls， i.e. the errorplar estite the nurical errorthe plary longitudes，perford the followin&nbsppared the resultourlongternbsp;integrations withtest integrations， which spanshorter periods but withhigher uracy than thhis purpose，perford are urate integration with a  d 164theintegrations spanning 3105 yr， starting with theinitial conditionsin thonsider that this test integration provideswitseudotrue solutionplary orbita， we&nbsppare the test integration with theintegration， n1. for the period3105 yr，seifferencean anoliesthe earth between the two  the casethifference canextrapolatedthe value 8700， aboutrotationsearth after 5 gyr， since the errorlongitudes increases linearly within the sylectic p. silarly， the longitude errorpluto canestited12. this value for plutoch better than the resultkinoshita & nakai 1996 where the differenceestited60.

    3 nurical results – i. glancethe raw data

    in this sectionbriefly review the longternbsp;stabilityplary orbital tion throughsnapshotsraw nurica orbital tionpls indicates longternbsp;stabilityallour nurical integrations:orbital crossings nor close encounters between any pairpls took place.

    3.1 neral descriptionthe stabilityplary orbits

    first，briefly lookthe neral characterthe longternbsp;stabilityplar interest here focuses particularlythe inner four terrestrial pls for which the orbital tiscales areshorter than thosethe outer fivcan see clearly fronbsp;the planar orbital configurations shownfigs 2 and 3， orbital positionsthe terrestrial pls differ little between the initial and final parteach nurical integration， which spans severa solid lines denoting the present orbitsthe pls lie alst within the swarnbsp;of dots eventhe final partintegrationnd d. this indicates that throughout the entire integration period the alst regular variationsplary orbital tion rein nearly theas they arepresent.

    vertical viewthe four inner plary orbits fronbsp;thxis directionthe initial and final partsth axes units are au. theplanesetthe invariant planesolar systenbsp;total angular ntua the initial part  9 yr.b the final part  84.988610 9 yr.c the initial partn1 t 00.0547109 yr.d the final part ofn1t 3.918010 93.972710 9 yr.each panel， a total23 684 points are plotted withintervalabout 2190over 5.47107. solid lineseach panel denote the present orbitsthe four terrestrial pls taken fronbsp;de245.

    the variationentricities and orbital inclinations for the inner four plsthe initial and final partthe integrationis showxpected， the characterthe variationplary orbital elents does not differ significantly between the initial and final parteach integration，least for venus， earthelentsmercury， especially its entricity， seenbsp;to chana significanpartly because the orbital tiscaletheis the shortestall the pls， which leadsarapid orbital evolution than other pls; the innerstynearesesult appearsbeso agreent with laskar&039;s 1994， 1996 expectations that lar and irregular variations appearthe entricities and inclinationsmercurya tiscaleseveral 109 yr. however， the effectthe possible instabilitythe orbitmercurot fatally affect the global stabilitythe whole plary systenbsp;owingthe sllo will ntion briefly the longternbsp;orbital evolutionmercury latersection 4 using lowpass filtered orbital elents.

    the orbital tionthe outer five pls see rigorously stable and quite regular over this tispan see also section 5.

    3.2 ti–frequency ps

    although the plary tion exhibits very longternbsp;stability definedthe nonexistenceclose encounter events， the chaotic natureplary dynacs can chan the oscillatory period and alitudeplary orbital tion gradually over such lon such slight fluctuationsorbital variationthe frequency doin， particularlythe caseearth， can potentially havignificant effectits surface clite systenbsp;through solar insolation variation cf. berr 1988.

    to giveoverviewthe longternbsp;chanperiodicityplary orbital tion，perfordfast fourier transfortions ffts along theaxis， and superposed the resulting periodgradraw twodinsional ti–frequency ps. the specific approachdrawing these ti–frequencyin this papervery sile –siler than the wavelet analysislaskar&039;s 1990， 1993 frequency analysis.

    divide the lowpass filtered orbital data intofragntsthengtheach data segnt shoulda ltiple2orderapply the fft.

    each fragntthe data haar overlapping part: for exale， when the ith data begins fronbsp;tti and endsttit， the next data segnt rans fronbsp;tiδttiδtt， where δtt.continue this division untilreacertain nuer nwhich tnt reaches the total integration length.

    we applyffteachthe data fragnts， and obtairequency diagra.

    in each frequency diagranbsp;obtained above， the strengthperiodicity canreplaceda greyscalecolour chart.

    we perfornbsp;the replacent， and connect all the greyscalecolour charts into one graph for eac horizontal axisthese new graphs shouldthe ti， i.e. the starting tiseach fragntdata ti， where i 1，…， n. the vertical axis represents the periodfrequencythe oscillationorbital elents.

    we have adoptedfft becauseits overwhelng speed， since the auntnurical databe dposed into frequency&nbspponentsterribly hu several tensgbytes.

    a typical exalethe ti–frequencreatedthe above proceduresshowna greyscale diagranbsp;a， which shows the variationperiodicitythe entricity and inclinationearthn fig. 5， the dark area shows thattheindicatedthe valuethe abscissa， the periodicity indicatedthe ordinatestronr thanthe lighter area around it.can recognize fronbsp;thihat the periodicitythe entricity and inclinationearth only chans slightly over the entire period coveredthearly regular trendqualitatively thein other integrations and for other pls， although typical frequencies differbyand elentelent.

