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，反正这本书写到现在，嚷嚷着本书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-term and stability of orbits in our system

Abstract

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

1

of the problem

The of the stability of our system has been debated over several hundred years， since the era of Newton. The problem has attracted many famous mathematicians over the years and has a central role in the development of non-linear dynamics and chaos theory. However，

we do not yet have a definite answer to the of whether our system is stable or not. This is partly a result of the fact that the of the term ‘stability’ is vague when it is used in to the problem of in the system. Actually it is not easy to give a clear， rigorous and physically meaningful of the stability of our system.　　Among many of stability， here we adopt the Hill ( 1993): actually this is not a of stability， but of instability. We define a system as unstable when a close somewhere in the system， starting from a certain initial (Chambers， Wetherill & Boss 1996; Ito & Tanikawa 1999). A system is defined as experiencing a close when two bodies approach one another within an area of the larger Hill radius. Otherwise the system is defined as being stable. Henceforward we state that our system is dynamically stable if no close happens during the age of our system， about ±5 Gyr. Incidentally， this may be by one in which an of any orbital crossing between either of a pair of takes This is because we know from experience that an orbital crossing is very likely to lead to a close in and systems (Yoshinaga， Kokubo & Makino 1999). Of course this statement cannot be simply applied to systems with stable orbital resonances such as the Neptune–Pluto system.

studies and aims of this research

In to the vagueness of the concept of stability， the in our system show a character typical of dynamical chaos (Sussman & Wisdom 1988， 1992). The cause of this chaotic is now partly understood as being a result of resonance (Murray & Holman 1999; Lecar， Franklin & Holman 2001). However， it would require integrating over an ensemble of systems including all nine for a covering several 10 Gyr to thoroughly understand the long-term of orbits， since chaotic dynamical systems are characterized by their strong dependence on initial

From that point of view， many of the long-term numerical included only the outer five (Sussman & Wisdom 1988; Kinoshita & Nakai 1996). This is because the orbital of the outer are so much longer than those of the inner four that it is much easier to follow the system for a given At present， the longest numerical published in journals are those of Duncan & Lissauer (1998). Although their main target was the effect of post-main-sequence mass loss on the stability of orbits， they performed many covering up to ～1011 yr of the orbital of the four jovian The initial orbital elements and masses of are the same as those of our system in Duncan & Lissauer's paper， but they decrease the mass of the Sun gradually in their numerical experiments. This is because they consider the effect of post-main-sequence mass loss in the paper. Consequently， they found that the crossing time-scale of orbits， which can 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 jovian remain stable over 1010 yr， or perhaps longer. Duncan & Lissauer also performed four experiments on the orbital of seven (Venus to Neptune)， which cover a span of ～109 yr. Their experiments on the seven are not yet comprehensive， but it seems that the terrestrial also remain stable during the ， maintaining almost

On the other hand， in his semi-analytical theory (Laskar 1988)， Laskar finds that large and can appear in the and of the terrestrial ， especially of Mercury and Mars on a time-scale of several 109 yr (Laskar 1996). The results of Laskar's theory should be confirmed and investigated by fully numerical

In this paper we present preliminary results of six long-term numerical on all nine orbits， covering a span of several 109 yr， and of two other covering a span of ± 5 × 1010 yr. The total time for all is more than 5 yr， using several dedicated PCs and One of the fundamental of our long-term is that system seems to be stable in terms of the Hill stability above， at least over a time-span of ± 4 Gyr. Actually， in our numerical the system was far more stable than what is defined by the Hill stability : not only did no close happen during the ， but also all the orbital elements have been confined in a narrow both in time and frequency domain， though are stochastic. Since the purpose of this paper is to exhibit and overview the results of our long-term numerical ， we show typical example figures as evidence of the very long-term stability of system For readers who have more specific and deeper interests in our numerical results， we have prepared a webpage ( )， where we show raw orbital elements， their low-pass filtered results， of elements and momentum deficit， and results of our simple time–frequency analysis on all of our

In 2 we briefly our dynamical model， numerical method and initial used in our 3 is devoted to a of the quick results of the numerical Very long-term stability of system is apparent both in and orbital elements. A rough of numerical errors is also given. 4 goes on to a of the longest-term of orbits using a low-pass filter and includes a of momentum deficit. In 5， we present a set of numerical for the outer five that spans ± 5 × 1010 yr. In 6 we also discuss the long-term stability of the and its possible cause.

