对火星轨道变化问题的最后解释

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-term iions and stability of plaary orbits in our Solar system

Abstract

We present the results of very long-term numeribsp; iions of plaary orbital motions over 109 -yr time-spans including all nine plas. A quibsp; iion of our numeribsp; data shows that the plaary motion， at least in our simple dynamibsp; model， seems to be quite stable even over this very long time-span. A closer look at the lowest-frequenbsp; oscillations using a low-pass filter shows us the potentially diffusive character of terrestrial plaary motion， especially that of Mercury. The behaviour of the etricity of Mercury in our iions is qualitatively similar to the results from Jacques Laskar's secular perturbation theory (e.g. emax～ 0.35 over ～± 4 Gyr). However， there are no apparent secular increases of etricity or ination in any orbital elements of the plas， whibsp; may be revealed by still loerm numeribsp; iions. We have also performed a couple of trial iions including motions of the outer five plas over the duration of ± 5 × 1010 yr. The result indicates that the three major resonanbsp; 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 Solar system has been debated over several hundred years， sinbsp; the era of on. The problem has attracted many famous mathematis over the years and has played a tral role in the development of non-linear dynamibsp; and chaos theory. However， we do not yet have a definite answer to the question of whether our Solar system is stable or not. This is partly a result of the fabsp; that the definition of the term ‘stability’ is vague when it is used in relation to the problem of plaary motion in the Solar system. Actually it is not easy to give a clear， rigorous and physically meaningful definition of the stability of our Solar system.

Among many definitions of stability， here we adopt the Hill definition (Gladman 1993): actually this is not a definition of stability， but of instability. We define a system as being unstable when a close enter occurs somewhere in the system， starting from a certain initial figuration (Chambers， Wetherill & Boss 1996; Ito & Tanikawa 1999). A system is defined as experieng a close enter when two bodies approabsp; one another within an area of the larger Hill radius. Otherwise the system is defined as being stable. Henceforward we state that our plaary system is dynamically stable if no close enter happens during the age of our Solar system， about ±5 Gyr. Ially， this definition may be replabsp; by one in whibsp; an occurrenbsp; of any orbital crossing between either of a pair of plas takes plabsp; This is because we know from experienbsp; that an orbital crossing is very likely to lead to a close enter in plaary and protoplaary systems (Yoshinaga， Kokubo & Makino 1999). Of course this statement ot be simply applied to systems with stable orbital resonanbsp; subsp; as the une–Pluto system.

1.2Previous studies and aims of this research

In addition to the vagueness of the cept of stability， the plas in our Solar system show a character typibsp; of dynamibsp; chaos (Sussman & Wisdom 1988， 1992). The cause of this chaotibsp; behaviour is now partly uood as being a result of resonanbsp; overlapping (Murray & Holman 1999; Lecar， Franklin & Holman 2001). However， it would require iing over an ensemble of plaary systems including all nine plas for a period c several 10 Gyr to thhly uand the long-term evolution of plaary orbits， sinbsp; chaotibsp; dynamibsp; systems are characterized by their strong dependenbsp; on initial ditions.

From that point of view， many of the previous long-term numeribsp; iions included only the outer five plas (Sussman & Wisdom 1988; Kinoshita & Nakai 1996). This is because the orbital periods of the outer plas are so mubsp; longer than those of the inner four plas that it is mubsp; easier to follow the system for a given iion period. At present， the lo numeribsp; iions published in journals are those of Dunbsp; & Lissauer (1998). Although their main target was the effebsp; of post-main-sequenbsp; solar mass loss on the stability of plaary orbits， they performed many iions c up to ～1011 yr of the orbital motions of the four jovian plas. The initial orbital elements and masses of plas are the same as those of our Solar system in Dunbsp; & Lissauer's paper， but they decrease the mass of the Sun gradually in their numeribsp; experiments. This is because they sider the effebsp; of post-main-sequenbsp; solar mass loss in the paper. sequently， they found that the crossing time-scale of plaary orbits， whibsp; bsp; be a typibsp; 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 jovian plas remain stable over 1010 yr， or perhaps longer. Dunbsp; & Lissauer also performed four similar experiments on the orbital motion of seven plas (Venus to une)， whibsp; cover a span of ～109 yr. Their experiments on the seven plas are not yet prehensive， but it seems that the terrestrial plas also remain stable during the iion period， maintaining almost regular oscillations.

