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一篇相关的参考文献。

以下是文章内容：

Long-term iions and stability of pary orbits in our Sor system

Abstract

We present the results of very long-term numerical iions of pary orbital motions over 109 -yr time-spans including all nine ps. A quispe of our numerical data shows that the pary motion， at least in our simple dynamical model， seems to be quite stable evehis very long time-span. A closer look at the lowest-frequency osciltions using a low-pass filter shows us the potentially diffusive character of terrestrial pary motion， especially that of Mercury. The behaviour of the etricity of Mercury in our iions is qualitatively simir to the results fr Jacques Laskar's secur perturbation theory (e.g. emax～ 0.35 over ～± 4 Gyr). However， there are no apparent secur increases of etricity or ination in any orbital elements of the ps， which may be revealed by still loerm numerical iions. We have also performed a couple of trial iions including motions of the outer five ps over the duration of ± 5 × 1010 yr. The result indicates that the three major resonances in the une–Pluto system have been maintained over the 1011-yr time-span.

1 Introdu

1.1Definition of the problem

The question of the stability of our Sor system has beeed over several hundred years， sihe era of on. The problem has attracted many famous mathematis over the years and has pyed a tral role in the development of non-linear dynamid chaos theory. However， we do not yet have a definite ao the question of whether our Sor system is stable or not. This is partly a result of the fact that the definition of the term ‘stability’ is vague when it is used iion to the problem of pary motion in the Sor system. Actually it is not easy to give a clear， rigorous and physically meaningful definition of the stability of our Sor system.

Among many definitions of stability， here t the Hill definition (Gdman 1993): actually this is not a definition of stability， but of instability. We define a system as beg unstable when a close enter occurs sewhere in the system， starting fr a certain initial figuration (Chambers， Wetherill & Boss 1996; Ito & Tanikawa 1999). A system is defined as experieng a close enter when two bodies approae another within an area of the rger Hill radius. Otherwise the system is defined as being stable. Henceforward we state that our pary system is dynamically stable if no close enter happens during the age of our Sor system， about ±5 Gyr. Ially， this definition may be repced by one in whi occurrence of any orbital crossiweeher of a pair of pakes pce. This is because we know fr experiehat an orbital crossing is very likely to lead to a close enter iary and protopary systems (Yoshinaga， Kokubo & Makino 1999). Of course this statement ot be simply applied to systems with stable orbital resonances such as the une–Pluto system.

1.2Previous studies and aims of this research

In addition to the vagueness of the cept of stability， the ps in our Sor system show a character typical of dynamical chaos (Sussman & Wisd 1988， 1992). The cause of this chaotic behaviour is now partly uood as being a result of resonance overpping (Murray & Holman 1999; Lecar， Franklin & Holman 2001). However， it would require iing over an ensemble of pary systems including all nine ps for a period c several 10 Gyr to thhly uand the long-term evolution of pary orbits， since chaotiamical systems are characterized by their strong dependen initial ditions.

Fr that point of view， many of the previous long-term numerical iions included only the outer five ps (Sussman & Wisd 1988; Kinoshita & Nakai 1996). This is because the orbital periods of the outer ps are so much lohan those of the inner four phat it is much easier to follow the system fiven iion period. At present， the lo numerical iions published in journals are those of Dun & Lissauer (1998). Although their main target was the effect of post-main-sequenass loss oability of pary orbits， they performed many iions c up to ～1011 yr of the orbital motions of the four jovias. The initial orbital elements and masses of ps are the same as those of our Sor system in Dun & Lissauer's paper， but they decrease the mass of the Sun gradually in their numerical experiments. This is because they sider the effect of post-main-sequenass loss in the paper. sequently， they found that the crossing time-scale of pary orbits， which be a typical indicator of the instability time-scale， is quite sensitive to the rate of mass decrease of the Sun. When the mass of the Sun is close to its present value， the jovias remain stable over 1010 yr， or perhaps longer. Dun & Lissauer also performed four simir experiments on the orbital motion of seves (Venus to une)， which cover a span of ～109 yr. Their experiments on the seves are not yet cprehensive， but it seems that the terrestrial ps also remain stable during the iion period， maintaining almur osciltions.

