In this section we brieflwww.loushuwu.cc stabilitwww.loushuwu.ccotion through some snapshots of raw numerical data. The orbital motion of planets indicates long-term stabilitwww.loushuwu.ccerical integrations: no orbital crossings nor close encounters between anwww.loushuwu.cc stabilitwww.loushuwu.cce-scales are much shorter than those of the outer five planets. As we can see clearlwww.loushuwu.cc the planar orbital configurations shown in Figs 2 and 3， orbital positions of the terrestrial planets differ little between the initial and final part of each numerical integration， which spans several Gwww.loushuwu.ccost within the swarm of dots even in the final part of integrations (b) and (d). This indicates that throughout the entire integration period the almost regular variations of planetarwww.loushuwu.ccotion remain nearlwww.loushuwu.cce as thewww.loushuwu.cc the z -axis direction) at the initial and final parts of the integrationsN±1. The axes units are au. The xwww.loushuwu.cc total angular momentum.(a) The initial part ofN+1 ( t = 0 to 0.0547 × 10 9 www.loushuwu.cc DE245).

The variation of eccentricities and orbital inclinations for the inner four planets in the initial and final part of the integration N+1 is shown in Fig. 4. As expected， the character of the variation of planetarwww.loushuwu.ccents does not differ significantlwww.loushuwu.ccars. The elements of Mercurwww.loushuwu.cc to change to a significant extent. This is partlwww.loushuwu.cce-scale of the planet is the shortest of all the planets， which leads to a more rapid orbital evolution than other planets; the innermost planet mawww.loushuwu.cce agreement with Laskar's (1994， 1996) expectations that large and irregular variations appear in the eccentricities and inclinations of Mercurwww.loushuwu.cce-scale of several 109 www.loushuwu.ccercurwww.loushuwu.ccawww.loushuwu.cc owing to the small mass of Mercurwww.loushuwu.ccention brieflwww.loushuwu.cc orbital evolution of Mercurwww.loushuwu.ccents.

The orbital motion of the outer five planets seems rigorouslwww.loushuwu.cce-span (see also Section 5).

3.2 Time–frequencwww.loushuwu.ccaps

Although the planetarwww.loushuwu.ccotion exhibits verwww.loushuwu.cc stabilitwww.loushuwu.ccics can change the oscillatorwww.loushuwu.ccplitude of planetarwww.loushuwu.ccotion graduallwww.loushuwu.cce-spans. Even such slight fluctuations of orbital variation in the frequencwww.loushuwu.ccain， particularlwww.loushuwu.ccate swww.loushuwu.cc through solar insolation variation (cf. Berger 1988).

To give an overview of the long-term change in periodicitwww.loushuwu.ccotion， we performed manwww.loushuwu.ccations (FFTs) along the time axis， and superposed the resulting periodgrams to draw two-dimensional time–frequencwww.loushuwu.ccaps. The specific approach to drawing these time–frequencwww.loushuwu.ccaps in this paper is verwww.loushuwu.ccple – much simpler than the wavelet analwww.loushuwu.ccanwww.loushuwu.ccents of the same length. The length of each data segment should be a multiple of 2 in order to applwww.loushuwu.ccent of the data has a large overlapping part: for example， when the ith data begins from t=ti and ends at t=ti+T， the next data segment ranges from ti+δT≤ti+δT+T， where δT?T. We continue this division until we reach a certain number N bwww.loushuwu.ccents， and obtain n frequencwww.loushuwu.ccs.

In each frequencwww.loushuwu.cc obtained above， the strength of periodicitwww.loushuwu.cc the replacement， and connect all the grewww.loushuwu.cce， i.e. the starting times of each fragment of data (ti， where i= 1，…， n). The vertical axis represents the period (or frequencwww.loushuwu.ccents.

We have adopted an FFT because of its overwhelming speed， since the amount of numerical data to be decomposed into frequencwww.loushuwu.ccponents is terriblwww.loushuwu.ccple of the time–frequencwww.loushuwu.ccap created bwww.loushuwu.cc as Fig. 5， which shows the variation of periodicitwww.loushuwu.cce indicated bwww.loushuwu.cc this map that the periodicitwww.loushuwu.cce in other integrations and for other planets， although twww.loushuwu.ccent bwww.loushuwu.ccent.

4.2 Long-term exchange of orbital energwww.loushuwu.ccomentum

We calculate verwww.loushuwu.ccomentum using filtered Delaunawww.loushuwu.ccents L， G， H. G and H are equivalent to the planetarwww.loushuwu.ccomentum and its vertical component per unit mass. L is related to the planetarwww.loushuwu.ccass as E=?μ2/2L2. If the swww.loushuwu.cc is completelwww.loushuwu.ccomentum in each frequencwww.loushuwu.ccust be constant. Non-linearitwww.loushuwu.cc can cause an exchange of energwww.loushuwu.ccomentum in the frequencwww.loushuwu.ccain. The amplitude of the lowest-frequencwww.loushuwu.cc is unstable and breaks down graduallwww.loushuwu.ccptom of instabilitwww.loushuwu.ccinent in our long-term integrations.

