The of and orbital for the inner four in the initial and final part of the N+1 is shown in Fig. 4. As expected， the character of the of orbital elements does not differ significantly between the initial and final part of each ， at least for Venus， Earth and Mars. The elements of Mercury， especially its ， seem to change to a significant extent. This is partly because the orbital time-scale of the is the shortest of all the ， which leads to a more rapid orbital than other ; the innermost may be nearest to instability. This result appears to be in some agreement with Laskar's (1994， 1996) that large and appear in the and 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 system owing to the small mass of Mercury. We will briefly the long-term orbital of Mercury later in 4 using low-pass filtered orbital elements.

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

3.2 Time–frequency maps

Although the exhibits very long-term stability defined as the non-existence of close events， the chaotic nature of dynamics can change the and amplitude of orbital gradually over such long time-spans. Even such slight of orbital in the frequency domain， in the case of Earth， can potentially have a significant effect on its surface climate system through (cf. Berger 1988).

To give an overview of the long-term change in in orbital ， we performed many fast Fourier (FFTs) along the time axis， and superposed the resulting to draw two- 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) frequency analysis.

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 large 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 until we reach a certain number N by which tn+T reaches the total length.

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

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

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

We have adopted an FFT because of its overwhelming speed， since the amount of numerical data to be into frequency components 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 of in the and of Earth in N+2 In Fig. 5， the dark area shows that at the time indicated by the value on the abscissa， the indicated by the ordinate is stronger than in the lighter area around it. We can from this map that the of the and of Earth only changes slightly over the entire covered by the N+2 This nearly trend is qualitatively the same in other and for other ， although typical frequencies differ by and element by element.

4.2 Long-term exchange of orbital energy and momentum

We very long- and exchange of orbital energy and momentum using filtered elements L， G， H. G and H are equivalent to the orbital momentum and its vertical component per unit mass. L is to the orbital energy E per unit mass as E=?μ2/2L2. If the system is completely linear， the orbital energy and the momentum in each frequency bin must be constant. Non-linearity in the system can cause an exchange of energy and momentum in the frequency domain. The amplitude of the lowest-frequency should increase if the system is unstable and breaks down gradually. However， such a symptom of instability is not prominent in our long-term

In Fig. 7， the total orbital energy and momentum of the four inner and all nine are shown for N+2. The upper three panels show the long- of total energy (denoted asE- E0)， total momentum ( G- G0)， and the vertical component ( H- H0) of the inner four from the low-pass filtered elements.E0， G0， H0 denote the initial values of each quantity. The absolute difference from 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 The shown in the lower panels is virtually entirely a result of the massive jovian

Comparing the of energy and momentum of the inner four and all nine ， it is apparent that the amplitudes of those of the inner are much smaller than those of all nine : the amplitudes of the outer five are much larger than those of the inner This does not mean that the inner terrestrial subsystem is more stable than the outer one: this is simply a result of the smallness of the masses of the four terrestrial compared with those of the outer jovian Another thing we notice is that the inner subsystem may unstable more rapidly than the outer one because of its shorter orbital time-scales. This can be seen in the panels denoted asinner 4 in Fig. 7 where the longer- and are more apparent than in the panels denoted astotal 9. Actually， the in theinner 4 panels are to a large extent as a result of the orbital of the Mercury. However， we cannot neglect the from other terrestrial ， as we will see in subsequent

4.4 Long-term coupling of several neighbouring pairs

Let us see some individual of orbital energy and momentum expressed by the low-pass filtered elements. Figs 10 and 11 show long-term of the orbital energy of each and the momentum in N+1 and N?2 We notice that some form apparent pairs in terms of orbital energy and momentum exchange. In ， Venus and Earth make a typical pair. In the figures， they show negative in exchange of energy and positive in exchange of momentum. The negative in exchange of orbital energy means that the two form a closed dynamical system in terms of the orbital energy. The positive in exchange of momentum means that the two are simultaneously under certain long-term Candidates for perturbers are Jupiter and Saturn. Also in Fig. 11， we can see that Mars shows a positive in the momentum to the Venus–Earth system. Mercury exhibits certain negative in the momentum versus the Venus–Earth system， which seems to be a caused by the of momentum in the terrestrial subsystem.

It is not clear at the moment why the Venus–Earth pair exhibits a negative in energy exchange and a positive in momentum exchange. We may possibly this through observing the general fact that there are no terms in semimajor axes up to -order theories (cf. Brouwer & Clemence 1961; & 1998). This means that the orbital energy (which is directly to the semimajor axis a) might be much less affected by perturbing than is the momentum exchange (which to e). Hence， the of Venus and Earth can be disturbed easily by Jupiter and Saturn， which results in a positive in the momentum exchange. On the other hand， the semimajor axes of Venus and Earth are less likely to be disturbed by the jovian Thus the energy exchange may be limited only within the Venus–Earth pair， which results in a negative in the exchange of orbital energy in the pair.