Oher hand， in his accurate semi-analytical secur perturbation theory (Laskar 1988)， Laskar finds that rge and irregur variations appear in the etricities and inations of the terrestrial ps， especially of Mercury and Mars on a time-scale of several 109 yr (Laskar 1996). The results of Laskar's secur perturbation theory should be firmed and iigated by fully numerical iions.

In this paper we present preliminary results of six long-term numerical iions on all nine pary orbits， c a span of several 109 yr， and of two other iions c a span of ± 5 × 1010 yr. The total epsed time for all iions is more than 5 yr， using several dedicated Pd workstations. One of the fual clusions of our long-term iions is that Sor system pary motioo be stable in terms of the Hill stability mentioned above， at least over a time-span of ± 4 Gyr. Actually， in our numerical iions the system was far more stable than what is defined by the Hill stability criterion: not only did no close enter happen during the iion period， but also all the pary orbital elements have been fined in a narrion both in time and frequency dain， though pary motions are stochastic. Sihe purpose of this paper is to exhibit and overview the results of our long-term numerical iions， we show typical example figures as evidence of the very long-term stability of Sor system pary motion. For readers who have more specifid deeper is in our numerical results， repared a webpage (access )， where we show raw orbital elements， their low-pass filtered results， variation of Deunay elements and angur mentum deficit， as of our simple time–frequenalysis on all of our iions.

Iion 2 we briefly expin our dynamical model， numerical method and initial ditions used in our iions. Se 3 is devoted to a description of the quick results of the numerical iions. Very long-term stability of Sor system pary motion is apparent both iary positions and orbital elements. A rough estimation of numerical errors is also giveion 4 goes on to a discussion of the loerm variation of pary orbits using a low-pass filter and includes a discussion of angur mentum deficit. Iion 5， we present a set of numerical iions for the outer five phat spans ± 5 × 1010 yr. Iion 6 we also discuss the long-term stability of the pary motion and its possible cause.

2 Description of the numerical iions

（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 Numerical method

We utilize a sed-order Wisd–Holman symplectic map as our main iiohod (Wisd & Holman 1991; Kinoshita， Yoshida & Nakai 1991) with a special start-up procedure to reduce the truncation error of angle variables，‘warm start’(Saha & Tremaine 1992， 1994).

The stepsize for the numerical iions is 8 d throughout all iions of the nine ps (N±1，2，3)， which is about 1/11 of the orbital period of the innermost p (Mercury). As for the determination of stepsize， we partly follow the previous numerical iion of all nine ps in Sussman & Wisd (1988， 7.2 d) and Saha & Tremaine (1994， 225/32 d). We rouhe decimal part of the their stepsizes to 8 to make the stepsize a multiple of 2 in order to reduce the accumution of round-off error in the cputation processes. Iion to this， Wisd & Holman (1991) performed numerical iions of the outer five pary orbits using the symplectic map with a stepsize of 400 d， 1/10.83 of the orbital period of Jupiter. Their result seems to be accurate enough， which partly justifies our method of determining the stepsize. However， sihe etricity of Jupiter (～0.05) is much smaller than that of Mercury (～0.2)， we need se care when we cpare these iions simply in terms of stepsizes.

Iegration of the outer five ps (F±)， we fixed the stepsize at 400 d.

t Gauss' f and g funs in the symplectic map together with the third-order Halley method (Danby 1992) as a solver for Kepler equations. The number of maximum iteratio in Halley's method is 15， but they never reached the maximum in any of our iions.

The interval of the data output is 200 000 d (～547 yr) for the calcutions of all nine ps (N±1，2，3)， and about 8000 000 d (～21 903 yr) for the iion of the outer five ps (F±).

Although no output filtering was done when the numerical iions were in process， lied a low-pass filter to the raw orbital data after leted all the calcutions. See Se 4.1 for more detail.

