最新网址：hx 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，反正这写到现在，嚷嚷着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-terrations and stability of pary orbits our r syste

abstract

e present the results of very long-ter nurations of pary orbital otions over 109 -yr tie-spans cdg all ne ps a ick spection of our nuerical data shos that the pary otion， at least our siple dynaical odel， sees to be ite stable even over this very long tie-span a closer look at the loest-freency osciltions g a lo-pass filter shos the potentially diffive character of terrestrial pary otion， especially that of ercury the behaviour of the eentricity of ercury our tegrations is alitatively siir to the results fro jaces skar's secur perturbation theory eg eax～ 035 over ～± 4 gyr hoever， there are no apparent secur creases of eentricity or ction any orbital eleents of the ps， hich ay be revealed by still lonr-ter nurations e have al perfored a uple of trial tegrations cdg otions of the outer five ps over the duration of ± 5 x 1010 yr the result dicates that the three ajor renances the neptune–pto syste have been ataed over the 1011-yr tie-span

1 troduction

11defition of the proble

the estion of the stability of our r syste has been debated over several hundred years， sce the era of on the proble has attracted any fao atheaticians over the years and has pyed a central role the developent of non-lear dynaics and chaos theory hoever， e do not yet have a defite anser to the estion of hether our r syste is stable or not this is partly a result of the fact that the defition of the ter ‘stability’ is vague hen it is ed retion to the proble of pary otion the r syste actually it is not easy to give a clear， rigoro and physically ful defition of the stability of our r syste

a any defitions of stability， here e adopt the hill defition gdan 1993: actually this is not a defition of stability， but of stability e defe a syste as be unstable hen a close enunter ours ehere the syste fro a certa itial nfiguration chabers， etherill apap boss 1996 ito apap tanikaa 1999 a syste is defed as experiencg a close enunter hen o bodies approach one another ith an area of the rr hill radi otherise the syste is defed as beg stable henceforard e state that our pary syste is dynaically stable if no close enunter happens durg the a of our r syste， about ±5 gyr cidentally， this defition ay be repced by one hich an ourrence of any orbital crossg beeen either of a pair of ps takes pce this is becae e kno fro experience that an orbital crossg is very likely to lead to a close enunter pary and proary systea， kokubo apap ako 1999 of urse this stateent cannot be siply applied to systes ith stable orbital renances such as the neptune–pto syste

12previo studies and ais of this research

addition to the vagueness of the ncept of stability， the ps our r syste sho a character typical of dynaical chaos ssan apap isdo 19， 1992 the cae of this chaotic behaviour is no partly understood as beg a result of renance overppg urray apap hon 1999 lecar， frankl apap hon 2001 hoever， it ould reire tegratg over an enseble of pary syste all ne ps for a period verg several 10 gyr to thoroughly understand the long-ter evotion of pary orbits， sce chaotic dynaical systes are characterized by their strong dependence on itial nditions

fro that pot of vie， any of the previo long-ter nurations cded only the outer five ps ssan apap isdo 19 koshita apap nakai 1996 this is becae the orbital periods of the outer ps are er than those of the ner four ps that it is uch easier to follo the syste for a given tegration period at present， the lonst nurations published journals are those of duncan apap lissauer 1998 although their et as the effect of post-a-seence r ass loss on the stability of pary orbits， they perfored rations verg up to ～1011 yr of the orbital otions of the four jovian ps the itial orbital eleents and asses of ps are the sae as those of our r syste duncan apap lissauer's paper， but they decrease the ass of the sun gradually their nuerical experients this is becae they nsider the effect of post-a-seence r ass loss the paper nseently， they found that the crossg tie-scale of pary orbits， hich can be a typical dicator of the stability tie-scale， is ite sensitive to the rate of ass decrease of the sun hen the ass of the sun is close to its present vae， the jovian ps rea stable over 1010 yr， or perha lonr duncan apap lissauer al perfored four siir experients on the orbital otion of seven ps ven to neptune， hich ver a span of ～109 yr their experients on the seven ps are not yet prehensive， but it sees that the terrestrial ps al rea stable durg the tegration period， ar osciltions

on the other hand， his aurate sei-analytical secur perturbation theory skar 19， skar fds that r and irregur variations can appear the eentricities and ctions of the terrestrial ps， especially of ercury and ars on a tie-scale of several 109 yr skar 1996 the results of skar's secur perturbation theory should be nfired and vestigated by fully nurations