On the other hand， in his accurate semi-analytibsp; secular perturbation theory (Laskar 1988)， Laskar finds that large and irregular variations bsp; appear in the etricities and inations of the terrestrial plas， especially of Mercury and Mars on a time-scale of several 109 yr (Laskar 1996). The results of Laskar's secular perturbation theory should be firmed and iigated by fully numeribsp; iions.

In this paper we present preliminary results of six long-term numeribsp; iions on all nine plaary orbits， c a span of several 109 yr， and of two other iions c a span of ± 5 × 1010 yr. The total elapsed time for all iions is more than 5 yr， using several dedicated Pbsp; and workstations. One of the fual clusions of our long-term iions is that Solar system plaary motion seems to be stable in terms of the Hill stability mentioned above， at least over a time-span of ± 4 Gyr. Actually， in our numeribsp; 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 plaary orbital elements have been fined in a narrow region both in time and frequenbsp; domain， though plaary motions are stochastibsp; Sinbsp; the purpose of this paper is to exhibit and overview the results of our long-term numeribsp; iions， we show typibsp; example figures as evidenbsp; of the very long-term stability of Solar system plaary motion. For readers who have more specifibsp; and deeper is in our numeribsp; results， we have prepared a webpage (access )， where we show raw orbital elements， their low-pass filtered results， variation of Delaunay elements and angular momentum deficit， and results of our simple time–frequenbsp; analysis on all of our iions.

In Se 2 we briefly explain our dynamibsp; model， numeribsp; method and initial ditions used in our iions. Se 3 is devoted to a description of the quibsp; results of the numeribsp; iions. Very long-term stability of Solar system plaary motion is apparent both in plaary positions and orbital elements. A rough estimation of numeribsp; errors is also given. Se 4 goes on to a discussion of the loerm variation of plaary orbits using a low-pass filter and includes a discussion of angular momentum deficit. In Se 5， we present a set of numeribsp; iions for the outer five plas that spans ± 5 × 1010 yr. In Se 6 we also discuss the long-term stability of the plaary motion and its possible cause.

2 Description of the numeribsp; 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 Numeribsp; method

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

The stepsize for the numeribsp; iions is 8 d throughout all iions of the nine plas (N±1，2，3)， whibsp; is about 1/11 of the orbital period of the innermost pla (Mercury). As for the determination of stepsize， we partly follow the previous numeribsp; iion of all nine plas in Sussman & Wisdom (1988， 7.2 d) and Saha & Tremaine (1994， 225/32 d). We rounded the decimal part of the their stepsizes to 8 to make the stepsize a multiple of 2 in order to redubsp; the accumulation of round-off error in the putation processes. In relation to this， Wisdom & Holman (1991) performed numeribsp; iions of the outer five plaary orbits using the symplectibsp; map with a stepsize of 400 d， 1/10.83 of the orbital period of Jupiter. Their result seems to be accurate enough， whibsp; partly justifies our method of determining the stepsize. However， sinbsp; the etricity of Jupiter (～0.05) is mubsp; smaller than that of Mercury (～0.2)， we need some care when we pare these iions simply in terms of stepsizes.

In the iion of the outer five plas (F±)， we fixed the stepsize at 400 d.

We adopt Gauss' f and g funs in the symplectibsp; map together with the third-order Halley method (Danby 1992) as a solver for Kepler equations. The number of maximum iterations we set 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 calculations of all nine plas (N±1，2，3)， and about 8000 000 d (～21 903 yr) for the iion of the outer five plas (F±).

