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
to give an overvie of the long-tere periodicity pary orbital otion， e perfored any fast fourier transforations ffts along the tie axis， and superposed the resultg periodgras to dra o-diensional tie–freency a the specific approach to drag these tie–freency a this paper is very siple – uch sipler than the avelet analysis or skar's 1990， 1993 freency analysis
divide the lo-pass filtered orbital data to ents of the sath the length of each data sent should be a ultiple of 2 order to apply the fft
each frant of the data has a r overppg part: for exaple， hen the ith data begs fro tti and ends at tti+t， the next data ses fro ti+δt≤ti+δt+t， here δt?t e ntue this division until e reach a certa nuber n by hich tn+t reaches the total tegration length
e apply an fft to each of the data frants， and obta n freency diagras
each freency diagra obtaed above， the strength of periodicity can be repced by a grey-scale or lour chart
e perfor the repceent， and nnect all the grey-scale or lour charts to one graph for each tegration the horizontal axis of these ne graphs should be the tie， ie the startg ties of each frant of data ti， here i 1，…， n the vertical axis represents the period or freency of the osciltion of orbital eleents
e have adopted an fft becae of its overhel speed， sce the aount of nuerical data to be deposed to freency ponents is terribly hu several tens of gbytes
a typical exaple of the tie–freency ap created by the above procedures is shon a grey-scale diagra 5， hich shos the variation of periodicity the eentricity and ction of earth n+2 tegration fig 5， the dark area shos that at the tie dicated by the vae on the abscissa， the periodicity dicated by the ordate is stronr than the lighter area around it e can regnize fro this ap that the periodicity of the eentricity and ction of earth only chans slightly over the entire period vered by the n+2 tegration this nearly regur trend is alitatively the sae other tegrations and for other ps， although typical freencies differ p by p and eleent by eleent
42 long-tere of orbital energy and angur oentu
e calcute very long-periodic variation and exchan of pary orbital energy and angur filtered deunay ele， h g and h are eivalent to the pary orbital angur oentu and its vertical ponent per unit ass l is reted to the pary orbital energy e per unit ass as e?μ22l2 if the syste is pletely lear， the orbital energy and the angur oentu each freency b t be nstant non-learity the pary syste can cae an exchan of energy and angur oentu the freency doa the aplitude of the loest-freency osciltion should crease if the syste is unstable and
eaks don gradually hoever， such a sypto of stability is not proent our long-terrations
fig 7， the total orbital energy and angur oentu of the four ner ps and all ne ps are shon for tegration n+2 the upper three panels sho the long-periodic variation of total energy denoted ase- e0， total angur o- g0， and the vertical ponent h- h0 of the ner four ps calcuted fro the lo-pass filtered deunay ele0， h0 denote the itial vaes of each antity the abte difference fro the itial vaes is plotted the panels the loer three panels each figure shoe-e0，g-g0 andh-h0 of the total of ne ps the fctuation shon the loer panels is virtually entirely a result of the assive jovian ps
the variations of energy and angur oentu of the ner four ps and all ne ps， it is apparent that the aplitudes of those of the ner ps are uch saller than those of all ne ps: the aplitudes of the outer five ps are er than those of the ner ps this does not ean that the ner terrestrial pary subsyste is ore stable than the outer one: this is siply a result of the retive sallness of the asses of the four terrestrial ps pared ith those of the outer jovian ps another thg e notice is that the ner pary subsyste ay bee unstable ore rapidly than the outer one becae of its shorter orbital tie-scales this can be seen the panels denoted asner 4 fig 7 here the lonr-periodic and irregur osciltions are ore apparent than the panels denoted astotal 9 actually， the fctuations thener 4 panels are to a r extent as a result of the orbital variation of the ercury hoever， e cannot neglect the ntribution fro other terrestrial ps， as e ill see subseent sections
44 long-ter of several neighbourg p pairs
let see e dividual variations of pary orbital energy and angur oentu expressed by the lo-pass filtered deunay eles 10 and 11 sho long-ter evotion of the orbital energy of each p and the angur oentu n+1 and n?2 tegrations e notice that e ps for apparent pairs ters of orbital energy and angur oe particur， ven and earth ake a typical pair the figures， they sho negative rretions exchan of energy and positive rretions exchan of angur oentuative rretion exchan of orbital energy eans that the o ps for a closed dynaical syste ters of the orbital energy the positive rretion exchan of angur oentu eans that the o ps are siultaneoly under certa long-ter perturbations candidates for perturbers are jupiter and saturn al fig 11， e can see that ars shos a positive rretion the angur oentu variation to the ven–earth syste ercury exhibits certa negative rretions the angur oentu vers the ven–earth syste， hich sees to be a reaction caed by the nservation of angur oentu the terrestrial pary subsyste
it is not clear at the oent hy the ven–earth pair exhibits a negative rretion energy exchan and a positive rretion angur oe e ay possibly exp this through observg the neral fact that there are no secur ters pary seiajor axes up to send-order perturbation theories cf
ouer apap cleence 1961 boaletti apap pucao 1998 this eans that the pary orbital energy hich is directly reted to the seiajor axis a ht be uch less affected by perturbg ps than is the angur oe hich retes to e hence， the eentricities of ven and earth can be disturbed easily by jupiter and saturn， hich results a positive rretion the angur oe on the other hand， the seiajor axes of ven and earth are less likely to be disturbed by the jovian ps th the energy exchan ay be liited only ith the ven–earth pair， hich results a negative rretion the exchan of orbital energy the pair