this paper e present preliary results of six long-ter nurations on all ne pary orbits， verg a span of several 109 yr， and of o other tegrations verg a span of ± 5 x 1010 yr the total eed tie for all tegrations is ore than 5 yr， g several dedicated pcs and orkstations one of the fundaental ncsions of our long-terrations is that r syste pary otion sees to be stable ters of the hill stability entioned above， at least over a tie-span of ± 4 gyr actually， our nurations the syste as far ore stable than hat is defed by the hill stability criterion: not only did no close enunter happen durg the tegration period， but al all the pary orbital eleents have been nfed a narro region both tie and freency doh pary otions are stochastic sce the purpose of this paper is to exhibit and overvie the results of our long-ter nurations， e sho typical exaures as evidence of the very long-ter stability of r syste pary otion for readers ho have ore specific and deeper terests our nuerical results， e have prepared a ebpa aess ， here e sho ra orbital eleents， their lo-pass filtered results， variation of deunay eleents and angur oentu deficit， and results of our siple tie–freency analysis on all of our tegrations

section 2 e

iefly exp our dynaical odel， nuerical ethod and itial nditions ed our tegrations section 3 is devoted to a description of the ick results of the nurations very long-ter stability of r syste pary otion is apparent both pary positions and orbital eleents a rough estiation of nuerical errors is al given section 4 goes on to a discsion of the lonst-ter variation of pary orbits g a lo-pass filter and cdes a discsion of angur oentu deficit section 5， e present a set of nurations for the outer five ps that spans ± 5 x 1010 yr section 6 e al discs the long-ter stability of the pary otion and its possible cae

2 description of the nurations

（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。）

23 nuerical ethod

e utilize a send-order isdo–hon syplectic ap as our ration ethod isdo apap hon 1991 koshita， yoshida apap nakai 1991 ith a special start-up procedure to reduce the truncation error of angle variables，‘ar start’saha apap treae 1992， 1994

the steize for the nurations is 8 d throughout all tegrations of the ne ps n±1，2，3， hich is about 111 of the orbital period of the nerost p ercury as for the deteration of steize， e partly follo the previo nuration of all ne ps ssan apap isdo 19， 72 d and saha apap treae 1994， 22532 d e rounded the decial part of the their steizes to 8 to ake the steize a ultiple of 2 order to reduce the auution of round-off error the putation processes retion to this， isdo apap hon 1991 perfored nurations of the outer five pary orbits g the syplectic ap ith a steize of 400 d， 11083 of the orbital period of jupiter their result sees to be aurate enough， hich partly jtifies our ethod of deter the steize hoever， sce the eentricity of jupiter ～005 is uch saller than that of ercury ～02， e need e care hen e pare these tegrations siply ters of steizes

the tegration of the outer five ps f±， e fixed the steize at 400 d

e adopt gas' f and g functions the syplectic ether ith the third-order halley ethod danby 1992 as a lver for kepler eations the nuber of axiu iterations e set halley's ethod is 15， but they never reached the axiu any of our tegrations

the terval of the data output is 200 000 d ～547 yr for the calcutions of all ne ps n±1，2，3， and about 8000 000 d ～21 903 yr for the tegration of the outer five ps f±

although no output filterg as done hen the nurations ere process， e applied a lo-pass filter to the ra orbital data after e had pleted all the calcutions see section 41 for ore detail

24 error estiation

241 retive errors total energy and angur oentu

aordg to one of the basic properties of syrators， hich nserve the physically nservative antities ell total orbital energy and angur oentu， our long-ter nurations see to have been perfored ith very sall errors the averad retive errors of total energy ～10?9 and of total angur oentu ～10?11 have reaed nearly nstant throughout the tegration period fig 1 the special startup procedure， ar start， ould have reduced the averad retive error total energy by about one order of nitude or ore

retive nuerical error of the total angur oentu δaa0 and the total energy δee0 our nurationsn± 1，2，3， here δe and δa are the abte chan of the total energy and total angur oentu， respectively， ande0anda0are their itial vaes the horizontal unit is gyr

note that different operatg systes， different atheatical li

aries， and different hardare architectures result different nuerical errors， through the variations round-off error handlg and nuoriths the upper panel of fig 1， e can regnize this situation the secur nuerical error the total angur oentu， hich should be rigoroly preserved up to ache-e precision

