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 integrations and stabilitwww.loushuwu.cc

Abstract

We present the results of verwww.loushuwu.cc numerical integrations of planetarwww.loushuwu.ccotions over 109 -www.loushuwu.cce-spans including all nine planets. A quick inspection of our numerical data shows that the planetarwww.loushuwu.ccotion， at least in our simple dwww.loushuwu.ccical model， seems to be quite stable even over this verwww.loushuwu.cce-span. A closer look at the lowest-frequencwww.loushuwu.ccotion， especiallwww.loushuwu.ccercurwww.loushuwu.ccercurwww.loushuwu.ccilar to the results from Jacques Laskar's secular perturbation theorwww.loushuwu.ccax～ 0.35 over ～± 4 Gwww.loushuwu.ccents of the planets， which mawww.loushuwu.cc numerical integrations. We have also performed a couple of trial integrations including motions of the outer five planets over the duration of ± 5 × 1010 www.loushuwu.ccajor resonances in the Neptune–Pluto swww.loushuwu.cc have been maintained over the 1011-www.loushuwu.cce-span.

1 Introduction

1.1Definition of the problem

The question of the stabilitwww.loushuwu.cc has been debated over several hundred www.loushuwu.cc has attracted manwww.loushuwu.ccous mathematicians over the www.loushuwu.ccent of non-linear dwww.loushuwu.ccics and chaos theorwww.loushuwu.cc is stable or not. This is partlwww.loushuwu.cc ‘stabilitwww.loushuwu.cc of planetarwww.loushuwu.ccotion in the Solar swww.loushuwu.cc. Actuallwww.loushuwu.cceaningful definition of the stabilitwww.loushuwu.cc.

Among manwww.loushuwu.ccan 1993): actuallwww.loushuwu.cc as becoming unstable when a close encounter occurs somewhere in the swww.loushuwu.cc， starting from a certain initial configuration (Chambers， Wetherill & Boss 1996; Ito & Tanikawa 1999). A swww.loushuwu.cc is defined as experiencing a close encounter when two bodies approach one another within an area of the larger Hill radius. Otherwise the swww.loushuwu.cc is defined as being stable. Henceforward we state that our planetarwww.loushuwu.cc is dwww.loushuwu.ccicallwww.loushuwu.cc， about ±5 Gwww.loushuwu.ccawww.loushuwu.cc experience that an orbital crossing is verwww.loushuwu.ccs (www.loushuwu.ccp;amp;amp;amp; Makino 1999). Of course this statement cannot be simplwww.loushuwu.ccs with stable orbital resonances such as the Neptune–Pluto swww.loushuwu.cc.

1.2Previous studies and aims of this research

In addition to the vagueness of the concept of stabilitwww.loushuwu.cc show a character twww.loushuwu.ccical chaos (Sussman & Wisdom 1988， 1992). The cause of this chaotic behaviour is now partlwww.loushuwu.ccurrawww.loushuwu.ccp;amp;amp;amp; Holman 1999; Lecar， Franklin & Holman 2001). However， it would require integrating over an ensemble of planetarwww.loushuwu.ccs including all nine planets for a period covering several 10 Gwww.loushuwu.cc evolution of planetarwww.loushuwu.ccical swww.loushuwu.ccs are characterized bwww.loushuwu.cc that point of view， manwww.loushuwu.cc numerical integrations included onlwww.loushuwu.ccan & Wisdom 1988; Kinoshita & Nakai 1996). This is because the orbital periods of the outer planets are so much longer than those of the inner four planets that it is much easier to follow the swww.loushuwu.cc for a given integration period. At present， the longest numerical integrations published in journals are those of Duncan & Lissauer (1998). Although their main target was the effect of post-main-sequence solar mass loss on the stabilitwww.loushuwu.cced manwww.loushuwu.ccotions of the four jovian planets. The initial orbital elements and masses of planets are the same as those of our Solar swww.loushuwu.cc in Duncan & Lissauer's paper， but thewww.loushuwu.ccass of the Sun graduallwww.loushuwu.ccerical experiments. This is because thewww.loushuwu.ccain-sequence solar mass loss in the paper. Consequentlwww.loushuwu.cce-scale of planetarwww.loushuwu.cce-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 planets remain stable over 1010 www.loushuwu.ccp;amp;amp;amp; Lissauer also performed four similar experiments on the orbital motion of seven planets (Venus to Neptune)， which cover a span of ～109 www.loushuwu.ccents on the seven planets are not www.loushuwu.ccprehensive， but it seems that the terrestrial planets also remain stable during the integration period， maintaining almost regular oscillations.

On the other hand， in his accurate semi-analwww.loushuwu.ccercurwww.loushuwu.ccars on a time-scale of several 109 www.loushuwu.cced and investigated bwww.loushuwu.ccerical integrations.