as for the outer jovian pary subsyste， jupiter–saturn and uran–neptune see to ake dynaical pairs hoever， the strength of their uplg is not as strong pared ith that of the ven–earth pair
5 ± 5 x 1010-yr tegrations of outer pary orbits
sce the jovian pary asses are er than the terrestrial pary asses， e treat the jovian pary syste as an dependent pary syste ters of the study of its dynaical stability hence， e added a uple of trial tegrations that span ± 5 x 1010 yr， cdg only the outer five ps the four jovian ps ps pto the results exhibit the rigoro stability of the outer pary syste over this long tie-span orbital nfigurations fig 12， and variation of eentricities and ctions fig 13 sho this very long-ter stability of the outer five ps both the tie and the freency doh e do not sho a here， the typical freency of the orbital osciltion of pto and the other outer ps is alost nstant durg these very long-terration periods， hich is deonstrated the tie–freency a on our ebpa
these o tegrations， the retive nuerical error the total energy as ～10?6 and that of the total angur oentu as ～10?10
51 renances the neptune–pto syste
koshita apap nakai 1996 tegrated the outer five pary orbits over ± 55 x 109 yr they found that four ajor renances beeen neptune and pto are ataed durg the hole tegration period， and that the renances ay be the a caes of the stability of the orbit of pto the ajor four renances found previo research are as follos the follog description，λ denotes the itude，Ω is the longitude of the ascendg node and ? is the longitude of perihelion subscripts p and n denote pto and neptune
ean otion renance beeen neptune and pto 3:2 the critical arguent θ1 3 λp? 2 λn??p li
ates around 180° ith an aplitude of about 80° and a li
ation period of about 2 x 104 yr
the arguent of perihelion of pto pθ2?p?Ωp li
ates around 90° ith a period of about 38 x 106 yr the doant periodic variations of the eentricity and ction of pto are synchronized ith the li
ation of its arguent of perihelion this is anticipated the secur perturbation theory nstructed by kozai 1962
the longitude of the node of pto referred to the longitude of the node of neptune，θ3Ωp?Ωn， circutes and the period of this circution is eal to the period of θ2 li
ation hen θ3 bees zero， ie the longitudes of ascendg nodes of neptune and pto overp， the ction of pto bees axiu， the eentricity bees iu and the arguent of perihelion bees 90° hen θ3 bees 180°， the ction of pto bees iu， the eentricity bees axiu and the arguent of perihelion bea illias apap benn 1971 anticipated this type of renance， ter nfired by i， nobili apap carpo 1989
an arguent θ4?p??n+ 3 Ωp?Ωn li
ates around 180° ith a long period，～ 57 x 108 yr
our nurations， the renances i–iii are ell ataed， and variation of the critical arguents θ1，θ2，θ3 rea si the hole tegration period figs 14–16 hoever， the fourth renance iv appears to be different: the critical arguent θ4 alternates li
ation and circution over a 1010-yr ti 17 this is an terestg fact that koshita apap nakai's 1995， 1996 shorter tegrations ere not able to disclose
6 discsion
hat kd of dynaical echanis atas this long-ter stability of the pary syste? e can iediately thk of o ajor features that ay be responsible for the long-ter stability first， there see to be no significant loer-order renances ean otion and secur beeen any pair a the ne ps jupiter and saturn are close to a 5:2 ean otion renance the fareat eality’， but not jt the renance zone higher-order renances ay cae the chaotic nature of the pary dynaical otion， but they are not strong as to destroy the stable pary otion ith the lifetie of the real r syste the send feature， hich e thk is ore iportant for the long-ter stability of our pary syste， is the difference dynaical distance beeen terrestrial and jovian pary subsystes ito apap tanikaa 1999， 2001 hen e easure pary separations by the utual hill radii r， separations a terrestrial ps are greater than 26rh， hereas those a jovian ps are less than 14rh this difference is directly reted to the difference beeen dynaical features of terrestrial and jovian ps terrestrial ps have saller asses， shorter orbital periods and ider dynaical separation they are strongly perturbed by jovian ps that have rr er orbital periods and narroer dynaical separation jovian ps are not perturbed by any other assive bodies
the present terrestrial pary syste is still beg disturbed by the assive jovian ps hoever， the ide separation and utual teraction a the terrestrial ps renders the disturbance effective the degree of disturbance by jovian ps is oejorder of nitude of the eentricity of jupiter， sce the disturbance caed by jovian ps is a forced osciltion havg an aplitude of oej heighteng of eentricity， for exaple oej～005， is far fro sufficient to provoke stability the terrestrial ps havg such a ide separation as 26rh th e assue that the present ide dynaical separation a terrestrial ps 26rh is probably one of the nificant nditions for the stability of the pary syste over a 109-yr tie-span our detailed analysis of the retionship beeen dynaical distance beeen ps and the stability tie-scale of r syste pary otion is no on-gog
although our nurations span the lifetie of the r syste， the nuber of tegrations is far fro sufficient to fill the itial phase space it is necessary to perfor ore and ore nurations to nfir and exae detail the long-ter stability of our pary dynaics
——以上文段引自 ito， tap tanikaa， k long-terrations and stability of pary orbits our r syste on not r astron c 336， 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。最新网址：hx
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