Although no output filtering was done when the numeribsp; iions were in process， we applied a low-pass filter to the raw orbital data after we had pleted all the calculations. See Se 4.1 for more detail.

2.4 Error estimation

2.4.1 Relative errors in total energy and angular momentum

Acc to one of the basibsp; properties of symplectibsp; iors， whibsp; serve the physically servative quantities well (total orbital energy and angular momentum)， our long-term numeribsp; iions seem to have been performed with very small errors. The averaged relative errors of total energy (～10?9) and of total angular momentum (～10?11) have remained nearly stant throughout the iion period (Fig. 1). The special startup procedure， warm start， would have redubsp; the averaged relative error in total energy by about one order of magnitude or more.

Relative numeribsp; error of the total angular momentum δA/A0 and the total energy δE/E0 in our numeribsp; iionsN± 1，2，3， where δE and δA are the absolute ge of the total energy and total angular momentum， respectively， andE0andA0are their initial values. The horizontal unit is Gyr.

Note that different operating systems， different mathematibsp; libraries， and different hardware architectures result in different numeribsp; errors， through the variations in round-off error handling and numeribsp; algorithms. In the upper panel of Fig. 1， we bsp; reize this situation in the secular numeribsp; error in the total angular momentum， whibsp; should be rigorously preserved up to mae-ε precision.

2.4.2 Error in plaary longitudes

Sinbsp; the symplectibsp; maps preserve total energy and total angular momentum of N-body dynamibsp; systems ily well， the degree of their preservation may not be a good measure of the accurabsp; of numeribsp; iions， especially as a measure of the positional error of plas， i.e. the error in plaary longitudes. To estimate the numeribsp; error in the plaary longitudes， we performed the following procedures. We pared the result of our main long-term iions with some test iions， whibsp; span mubsp; shorter periods but with mubsp; higher accurabsp; than the main iions. For this purpose， we performed a mubsp; 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 plaary orbital evolution. ， we pare the test iion with the main iion， N?1. For the period of 3 × 105 yr， we see a differenbsp; in mean anomalies of the Earth between the two iions of ～0.52°(in the case of the N?1 iion). This differenbsp; bsp; be extrapolated to the value ～8700°， about 25 rotations of Earth after 5 Gyr， sinbsp; the error of longitudes increases linearly with time in the symplectibsp; map. Similarly， the longitude error of Pluto bsp; be estimated as ～12°. This value for Pluto is mubsp; better than the result in Kinoshita & Nakai (1996) where the differenbsp; is estimated as ～60°.

3 Numeribsp; results – I. Glanbsp; at the raw data

In this se we briefly review the long-term stability of plaary orbital motion through some snapshots of raw numeribsp; data. The orbital motion of plas indicates long-term stability in all of our numeribsp; iions: no orbital crossings nor close enters between any pair of plas took place.

3.1 General description of the stability of plaary orbits

First， we briefly look at the general character of the long-term stability of plaary orbits. Our i here focuses particularly on the inner four terrestrial plas for whibsp; the orbital time-scales are mubsp; shorter than those of the outer five plas. As we bsp; see clearly from the planar orbital figurations shown in Figs 2 and 3， orbital positions of the terrestrial plas differ little between the initial and final part of eabsp; numeribsp; iion， whibsp; spans several Gyr. The solid lines denoting the present orbits of the plas lie almost within the swarm of dots even in the final part of iions (b) and (d). This indicates that throughout the entire iion period the almost regular variations of plaary orbital motion remain nearly the same as they are at present.

Vertibsp; view of the four inner plaary orbits (from the z -axis dire) at the initial and final parts of the iionsN±1. The axes units are au. The xy -plane is set to the invariant plane of Solar system total angular momentum.(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).(bsp; 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 eabsp; panel， a total of 23 684 points are plotted with an interval of about 2190 yr over 5.47 × 107 yr . Solid lines in eabsp; panel denote the present orbits of the four terrestrial plas (taken from DE245).