对火星轨道变化问题的最后解释（1 / 2）

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。

那么作者君在此列出相关参考文献中的一篇开源论文。

以下是文章内容：

long-ter tegrations and stability of pary orbitsour sor syste

abstract

we present the results of very long-ter nurical tegrations of pary orbital otions over 109 -yr ti-spans cdg all ne ps a quick spection of our nurical data shows that the pary otion， at leastour siple dynaical odel， sees to be quite stable even over this very long ti-span a closer look at the lowest-frequency osciltions g a low-pass filter showsthe potentially diffive character of terrestrial pary otion， especially that of rcury the behaviour of the eentricity of rcuryour tegrations is qualitatively siir to the results fro jacques skar&039;s secur perturbation theory (eg eax～ 035 over ～± 4 gyr) however， there are no apparent secur creases of eentricity or clationany orbital elents of the ps， which ay be revealed by still longer-ter nurical tegrations we have also perford 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 resonancesthe neptune–pto syste have been ataed over the 1011-yr ti-span

1 troduction

11defition of the proble

the question of the stability of our sor syste has been debated over several hundred years， sce the era of newton the proble has attracted any fao atheaticians over the years and has pyed a central rolethe developnt of non-lear dynaics and chaos theory however， we do not yet have a defite answer to the question of whether our sor syste is stable or not this is partly a result of the fact that the defition of the ter ‘stability’ is vague when it is edretion to the proble of pary otionthe sor syste actually it is not easy to give a clear， rigoro and physically angful defition of the stability of our sor syste

aong any defitions of stability， here we adopt the hill defition (gdan 1993): actually this is not a defition of stability， but of stability we defe a syste as beg unstable when a close enunter ours sowherethe syste， startg fro a certa itial nfiguration (chabers， wetherill ≈ap;ap;ap; boss 1996; ito ≈ap;ap;ap; tanikawa 1999) a syste is defed as experiencg a close enunter when o bodies approach one another with an area of the rger hill radi otherwise the syste is defed as beg stable henceforward we state that our pary syste is dynaically stable if no close enunter happens durg the age of our sor syste， about ±5 gyr cidentally， this defition ay be repced by onewhich an ourrence of any orbital crossg beeen either of a pair of ps takes pce this is becae we know fro experience that an orbital crossg is very likely to lead to a close enunterpary and proary systes (yoshaga， kokubo ≈ap;ap;ap; ako 1999) of urse this statent cannot be siply applied to systes with stable orbital resonances such as the neptune–pto syste

12previo studies and ais of this research

addition to the vagueness of the ncept of stability， the psour sor syste show a character typical of dynaical chaos (ssan ≈ap;ap;ap; wisdo 1988， 1992) the cae of this chaotic behaviour is now partly understood as beg a result of resonance overppg (urray ≈ap;ap;ap; holan 1999; lecar， frankl ≈ap;ap;ap; holan 2001) however， it would require tegratg over an enseble of pary systes cdg 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 view， any of the previo long-ter nurical tegrations cded only the outer five ps (ssan ≈ap;ap;ap; wisdo 1988; koshita ≈ap;ap;ap; nakai 1996) this is becae the orbital periods of the outer ps are so uch longer than those of the ner four ps that it is uch easier to follow the syste for a given tegration period at present， the longest nurical tegrations publishedjournals are those of duncan ≈ap;ap;ap; lissauer (1998) although their a target was the effect of post-a-sequence sor ass loss on the stability of pary orbits， they perford any tegrations verg up to ～1011 yr of the orbital otions of the four jovian ps the itial orbital elents and asses of ps are the sa as those of our sor systeduncan ≈ap;ap;ap; lissauer&039;s paper， but they decrease the ass of the sun graduallytheir nurical experints this is becae they nsider the effect of post-a-sequence sor ass lossthe paper nsequently， they found that the crossg ti-scale of pary orbits， which can be a typical dicator of the stability ti-scale， is quite sensitive to the rate of ass decrease of the sun when the ass of the sun is close to its present vae， the jovian ps rea stable over 1010 yr， or perhaps longer duncan ≈ap;ap;ap; lissauer also perford four siir experints on the orbital otion of seven ps (ven to neptune)， which ver a span of ～109 yr their experints on the seven ps are not yet prehensive， but it sees that the terrestrial ps also rea stable durg the tegration period， atag alost regur osciltions

on the other hand，his aurate sei-analytical secur perturbation theory (skar 1988)， skar fds that rge and irregur variations can appearthe eentricities and clations of the terrestrial ps， especially of rcury and ars on a ti-scale of several 109 yr (skar 1996) the results of skar&039;s secur perturbation theory should be nfird and vestigated by fully nurical tegrations

this paper we present preliary results of six long-ter nurical tegrations 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 epsed ti for all tegrations is ore than 5 yr， g several dedicated pcs and workstations one of the fundantal ncsions of our long-ter tegrations is that sor syste pary otion sees to be stableters of the hill stability ntioned above， at least over a ti-span of ± 4 gyr actually，our nurical tegrations the syste was far ore stable than what is defed by the hill stability criterion: not only did no close enunter happen durg the tegration period， but also all the pary orbital elents have been nfeda narrow region bothti and frequency doa， though pary otions are stochastic sce the purpose of this paper is to exhibit and overview the results of our long-ter nurical tegrations， we show typical exaple figures as evidence of the very long-ter stability of sor syste pary otion for readers who have ore specific and deeper terestsour nurical results， we have prepared a webpage (aess )， where we show raw orbital elents， their low-pass filtered results， variation of deunay elents and angur ontu deficit， and results of our siple ti–frequency analysis on all of our tegrations

section 2 we briefly exp our dynaical odel， nurical thod and itial nditions edour tegrations section 3 is devoted to a description of the quick results of the nurical tegrations very long-ter stability of sor syste pary otion is apparent bothpary positions and orbital elents a rough estiation of nurical errors is also given section 4 goes on to a discsion of the longest-ter variation of pary orbits g a low-pass filter and cdes a discsion of angur ontu deficitsection 5， we present a set of nurical tegrations for the outer five ps that spans ± 5 x 1010 yrsection 6 we also discs the long-ter stability of the pary otion and its possible cae