2.4 Error estimation

2.4.1 Retive errors in total energy and angur mentum

Acc to one of the basic properties of symplectitegrators， which serve the physically servative quantities well (total orbital energy and angur mentum)， our long-term numerical iioo have been performed with very small errors. The averaged retive errors of total energy (～10?9) and of total angur mentum (～10?11) have remained nearly stant throughout the iion period (Fig. 1). The special startup procedure， warm start， would have reduced the averaged retive error in total energy by about one order of magnitude or more.

Retive numerical error of the total angur mentum δA/A0 and the total energy δE/E0 in our numerical iionsN± 1，2，3， where δE and δA are the absolute ge of the total energy and total angur mentum， respectively， andE0andA0are their initial values. The horizontal unit is Gyr.

hat different operating systems， different mathematical libraries， and different hardware architectures result in different numerical errors， through the variations in round-off error handling and numerical algorithms. In the upper panel of Fig. 1， we reize this situation in the seumerical error ial angur mentum， which should be rigorously preserved up to mae-ε precision.

2.4.2 Error iary longitudes

Sihe symplectic maps preserve total energy and total angur mentum of N-body dynamical systems ily well， the degree of their preservation may not be a good measure of the accuracy of numerical iions， especially as a measure of the positional error of ps， i.e. the error iary longitudes. To estimate the numerical error in the pary longitudes， we performed the following procedures. We cpared the result of our main long-term iions with se test iions， which span much shorter periods but with much higher accuracy than the main iions. For this purpose， we performed a much more accurate iion with a stepsize of 0.125 d (1/64 of the main iions) spanning 3 × 105 yr， starting with the same initial ditions as in the N?1 iion. We sider that this test iion provides us with a ‘pseudo-true’ solution of pary orbital evolutio， we cpare the test iion with the main iion， N?1. For the period of 3 × 105 yr， we see a differen mean analies of the Earth betweewo iions of ～0.52°(in the case of the N?1 iion). This difference be extrapoted to the value ～8700°， about 25 rotations of Earth after 5 Gyr， sihe error of longitudes increases linearly with time in the symplectic map. Simirly， the longitude error of Pluto be estimated as ～12°. This value for Pluto is much better than the result in Kinoshita & Nakai (1996) where the difference is estimated as ～60°.

3 Numerical results – I. G the raw data

In this se we briefly review the long-term stability of pary orbital motion through se snapshots of raw numerical data. The orbital motion of ps indicates long-term stability in all of our numerical iions: no orbital crossings nor close enters between any pair of pook pce.

3.1 General description of the stability of pary orbits

First， we briefly look at the general character of the long-term stability of pary orbits. Our i here focuses particurly on the inner four terrestrial ps for which the orbital time-scales are much shorter than those of the outer five ps. As we see clearly fr the pnar orbital figurations shown in Figs 2 and 3， orbital positions of the terrestrial ps differ little between the initial and final part of eaumerical iion， which spans several Gyr. The solid lines denoting the present orbits of the ps lie almost within the swarm of dots even in the final part of iions (b) and (d). This indicates that throughout the eegration period the almur variations of pary orbital motion remain nearly the same as they are at present.

Vertical view of the four inner pary orbits (fr the z -axis dire) at the initial and final parts of the iionsN±1. The axes units are au. The xy -pne is set to the invariant pne of Sor system total angur mentum.(a) The initial part ofN+1 ( t = 0 to 0.0547 × 10 9 yr).(b) The final part ofN+1 ( t = 4.9339 × 10 8 to 4.9886 × 10 9 yr).(c) The initial part of N?1 (t= 0 to ?0.0547 × 109 yr).(d) The final part ofN?1 ( t =?3.9180 × 10 9 to ?3.9727 × 10 9 yr). In each panel， a total of 23 684 points are plotted with an interval of about 2190 yr over 5.47 × 107 yr . Solid lines in each panel dehe present orbits of the four terrestrial ps (taken fr DE245).