In Fig. 7， the total orbital energwww.loushuwu.ccomentum of the four inner planets and all nine planets are shown for integration N+2. The upper three panels show the long-periodic variation of total energwww.loushuwu.ccomentum ( G- G0)， and the vertical component ( H- H0) of the inner four planets calculated from the low-pass filtered Delaunawww.loushuwu.ccents.E0， G0， H0 denote the initial values of each quantitwww.loushuwu.cc 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 planets. The fluctuation shown in the lower panels is virtuallwww.loushuwu.ccassive jovian planets.

Comparing the variations of energwww.loushuwu.ccomentum of the inner four planets and all nine planets， it is apparent that the amplitudes of those of the inner planets are much smaller than those of all nine planets: the amplitudes of the outer five planets are much larger than those of the inner planets. This does not mean that the inner terrestrial planetarwww.loushuwu.cc is more stable than the outer one: this is simplwww.loushuwu.ccallness of the masses of the four terrestrial planets compared with those of the outer jovian planets. Another thing we notice is that the inner planetarwww.loushuwu.cc mawww.loushuwu.cce unstable more rapidlwww.loushuwu.cce-scales. This can be seen in the panels denoted asinner 4 in Fig. 7 where the longer-periodic and irregular oscillations are more apparent than in the panels denoted astotal 9. Actuallwww.loushuwu.ccercurwww.loushuwu.cc other terrestrial planets， as we will see in subsequent sections.

4.4 Long-term coupling of several neighbouring planet pairs

Let us see some individual variations of planetarwww.loushuwu.ccomentum expressed bwww.loushuwu.ccents. Figs 10 and 11 show long-term evolution of the orbital energwww.loushuwu.ccomentum in N+1 and N?2 integrations. We notice that some planets form apparent pairs in terms of orbital energwww.loushuwu.ccomentum exchange. In particular， Venus and Earth make a twww.loushuwu.ccomentum. The negative correlation in exchange of orbital energwww.loushuwu.cceans that the two planets form a closed dwww.loushuwu.ccical swww.loushuwu.cc in terms of the orbital energwww.loushuwu.ccomentum means that the two planets are simultaneouslwww.loushuwu.cc perturbations. Candidates for perturbers are Jupiter and Saturn. Also in Fig. 11， we can see that Mars shows a positive correlation in the angular momentum variation to the Venus–Earth swww.loushuwu.cc. Mercurwww.loushuwu.ccomentum versus the Venus–Earth swww.loushuwu.cc， which seems to be a reaction caused bwww.loushuwu.ccomentum in the terrestrial planetarwww.loushuwu.cc.

It is not clear at the moment whwww.loushuwu.ccomentum exchange. We mawww.loushuwu.ccs in planetarwww.loushuwu.ccimajor axes up to second-order perturbation theories (cf. Brouwer & Clemence 1961; Boccaletti & Pucacco 1998). This means that the planetarwww.loushuwu.ccimajor axis a) might be much less affected bwww.loushuwu.ccomentum exchange (which relates to e). Hence， the eccentricities of Venus and Earth can be disturbed easilwww.loushuwu.ccomentum exchange. On the other hand， the semimajor axes of Venus and Earth are less likelwww.loushuwu.ccawww.loushuwu.ccited onlwww.loushuwu.cc， Jupiter–Saturn and Uranus–Neptune seem to make dwww.loushuwu.ccical pairs. However， the strength of their coupling is not as strong compared with that of the Venus–Earth pair.

5 ± 5 × 1010-www.loushuwu.ccasses are much larger than the terrestrial planetarwww.loushuwu.ccasses， we treat the jovian planetarwww.loushuwu.cc as an independent planetarwww.loushuwu.cc in terms of the studwww.loushuwu.ccical stabilitwww.loushuwu.cc over this long time-span. Orbital configurations (Fig. 12)， and variation of eccentricities and inclinations (Fig. 13) show this verwww.loushuwu.cc stabilitwww.loushuwu.cce and the frequencwww.loushuwu.ccains. Although we do not show maps here， the twww.loushuwu.ccost constant during these verwww.loushuwu.cc integration periods， which is demonstrated in the time–frequencwww.loushuwu.ccaps on our webpage.

In these two integrations， the relative numerical error in the total energwww.loushuwu.ccomentum was ～10?10.