2 of the numerical

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

The stepsize for the numerical is 8 d throughout all of the nine (N±1，2，3)， which is about 1/11 of the orbital of the innermost (Mercury). As for the of stepsize， we partly follow the numerical of all nine 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 reduce the of round-off error in the processes. In to this， Wisdom & Holman (1991) performed numerical of the outer five orbits using the symplectic map with a stepsize of 400 d， 1/10.83 of the orbital of Jupiter. Their result seems to be enough， which partly justifies our method of determining the stepsize. However， since the of Jupiter (～0.05) is much smaller than that of Mercury (～0.2)， we need some care when we compare these simply in terms of stepsizes.

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

We adopt Gauss' f and g in the symplectic map together with the third-order Halley method (Danby 1992) as a solver for Kepler The number of maximum we set in Halley's method is 15， but they never reached the maximum in any of our

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

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

2.4 Error

2.4.1 errors in total energy and momentum

to one of the basic properties of symplectic integrators， which conserve the physically conservative quantities well (total orbital energy and momentum)， our long-term numerical seem to have been performed with very small errors. The averaged errors of total energy (～10?9) and of total momentum (～10?11) have remained nearly constant throughout the (Fig. 1). The special startup procedure， warm start， would have reduced the averaged error in total energy by about one order of magnitude or more.

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

Note that different operating systems， different mathematical libraries， and different hardware architectures result in different numerical errors， through the in round-off error handling and numerical algorithms. In the upper panel of Fig. 1， we can this in the numerical error in the total momentum， which should be rigorously preserved up to machine-ε

2.4.2 Error in longitudes

Since the symplectic maps preserve total energy and total momentum of N-body dynamical systems inherently well， the degree of their may not be a good measure of the of numerical ， especially as a measure of the error of ， i.e. the error in longitudes. To estimate the numerical error in the longitudes， we performed the following procedures. We compared the result of our main long-term with some test ， which span much shorter but with much higher than the main For this purpose， we performed a much more with a stepsize of 0.125 d (1/64 of the main ) spanning 3 × 105 yr， starting with the same initial as in the N?1 We consider that this test provides us with a ‘pseudo-true’ of orbital Next， we compare the test with the main ， N?1. For the of 3 × 105 yr， we see a difference in mean anomalies of the Earth between the two of ～0.52°(in the case of the N?1 ). This difference can be to the value ～8700°， about 25 of Earth after 5 Gyr， since the error of longitudes increases linearly with time in the symplectic map. ， the longitude error of Pluto can 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. at the raw data

In this we briefly review the long-term stability of orbital through some snapshots of raw numerical data. The orbital of indicates long-term stability in all of our numerical : no orbital crossings nor close between any pair of took

3.1 General of the stability of orbits

First， we briefly look at the general character of the long-term stability of orbits. Our interest here focuses on the inner four terrestrial for which the orbital time-scales are much shorter than those of the outer five As we can see clearly from the orbital shown in Figs 2 and 3， orbital of the terrestrial differ little between the initial and final part of each numerical ， which spans several Gyr. The solid lines denoting the present orbits of the lie almost within the swarm of dots even in the final part of (b) and (d). This indicates that throughout the entire the almost of orbital remain nearly the same as they are at present.

Vertical view of the four inner orbits (from the z -axis ) at the initial and final parts of the ±1. The axes units are au. The xy - is set to the invariant of system total 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).(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 denote the present orbits of the four terrestrial (taken from DE245).