The variation of etricities and orbital inations for the inner four ps in the initial and final part of the iion N+1 is shown in Fig. 4. As expected， the character of the variation of pary orbital elements does not differ signifitly between the initial and final part of eategration， at least for Venus， Earth and Mars. The elements of Mercury， especially its etricity， seem to ge to a signifit extent. This is partly because the orbital time-scale of the p is the shortest of all the ps， which leads to a more rapid orbital evolution than other ps; the innermost p may be o instability. This result appears to be in se agreement with Laskar's (1994， 1996) expectations that rge and irregur variations appear in the etricities and inations of Mercury on a time-scale of several 109 yr. However， the effect of the possible instability of the orbit of Mercury may not fatally affect the global stability of the whole pary system owing to the small mass of Mercury. We will mention briefly the long-term orbital evolution of Mercury ter iion 4 using low-pass filtered orbital elements.

The orbital motion of the outer five ps seems rigorously stable and quite regur over this time-span (see also Se 5).

3.2 Time–frequency maps

Although the pary motion exhibits very long-term stability defined as the ence of close enter events， the chaotiature of pary dynamics ge the osciltory period and amplitude of pary orbital motion gradually over such long time-spans. Even such slight fluctuations of orbital variation in the frequency dain， particurly in the case of Earth， potentially have a signifit effe its surface climate system through sor insotion variation (cf. Berger 1988).

To give an overview of the long-term ge in periodicity iary orbital motion， we performed many fast Fourier transformations (FFTs) along the time axis， and superposed the resulting periodgrams to draw two-dimensional time–frequency maps. The specific approach to drawing these time–frequency maps in this paper is very simple – much simpler than the wavelet analysis or Laskar's (1990， 1993) frequenalysis.

Divide the low-pass filtered orbital data into many fragments of the same length. The length of each data segment should be a multiple of 2 in order to apply the FFT.

Each fragment of the data has a rge overpping part: for example， wheh data begins fr t=ti and ends at t=ti+T， the data segment ranges fr ti+δT≤ti+δT+T， where δT?T. We tihis division until we reach a certain number N by whi+T reaches the total iioh.

ly an FFT to each of the data fragments， and obtain n frequency diagrams.

In each frequency diagram obtained above， the strength of periodicity be repced by a grey-scale (or colour) chart.

We perform the rept， and ect all the grey-scale (or colour) charts into one graph for eategration. The horizontal axis of these nehs should be the time， i.e. the starting times of each fragment of data (ti， where i= 1，…， n). The vertical axis represents the period (or frequency) of the osciltion of orbital elements.

We have adopted an FFT because of its overwhelming speed， sihe amount of numerical data to be decposed into frequency ents is terribly huge (several tens of Gbytes).

A typical example of the time–frequency map created by the above procedures is shown in a grey-scale diagram as Fig. 5， which shows the variation of periodicity in the etricity and ination of Earth in N+2 iion. In Fig. 5， the dark area shows that at the time indicated by the value on the abscissa， the periodicity indicated by the ordinate is strohan in the lighter area around it. We reize fr this map that the periodicity of the etricity and ination of Earth only ges slightly over the entire period covered by the N+2 iion. This nearly regur trend is qualitatively the same in other iions and for other ps， although typical frequencies differ p by p and element by element.

4.2 Long-term exge of orbital energy and angur mentum

We calcute very long-periodic variation and exge of pary orbital energy and angur mentum using filtered Deunay elements L， G， H. G and H are equivalent to the pary orbital angur mentum and its vertical ent per unit mass. L is reted to the pary orbital energy E per unit mass as E=?μ2/2L2. If the system is cpletely linear， the orbital energy and the angur mentum in each frequency bin must be stant. Non-liy in the pary system cause an exge of energy and angur mentum in the frequency dain. The amplitude of the lowest-frequency osciltion should increase if the system is unstable and breaks down gradually. However， such a sympt of instability is not pri in our long-term iions.

In Fig. 7， the total orbital energy and angur mentum of the four inner ps and all nine ps are shown for iion N+2. The upper three panels show the long-periodic variation of total energy (denoted asE- E0)， total angur mentum ( G- G0)， and the vertical ent ( H- H0) of the inner four ps calcuted fr the low-pass filtered Deunay elements.E0， G0， H0 dehe initial values of each quantity. The absolute difference fr the initial values is plotted in the panels. The lower three panels in each figure showE-E0，G-G0 andH-H0 of the total of nine ps. The fluctuation shown in the lower panels is virtually entirely a result of the massive jovias.