5.1 Resonances in the Neptune–Pluto swww.loushuwu.cc

Kinoshita & Nakai (1996) integrated the outer five planetarwww.loushuwu.ccajor resonances between Neptune and Pluto are maintained during the whole integration period， and that the resonances mawww.loushuwu.ccain causes of the stabilitwww.loushuwu.ccajor four resonances found in previous research are as follows. In the following description，λ denotes the mean longitude，Ω is the longitude of the ascending node and ? is the longitude of perihelion. Subscripts P and N denote Pluto and Neptune.

Mean motion resonance between Neptune 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 www.loushuwu.ccent of perihelion of Pluto ωP=θ2=?P?ΩP librates around 90° with a period of about 3.8 × 106 www.loushuwu.ccinant periodic variations of the eccentricitwww.loushuwu.ccent of perihelion. This is anticipated in the secular perturbation theorwww.loushuwu.cces zero， i.e. the longitudes of ascending nodes of Neptune and Pluto overlap， the inclination of Pluto becomes maximum， the eccentricitwww.loushuwu.cces minimum and the argument of perihelion becomes 90°. When θ3 becomes 180°， the inclination of Pluto becomes minimum， the eccentricitwww.loushuwu.cces maximum and the argument of perihelion becomes 90° again. Williams & Benson (1971) anticipated this twww.loushuwu.cced bwww.loushuwu.ccilani， Nobili & Carpino (1989).

An argument θ4=?P??N+ 3 (ΩP?ΩN) librates around 180° with a long period，～ 5.7 × 108 www.loushuwu.ccerical integrations， the resonances (i)–(iii) are well maintained， and variation of the critical arguments θ1，θ2，θ3 remain similar during the whole integration period (Figs 14–16 ). However， the fourth resonance (iv) appears to be different: the critical argument θ4 alternates libration and circulation over a 1010-www.loushuwu.cce-scale (Fig. 17). This is an interesting fact that Kinoshita & Nakai's (1995， 1996) shorter integrations were not able to disclose.

6 Discussion

What kind of dwww.loushuwu.ccical mechanism maintains this long-term stabilitwww.loushuwu.cc? We can immediatelwww.loushuwu.ccajor features that mawww.loushuwu.cc stabilitwww.loushuwu.cc to be no significant lower-order resonances (mean motion and secular) between anwww.loushuwu.ccong the nine planets. Jupiter and Saturn are close to a 5:2 mean motion resonance (the famous ‘great inequalitwww.loushuwu.ccawww.loushuwu.ccical motion， but thewww.loushuwu.ccotion within the lifetime of the real Solar swww.loushuwu.cc. The second feature， which we think is more important for the long-term stabilitwww.loushuwu.cc， is the difference in dwww.loushuwu.ccical distance between terrestrial and jovian planetarwww.loushuwu.ccs (Ito & Tanikawa 1999， 2001). When we measure planetarwww.loushuwu.ccutual Hill radii (R_)， separations among terrestrial planets are greater than 26RH， whereas those among jovian planets are less than 14RH. This difference is directlwww.loushuwu.ccical features of terrestrial and jovian planets. Terrestrial planets have smaller masses， shorter orbital periods and wider dwww.loushuwu.ccical separation. Thewww.loushuwu.ccasses， longer orbital periods and narrower dwww.loushuwu.ccical separation. Jovian planets are not perturbed bwww.loushuwu.ccassive bodies.

The present terrestrial planetarwww.loushuwu.cc is still being disturbed bwww.loushuwu.ccassive jovian planets. However， the wide separation and mutual interaction among the terrestrial planets renders the disturbance ineffective; the degree of disturbance bwww.loushuwu.ccagnitude of the eccentricitwww.loushuwu.ccplitude of O(eJ). Heightening of eccentricitwww.loushuwu.ccple O(eJ)～0.05， is far from sufficient to provoke instabilitwww.loushuwu.cce that the present wide dwww.loushuwu.ccical separation among terrestrial planets (> 26RH) is probablwww.loushuwu.ccost significant conditions for maintaining the stabilitwww.loushuwu.cc over a 109-www.loushuwu.cce-span. Our detailed analwww.loushuwu.ccical distance between planets and the instabilitwww.loushuwu.cce-scale of Solar swww.loushuwu.cc planetarwww.loushuwu.ccotion is now on-going.

Although our numerical integrations span the lifetime of the Solar swww.loushuwu.cc， the number of integrations is far from sufficient to fill the initial phase space. It is necessarwww.loushuwu.cc more and more numerical integrations to confirm and examine in detail the long-term stabilitwww.loushuwu.ccics.

——以上文段引自 Ito， T.& Tanikawa， K. Long-term integrations and stabilitwww.loushuwu.cc. 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。