2 description of the nurical tegrations

（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。）

23 nurical thod

we utilize a send-order wisdo–holan syplectic ap as our a tegration thod (wisdo ≈ap;ap;ap; holan 1991; koshita， yoshida ≈ap;ap;ap; nakai 1991) with a special start-up procedure to reduce the truncation error of angle variables，‘war start’(saha ≈ap;ap;ap; treae 1992， 1994)

the stepsize for the nurical tegrations is 8 d throughout all tegrations of the ne ps (n±1，2，3)， which is about 1/11 of the orbital period of the nerost p (rcury) as for the deteration of stepsize， we partly follow the previo nurical tegration of all ne psssan ≈ap;ap;ap; wisdo (1988， 72 d) and saha ≈ap;ap;ap; treae (1994， 225/32 d) we rounded the decial part of the their stepsizes to 8 to ake the stepsize a ultiple of 2order to reduce the auution of round-off errorthe putation processesretion to this， wisdo ≈ap;ap;ap; holan (1991) perford nurical tegrations of the outer five pary orbits g the syplectic ap with a stepsize of 400 d， 1/1083 of the orbital period of jupiter their result sees to be aurate enough， which partly jtifies our thod of deterg the stepsize however， sce the eentricity of jupiter (～005) is uch saller than that of rcury (～02)， we need so care when we pare these tegrations siplyters of stepsizes

the tegration of the outer five ps (f±)， we fixed the stepsize at 400 d

we adopt gas&039; f and g functionsthe syplectic ap together with the third-order halley thod (danby 1992) as a solver for kepler equations the nuber of axiu iterations we sethalley&039;s thod is 15， but they never reached the axiuany 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 was done when the nurical tegrations wereprocess， we applied a low-pass filter to the raw orbital data after we had pleted all the calcutions see section 41 for ore detail

24 error estiation

241 retive errorstotal energy and angur ontu

aordg to one of the basic properties of syplectic tegrators， which nserve the physically nservative quantities well (total orbital energy and angur ontu)， our long-ter nurical tegrations see to have been perford with very sall errors the averaged retive errors of total energy (～10?9) and of total angur ontu (～10?11) have reaed nearly nstant throughout the tegration period (fig 1) the special startup procedure， war start， would have reduced the averaged retive errortotal energy by about one order of agnitude or ore

retive nurical error of the total angur ontu δa/a0 and the total energy δe/e0our nurical tegrationsn± 1，2，3， where δe and δa are the absote change of the total energy and total angur ontu， respectively， ande0anda0are their itial vaes the horizontal unit is gyr

note that different operatg systes， different atheatical libraries， and different hardware architectures resultdifferent nurical errors， through the variationsround-off error handlg and nurical algorithsthe upper panel of fig 1， we can regnize this situationthe secur nurical errorthe total angur ontu， which should be rigoroly preserved up to ache-e precision

242 errorpary longitudes

sce the syplectic aps preserve total energy and total angur ontu of n-body dynaical systes herently well， the degree of their preservation ay not be a good asure of the auracy of nurical tegrations， especially as a asure of the positional error of ps， ie the errorpary longitudes to estiate the nurical errorthe pary longitudes， we perford the follog procedures we pared the result of our a long-ter tegrations with so test tegrations， which span uch shorter periods but with uch higher auracy than the a tegrations for this purpose， we perford a uch ore aurate tegration with a stepsize of 0125 d (1/64 of the a tegrations) spanng 3 x 105 yr， startg with the sa itial nditions asthe n?1 tegration we nsider that this test tegration provideswith a ‘pseudo-true’ sotion of pary orbital evotion next， we pare the test tegration with the a tegration， n?1 for the period of 3 x 105 yr， we see a differencean 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 with tithe syplectic ap siirly， the longitude error of pto can be estiated as ～12° this vae for pto is uch better than the resultkoshita ≈ap;ap;ap; nakai (1996) where the difference is estiated as ～60°

3 nurical results – i gnce at the raw data

this section we briefly review the long-ter stability of pary orbital otion through so snapshots of raw nurical data the orbital otion of ps dicates long-ter stabilityall of our nurical tegrations: no orbital crossgs nor close enunters beeen any pair of ps took pce

31 general description of the stability of pary orbits

first， we briefly look at the general character of the long-ter stability of pary orbits our terest here foces particurly on the ner four terrestrial ps for which the orbital ti-scales are uch shorter than those of the outer five ps as we can see clearly fro the pnar orbital nfigurations shownfigs 2 and 3， orbital positions of the terrestrial ps differ little beeen the itial and fal part of each nurical tegration， which spans several gyr the solid les denotg the present orbits of the ps lie alost with the swar of dots eventhe fal part of tegrations (b) and (d) this dicates that throughout the entire tegration period the alost regur variations of pary orbital otion rea nearly the sa as they are at present

vertical view 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 sor syste total angur ontu(a) the itial part ofn+1 ( t = 0 to 00547 x 10 9 yr)(b) the fal part ofn+1 ( t = 49339 x 10 8 to 49886 x 10 9 yr)(c) the itial part of n?1 (t= 0 to ?00547 x 109 yr)(d) 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 with an terval of about 2190 yr over 547 x 107 yrsolid leseach panel denote the present orbits of the four terrestrial ps (taken fro de245)