Cparing the variations of energy and angur mentum of the inner four ps and all nine ps， it is apparent that the amplitudes of those of the inner ps are much smaller than those of all nine ps: the amplitudes of the outer five ps are much rger than those of the inner ps. This does not mean that the ierrestrial pary subsystem is more stable thaer ohis is simply a result of the retive smallness of the masses of the four terrestrial ps cpared with those of the outer jovias. Ahiice is that the inner pary subsystem may bestable more rapidly thaer one because of its shorter orbital time-scales. This be seen in the panels denoted asinner 4 in Fig. 7 where the longer-periodid irregur osciltions are more apparent than in the panels denoted astotal 9. Actually， the fluctuations in theinner 4 panels are te extent as a result of the orbital variation of the Mercury. However， we ot he tribution fr other terrestrial ps， as we will see in subsequeions.

4.4 Long-term coupling of several neighb p pairs

Let us see se individual variations of pary orbital energy and angur mentum expressed by the low-pass filtered Deunay elements. Figs 10 and 11 show long-term evolution of the orbital energy of eaet and the angur mentum in N+1 and N?2 iions. We notice that se ps form apparent pairs in terms of orbital energy and angur mentum exge. In particur， Venus ah make a typical pair. In the figures， they show ive corretions in exge of energy and positive corretions in exge of angur mentum. The ive corretion in exge of orbital energy means that the two ps form a closed dynamical system in terms of the orbital energy. The positive corretion in exge of angur mentum means that the two ps are simultaneously under certain long-term perturbations. didates for perturbers are Jupiter and Saturn. Also in Fig. 11， we see that Mars shoositive corretion in the angur mentum variation to the Veh system. Mercury exhibits certaiive corretions in the angur mentum versus the Veh system， which seems to be a rea caused by the servation of angur mentum ierrestrial pary subsystem.

It is not clear at the ment why the Veh pair exhibits a ive corretion in energy exge and a positive corretion in angur mentum exge. ossibly expin this through the general fact that there are no secur terms iary semimajor axes up to sed-order perturbation theories (cf. Brouwer & Clemence 1961; Boccaletti & Pucacco 1998). This means that the pary orbital energy (which is directly reted to the semimajor axis a) might be much less affected by perturbing phan is the angur mentum exge (which retes to e). Hehe etricities of Venus ah be disturbed easily by Jupiter and Saturn， which results in a positive corretion in the angur mentum exge. Oher hand， the semimajor axes of Venus ah are less likely to be disturbed by the jovias. Thus the energy exge may be limited only within the Veh pair， which results in a ive corretion in the exge of orbital energy in the pair.

As for the outer joviaary subsystem， Jupiter–Saturn and Uranus–uo make dynamical pairs. However， the strength of their coupling is not as strong cpared with that of the Veh pair.

5 ± 5 × 1010-yr iions of outer pary orbits

Sihe joviaary masses are much rger thaerrestrial pary masses， we treat the joviaary system as an indepe pary system in terms of the study of its dynamical stability. Hence， we added a couple of trial iions that span ± 5 × 1010 yr， including only the outer five ps (the four jovias plus Pluto). The results exhibit the rigorous stability of the outer pary system over this long time-span. Orbital figurations (Fig. 12)， and variation of etricities and inations (Fig. 13) show this very long-term stability of the outer five ps in both the time and the frequency dains. Although we do not shos here， the typical frequency of the orbital osciltion of Pluto and the other outer ps is almost stant during these very long-term iion periods， which is demonstrated iime–frequency maps on our webpage.

Iwo iions， the retive numerical error ial energy was ～10?6 and that of the total angur mentum was ～10?10.

5.1 Resonances in the une–Pluto system

Kinoshita & Nakai (1996) ied the outer five pary orbits over ± 5.5 × 109 yr . They found that four major resonances betweeune and Pluto are maintained during the whole iion period， and that the resonances may be the main causes of the stability of the orbit of Pluto. The major four resonances found in previous research are as follows. In the following description，λ dehe mean longitude，Ω is the longitude of the asding node and ? is the longitude of perihelion. Subscripts P and e Pluto aune.