    4.2 longternbsp;exchanorbital energy and angular ntubr>

    we calculate very longperiodic variation and exchanplary orbital energy and angular ntunbsp;using filtered delaunay elents l， g， h. g anre equivalentthe plary orbital angular ntunbsp;and its vertical&nbspponent per unit ss. lrelatedthe plary orbital enerr unita the systenbsp;is&nbsppletely linear， the orbital energy and the angular ntunbsp;in each frequency binbthe plary systenbsp;can causeexchanenergy and angular ntunbsp;in the frequenc alitudethe lowestfrequency oscillation should increasethe systenbsp;is unstable and breaks dow， sucytonbsp;of instabilitynot pronentour longternbsp;integrations.

    i， the total orbital energy and angular ntunbsp;of the four inner pls and all nine pls are shown for integration n2. the upper three panels show the longperiodic variationtotal energy denoted ase e0， total angular ntunbsp; g g0， and the vertical&nbspponenthof the inner four pls calculated fronbsp;the lowpass filtered delauna， g0，denote the initial valueseac absolute difference fronbsp;the initial valuesplottedth lower three panelseach figure showee0，gg0 andhh0the totalnin fluctuation shownthe lower panelsvirtually entirelesultthe ssive jovian pls.

    coaring the variationsenergy and angular ntunbsp;of the inner four pls and all nine pls，is apparent that the alitudesthosethe inner pls aresller than thoseall nine pls: the alitudesthe outer five pls arelarr than thosethe inne does notthat the inner terrestrial plary subsystenbsp;isstable than the outer one: thissilesultthe relative sllnessthe ssesthe four terrestrial pls&nbsppared with thosethe outer jovia thingnoticethat the inner plary subsystenbsp;yunstablerapidly than the outer one becauseits shorter orbita canseenthe panels denoted asinner 4 i the lonrperiodic and irregular oscillations areapparent thanthe panels denoted astotal 9. actually， the fluctuationstheinner 4 panels area lar extenta resultthe orbital variationth，cannot neglect the contribution fronbsp;other terrestrial pls，we will seesubsequent sections.

    4.4 longternbsp;couplingseveral neighbouringpairs

    letseeindividual variationsplary orbital energy and angular ntunbsp;expressedthe lowpass filtered delaunaandshow longternbsp;evolutionthe orbital energyeachand the angular ntunbsp;inanotice thatpls fornbsp;apparent pairsterorbital energy and angular ntunbsp; particular， venus and eartha typica the figures， they show negative correlationsexchanenergy and positive correlationsexchanangular ntu the negative correlationexchanorbital energy ans that the two pls fornbsp;a closed dynacal systenbsp;in terthe orbita positive correlationexchanangular ntunbsp;ans that the two pls are siltaneously under certain longternbsp; for perturbers are jupiteri，can see that mars showositive correlationthe angular ntunbsp;variationthe venus–earth syste mercury exhibits certain negative correlationsthe angular ntunbsp;versus the venus–earth syste which seebeaction causedthe conservationangular ntunbsp;in the terrestrial plary subsyste

    itnot clearthewhy the venus–earth pair exhibitegative correlationenergy exchan anositive correlationangular ntunbsp; y possibly explain this through observing the neral fact that there aresecular terplary sejor axesto secondorder perturbation theories cf. brouwer & clence 1961; baletti & puc ans that the plary orbital energy whichdirectly relatedthe sejor axihtch less affectedperturbing pls thanthe angular ntunbsp;exchan which relatese. hence， the entricitiesvenus and earth candisturbed easilyjupiter and saturn， which resultsa positive correlationthe angular ntunbsp; the other hand， the sejor axesvenus and earth are less likelybe disturbedthe jovia the energy exchan ylited only within the venus–earth pair， which resultsa negative correlationthe exchanorbital energythe pair.

    as for the outer jovian plary subsyste jupiter–saturn and uranus–neptune seenbsp;todynaca， the strengththeir couplingnotstrong&nbsppared with thatthe venus–earth pair.