242 error pary longitudes

sce the syplectic a preserve total energy and total angur oentu of n-body dynaical systes herently ell， the degree of their preservation ay not be a good easure of the auracy of nurations， especially as a easure of the positional error of ps， ie the error pary longitudes to estiate the nuerical error the pary longitudes， e perfored the follog procedures e pared the result of our -terrations ith e test tegrations， hich span uch shorter periods but ith her auracy than the rations for this purpose， e perfored a uch ore aurate tegration ith a steize of 0125 d 164 of the rations spanng 3 x 105 yr， startg ith the sae itial nditions as the n?1 tegration e nsider that this test tegration provides ith a ‘eudo-true’ tion of pary orbital evotion next， e pare the test tegration ith the ration， n?1 for the period of 3 x 105 yr， e see a difference ean anoalies of the earth beeen the o tegrations of ～052° the case of the n?1 tegration this difference can be extrapoted to the vae ～8700°， about 25 rotations of earth after 5 gyr， sce the error of longitudes creases learly ith tie the syplectic ap siirly， the longitude error of pto can be estiated as ～12° this vae for pto is uch better than the result koshita apap nakai 1996 here the difference is estiated as ～60°

3 nuerical results – i gnce at the ra data

this section e

iefly revie the long-ter stability of pary orbital h e snahots of ra nuerical data the orbital otion of ps dicates long-ter stability all of our nurations: no orbital crossgs nor close enunters beeen any pair of ps took pce

31 neral description of the stability of pary orbits

first， e

iefly look at the neral character of the long-ter stability of pary orbits our terest here foces particurly on the ner four terrestrial ps for hich the orbital tie-scales are uch shorter than those of the outer five ps as e can see clearly fro the pnar orbital nfigurations shon figs 2 and 3， orbital positions of the terrestrial ps differ little beeen the itial and fal part of each nuration， hich spans several gyr the lid les denotg the present orbits of the ps lie alost ith the sar of dots even the fal part of tegrations b and d this dicates that throughout the entire tegration period the ar variations of pary orbital otion rea nearly the sae as they are at present

vertical vie of the four ner pary orbits fro the z -axis direction at the itial and fal parts of the tegrationsn±1 the axes units are au the xy -pne is set to the variant pne of r syste total angur oentua the itial part ofn+1 t 0 to 00547 x 10 9 yrb the fal part ofn+1 t 49339 x 10 8 to 496 x 10 9 yrc the itial part of n?1 t 0 to ?00547 x 109 yrd the fal part ofn?1 t ?39180 x 10 9 to ?39727 x 10 9 yr each panel， a total of 23 684 pots are plotted ith an terval of about 2190 yr over 547 x 107 yr lid les each panel denote the present orbits of the four terrestrial ps taken fro de245

the variation of eentricities and orbital ctions for the ner four ps the itial and fal part of the tegration n+1 is shon fig 4 as expected， the character of the variation of pary orbital eleents does not differ significantly beeen the itial and fal part of each tegration， at least for ven， earth and ars the eleents of ercury， especially its eentricity， seee to a significant extent this is partly becae the orbital tie-scale of the p is the shortest of all the ps， hich leads to a ore rapid orbital evotion than other ps the nerost p ay be nearest to stability this result appears to be reeent ith skar's 1994， 1996 expectations that r and irregur variations appear the eentricities and ctions of ercury on a tie-scale of several 109 yr hoever， the effect of the possible stability of the orbit of ercury ay not fatally affect the global stability of the hole pary syste to the sall ass of ercury e ill ention

iefly the long-ter orbital evotion of ercury ter section 4 g lo-pass filtered orbital eleents

the orbital otion of the outer five ps seeoroly stable and ite regur over this tie-span see al section 5

32 tie–freency a

although the pary otion exhibits very long-ter stability defed as the non-existence of close enunter events， the chaotic nature of pary dynaics can chan the osciltory period and aplitude of pary orbital radually over such long tie-spans even such slight fctuations of orbital variation the freency doa， particurly the case of earth， can potentially have a significant effect on its surface cliate systeh r tion variation cf berr 19