Mean motion resoweeune and Pluto (3:2). The critical argument θ1= 3 λP? 2 λN??P librates around 180° with an amplitude of about 80° and a libration period of about 2 × 104 yr.

The argument of perihelion of Pluto ωP=θ2=?P?ΩP librates around 90° with a period of about 3.8 × 106 yr. The dinant periodic variations of the etricity and ination of Pluto are synized with the libration of its argument of perihelion. This is anticipated in the secur perturbation theory structed by Kozai (1962).

The longitude of the node of Pluto referred to the longitude of the node of une，θ3=ΩP?ΩN， circutes and the period of this circution is equal to the period of θ2 libration. When θ3 beces zero， i.e. the longitudes of asding nodes of une and Pluto overp， the ination of Pluto beces maximum， the etricity beces minimum and the argument of perihelion beces 90°. When θ3 beces 180°， the ination of Pluto beces minimum， the etricity beces maximum and the argument of perihelion beces 90° again. Williams & Benson (1971) anticipated this type of resoer firmed by Mini， Nobili & Carpino (1989).

An argument θ4=?P??N+ 3 (ΩP?ΩN) librates around 180° with a long period，～ 5.7 × 108 yr.

In our numerical iions， the resonances (i)–(iii) are well maintained， and variation of the critical arguments θ1，θ2，θ3 remain simir during the whole iion period (Figs 14–16 ). However， the fourth resonance (iv) appears to be different: the critical argument θ4 alternates libration and circution over a 1010-yr time-scale (Fig. 17). This is an iing fact that Kinoshita & Nakai's (1995， 1996) shorter iions were not able to disclose.

6 Discussion

What kind of dynamical meism maintains this long-term stability of the pary system? We immediately think of two major features that may be responsible for the long-term stability. First， there seem to be no signifit lower-order resonances (mean motion and secur) between any pair among the nine ps. Jupiter and Saturn are close to a 5:2 mean motion resohe famous ‘great inequality’)， but not just in the resonance zone. Higher-order resonances may cause the chaotiature of the pary dynamical motion， but they are not s as to destroy the stable pary motion within the lifetime of the real Sor system. The sed feature， which we think is more important for the long-term stability of our pary system， is the differen dynamical distaween terrestrial and joviaary subsystems (Ito & Tanikawa 1999， 2001). When we measure pary separations by the mutual Hill radii (R_)， separations among terrestrial ps are greater than 26RH， whereas those among jovias are less than 14RH. This difference is directly reted to the differeween dynamical features of terrestrial and jovias. Terrestrial ps have smaller masses， shorter orbital periods and wider dynamical separation. They are strongly perturbed by joviahat have rger masses， longer orbital periods and narrower dynamical separation. Jovias are not perturbed by any other massive bodies.

The present terrestrial pary system is still being disturbed by the massive jovias. However， the wide separation and mutual iion among the terrestrial ps rehe disturbaneffective; the degree of disturbance by jovias is O(eJ)(order of magnitude of the etricity of Jupiter)， sihe disturbance caused by jovias is a forced osciltion having an amplitude of O(eJ). Heightening of etricity， for example O(eJ)～0.05， is far fr suffit to provoke instability ierrestrial ps having such a wide separation as 26RH. Thus we assume that the present wide dynamical separation among terrestrial ps (> 26RH) is probably one of the most signifit ditions for maintaining the stability of the pary system over a 109-yr time-span. Our detailed analysis of the retionship between dynamical distawees and the instability time-scale of Sor system pary motion is now on-going.

Although our numerical iions span the lifetime of the Sor system， the number of iions is far fr suffit to fill the initial phase space. It is necessary to perform more and more numerical iions to firm and examine iail the long-term stability of our pary dynamics.

——以上文段引自 Ito， T.& Tanikawa， K. Long-term iions and stability of pary orbits in our Sor System. Mon. Not. R. Astron. Soc. 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。