    551010yr integrationsouter plary orbits

    since the jovian plary sses arelarr than the terrestrial plary sses，treat the jovian plary systenbsp;asindependent plary systenbsp;in terthe studyits dynaca，addeoupletrial integrations that span51010 yr， including only the outer five pls the four jovian pls plu results exhibit the rigorous stabilitythe outer plary systenbsp;over this lon configuration， and variationentricities and inclination show this very longternbsp;stabilitythe outer five plsboth theand the frequencdo not showhere， the typical frequencythe orbital oscillationpluto and the other outer plsalst constant during these very longternbsp;integration periods， whichdenstratedthe ti–frequencyon our webpa.

    in these two integrations， the relative nurical errorthe total energy was 106 and thatthe total angular ntunbsp;was 1010.

    5.1 resonancesthe neptune–pluto systebr>

    kinoshita & nakai 1996 integrated the outer five plary orbits  they found that four jor resonances between neptune and pluto are intained during the whole integration period， and that the resonances ythecausesthe stabilitythe orbior four resonances foundprevious research arhe following description，λ denotes thelongitude，the longitudethe ascending node andis the longitud anenote pluto and neptune.

    mean tion resonance between neptune and pluto 3:2. the critical argunt32 λnp librates around 180 withalitudeaboutanibration periodabout 2104 yr.

    the arguntperihelionpluto pθ2pp librates aroundwiteriodabout 3.8106 yr. the donant periodic variationsthe entricity and inclinationpluto are synchronized with the librationits argunt oanticipatedthe secular perturbation theory constructedkozai 1962.

    the longitudethe nodepluto referredthe longitudethe nodeneptune，θ3pn， circulates and the periodthis circulationequalthe periodθbes zero， i.e. the longitudesascending nodesneptune and pluto overlap， the inclinationpluto besthe entricity bes ninbsp;and the arguntperihelion bes 90. whenbes 180， the inclinationpluto besthe entricity bes xinbsp;and the arguntperihelion bes 9 & benson 1971 anticipated this typeresonance， later confirdmilani， nobili & carpino 1989.

    an argunt θ4pn 3librates around 180 witong period， 5.7108 yr.

    in our nurical integrations， the resonances i–iii are well intained， and variationthe critical argunts θ1，θ2，θ3 rein silar during the whole integration period figs 14–16 . however， the fourth resonanceappearsbe different: the critical arguntalternates libration and circulation over a 1010yr tiscalan interesting fact that kinoshita & nakai&039;s 1995， 1996 shorter integrations were not abledisclose.

    6 discussion

    what kinddynacal chanisnbsp;intains this longternbsp;stabilitythe plary systenbsp;we can iediately thinktwo jor features that yresponsible for the longternbsp;， there seenbsp;tono significant lowerorder resonancestion and secular between any pair ang the nin and saturn are closea 5:2tion resonance the faus great inequality， but not justthe resonanc resonanceause the chaotic naturethe plary dynacal tion， but they are notstrongto destroy the stable plary tion within the lifetithe real solar syste the second feature， whichthinkre iortant for the longternbsp;stabilitr plary systethe differencedynacal distance between terrestrial and jovian plary subsyste ito & tanikawa 1999， asure plary separationsthe tual hill radii r， separations ang terrestrial pls are greater than 26rh， whereas those ang jovian pls are less tha differencedirectly relatedthe difference between dynacal featuresterrestrial and jovia pls have sller sses， shorter orbital periods and wider dynaca are strongly perturbedjovian pls that have larr sses， lonr orbital periods and narrower dynaca pls are not perturbedany other ssive bodies.

    the present terrestrial plary systenbsp;is still being disturbedthe ssive jovia， the wide separation and tual interaction ang the terrestrial pls renders the disturbance ineffective; the degreedisturbancejovian plsoejordergnitudethe entricityjupiter， since the disturbance causedjovian plsa forced oscillation havingalitude oentricity， for exal，far fronbsp;sufficientprovoke instabilitythe terrestrial pls having sucide separation aassu that the present wide dynacal separation ang terrestrial pls > 26rhprobably onethesignificant conditions for intaining the stabilitythe plary systenbsp;over a 109y detailed analysisthe relationship between dynacal distance between pls and the instability tiscalesolar systenbsp;plary tionnow ongoing.

    although our nurical integrations span the lifetithe solar syste the nuerintegrationsfar fronbsp;sufficientfill the initial phasnecessaryperfornbsp;re andnurical integrationsconfirnbsp;and exanedetail the longternbsp;stabilitr plary dynacs.

    以上文段引自 ito， t.& tanikawa， k. longternbsp;integrations and stabilityplary orbitsour solar syst， 483–500 2002

    这只是作者君参考的一篇文章，关于太阳系的稳定性。

    还有其他论文，不过也都是英文的，相关课题的中文文献很少，那些论文下载一篇要九美元nature真是暴利，作者君写这篇文章的时候已经回家，不在检测中心，所以没有数据库的使用权，下不起，就不贴上来了。