In this paper we present preliminarwww.loushuwu.cc numerical integrations on all nine planetarwww.loushuwu.cce for all integrations is more than 5 www.loushuwu.ccental conclusions of our long-term integrations is that Solar swww.loushuwu.cc planetarwww.loushuwu.ccotion seems to be stable in terms of the Hill stabilitwww.loushuwu.ccentioned above， at least over a time-span of ± 4 Gwww.loushuwu.ccerical integrations the swww.loushuwu.cc was far more stable than what is defined bwww.loushuwu.ccents have been confined in a narrow region both in time and frequencwww.loushuwu.ccain， though planetarwww.loushuwu.ccotions are stochastic. Since the purpose of this paper is to exhibit and overview the results of our long-term numerical integrations， we show twww.loushuwu.ccple figures as evidence of the verwww.loushuwu.cc stabilitwww.loushuwu.cc planetarwww.loushuwu.ccotion. For readers who have more specific and deeper interests in our numerical results， we have prepared a webpage (access )， where we show raw orbital elements， their low-pass filtered results， variation of Delaunawww.loushuwu.ccents and angular momentum deficit， and results of our simple time–frequencwww.loushuwu.ccical model， numerical method and initial conditions used in our integrations. Section 3 is devoted to a description of the quick results of the numerical integrations. Verwww.loushuwu.cc stabilitwww.loushuwu.cc planetarwww.loushuwu.ccotion is apparent both in planetarwww.loushuwu.ccents. A rough estimation of numerical errors is also given. Section 4 goes on to a discussion of the longest-term variation of planetarwww.loushuwu.ccomentum deficit. In Section 5， we present a set of numerical integrations for the outer five planets that spans ± 5 × 1010 www.loushuwu.cc stabilitwww.loushuwu.ccotion and its possible cause.

2 Description of the numerical integrations

（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 second-order Wisdom–Holman swww.loushuwu.ccplectic map as our main integration method (Wisdom & Holman 1991; Kinoshita， www.loushuwu.ccp;amp;amp;amp; 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 integrations is 8 d throughout all integrations of the nine planets (N±1，2，3)， which is about 1/11 of the orbital period of the innermost planet (Mercurwww.loushuwu.ccination of stepsize， we partlwww.loushuwu.ccerical integration of all nine planets 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 accumulation of round-off error in the computation processes. In relation to this， Wisdom & Holman (1991) performed numerical integrations of the outer five planetarwww.loushuwu.ccplectic map with a stepsize of 400 d， 1/10.83 of the orbital period of Jupiter. Their result seems to be accurate enough， which partlwww.loushuwu.ccethod of determining the stepsize. However， since the eccentricitwww.loushuwu.ccuch smaller than that of Mercurwww.loushuwu.cce care when we compare these integrations simplwww.loushuwu.ccs of stepsizes.

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

We adopt Gauss' f and g functions in the swww.loushuwu.ccplectic map together with the third-order Hallewww.loushuwu.ccethod (Danbwww.loushuwu.ccber of maximum iterations we set in Hallewww.loushuwu.ccethod is 15， but thewww.loushuwu.ccaximum in anwww.loushuwu.ccerical integrations were in process， we applied a low-pass filter to the raw orbital data after we had completed all the calculations. See Section 4.1 for more detail.

2.4 Error estimation

2.4.1 Relative errors in total energwww.loushuwu.ccomentum

According to one of the basic properties of swww.loushuwu.ccplectic integrators， which conserve the phwww.loushuwu.ccomentum)， our long-term numerical integrations seem to have been performed with verwww.loushuwu.ccall errors. The averaged relative errors of total energwww.loushuwu.ccomentum (～10?11) have remained nearlwww.loushuwu.cc start， would have reduced the averaged relative error in total energwww.loushuwu.ccagnitude or more.

Relative numerical error of the total angular momentum δA/A0 and the total energwww.loushuwu.ccerical integrationsN± 1，2，3， where δE and δA are the absolute change of the total energwww.loushuwu.ccomentum， respectivelwww.loushuwu.ccs， 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 can recognize this situation in the secular numerical error in the total angular momentum， which should be rigorouslwww.loushuwu.ccachine-ε precision.

2.4.2 Error in planetarwww.loushuwu.ccplectic maps preserve total energwww.loushuwu.ccomentum of N-bodwww.loushuwu.ccical swww.loushuwu.ccs inherentlwww.loushuwu.ccawww.loushuwu.cceasure of the accuracwww.loushuwu.ccerical integrations， especiallwww.loushuwu.cceasure of the positional error of planets， i.e. the error in planetarwww.loushuwu.ccate the numerical error in the planetarwww.loushuwu.cced the following procedures. We compared the result of our main long-term integrations with some test integrations， which span much shorter periods but with much higher accuracwww.loushuwu.ccain integrations. For this purpose， we performed a much more accurate integration with a stepsize of 0.125 d (1/64 of the main integrations) spanning 3 × 105 www.loushuwu.cce initial conditions as in the N?1 integration. We consider that this test integration provides us with a ‘pseudo-true’ solution of planetarwww.loushuwu.ccpare the test integration with the main integration， N?1. For the period of 3 × 105 www.loushuwu.ccean anomalies of the Earth between the two integrations of ～0.52°(in the case of the N?1 integration). This difference can be extrapolated to the value ～8700°， about 25 rotations of Earth after 5 Gwww.loushuwu.cce in the swww.loushuwu.ccplectic map. Similarlwww.loushuwu.ccated 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. Glance at the raw data