As for the outer jovian subsystem， Jupiter–Saturn and Uranus–Neptune seem to make dynamical pairs. However， the strength of their coupling is not as strong compared with that of the Venus–Earth pair.

5 ± 5 × 1010-yr of outer orbits

Since the jovian masses are much larger than the terrestrial masses， we treat the jovian system as an independent system in terms of the study of its dynamical stability. Hence， we added a couple of trial that span ± 5 × 1010 yr， including only the outer five (the four jovian plus Pluto). The results exhibit the rigorous stability of the outer system over this long time-span. Orbital (Fig. 12)， and of and (Fig. 13) show this very long-term stability of the outer five in both the time and the frequency domains. Although we do not show maps here， the typical frequency of the orbital of Pluto and the other outer is almost constant during these very long-term ， which is demonstrated in the time–frequency maps on our webpage.

In these two ， the numerical error in the total energy was ～10?6 and that of the total momentum was ～10?10.

5.1 Resonances in the Neptune–Pluto system

Kinoshita & Nakai (1996) integrated the outer five orbits over ± 5.5 × 109 yr . They found that four major resonances between Neptune and Pluto are maintained during the whole ， and that the resonances may be the main causes of the stability of the orbit of Pluto. The major four resonances found in research are as follows. In the following ，λ denotes the mean longitude，Ω is the longitude of the ascending node and ? is the longitude of Subs P and N denote Pluto and Neptune.

Mean 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 of about 2 × 104 yr.

The argument of of Pluto ωP=θ2=?P?ΩP librates around 90° with a of about 3.8 × 106 yr. The dominant of the and of Pluto are synchronized with the of its argument of This is anticipated in the theory constructed by Kozai (1962).

The longitude of the node of Pluto referred to the longitude of the node of Neptune，θ3=ΩP?ΩN， and the of this is equal to the of θ2 When θ3 zero， i.e. the longitudes of ascending nodes of Neptune and Pluto ， the of Pluto maximum， the minimum and the argument of 90°. When θ3 180°， the of Pluto minimum， the maximum and the argument of 90° again. Williams & Benson (1971) anticipated this type of resonance， later confirmed by ， Nobili & Carpino (1989).

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

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

6

What kind of dynamical mechanism maintains this long-term stability of the system? We can immediately think of two major features that may be responsible for the long-term stability. First， there seem to be no significant lower-order resonances (mean and ) between any pair among the nine Jupiter and Saturn are close to a 5:2 mean resonance (the famous ‘great inequality’)， but not just in the resonance zone. Higher-order resonances may cause the chaotic nature of the dynamical ， but they are not so strong as to destroy the stable within the lifetime of the real system. The feature， which we think is more important for the long-term stability of our system， is the difference in dynamical distance between terrestrial and jovian subsystems (Ito & Tanikawa 1999， 2001). When we measure by the mutual Hill radii (R_)， among terrestrial are greater than 26RH， whereas those among jovian are less than 14RH. This difference is directly to the difference between dynamical features of terrestrial and jovian Terrestrial have smaller masses， shorter orbital and wider dynamical They are strongly perturbed by jovian that have larger masses， longer orbital and narrower dynamical Jovian are not perturbed by any other massive bodies.

The present terrestrial system is still being disturbed by the massive jovian However， the wide and mutual among the terrestrial renders the disturbance ineffective; the degree of disturbance by jovian is O(eJ)(order of magnitude of the of Jupiter)， since the disturbance caused by jovian is a forced having an amplitude of O(eJ). Heightening of ， for example O(eJ)～0.05， is far from sufficient to provoke instability in the terrestrial having such a wide as 26RH. Thus we assume that the present wide dynamical among terrestrial (> 26RH) is probably one of the most significant for maintaining the stability of the system over a 109-yr time-span. Our detailed analysis of the between dynamical distance between and the instability time-scale of system is now on-going.

Although our numerical span the lifetime of the system， the number of is far from sufficient to fill the initial phase space. It is necessary to perform more and more numerical to confirm and examine in detail the long-term stability of our dynamics.

——以上文段引自 Ito， T.& Tanikawa， K. Long-term and stability of orbits in our System. Mon. Not. R. Astron. Soc. 336， 483–500 (2002)

这只是作者君参考的一篇文章，关於太阳系的稳定性。

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。