Vertical view of the four inner planetary orbits (from the z -axis direction) at the initial and final parts of the integrationsN±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).(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 planets (taken from 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 planetary orbital elements does not differ significantly between the initial and final part of each integration， at least for Venus， Earth and Mars. The elements of Mercury， especially its eccentricity， seem to change to a significant extent. This is partly because the orbital time-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 may be nearest to instability. This result appears to be in some agreement with Laskar's (1994， 1996) expectations that large and irregular variations appear in the eccentricities and inclinations 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 planetary system owing to the small mass of Mercury. We will mention briefly the long-term orbital evolution of Mercury later in Section 4 using low-pass filtered orbital elements.

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

3.2 Time–frequency maps

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

To give an overview of the long-term change in periodicity in planetary orbital motion， we performed many fast Fourier transformations (FFTs) along the time axis， and superposed the resulting periodgrams to draw two-dimensional 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 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 by which tn+T reaches the total integration 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 periodicity can be replaced by a grey-scale (or colour) chart.

We perform the replacement， and connect all the grey-scale (or colour) charts into one graph for each integration. 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 period (or frequency) of the oscillation of orbital elements.

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

4.2 Long-term exchange of orbital energy and angular momentum

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

In Fig. 7， the total orbital energy and angular momentum 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 energy (denoted asE- E0)， total angular momentum ( G- G0)， and the vertical component ( H- H0) of the inner four planets calculated from the low-pass filtered Delaunay 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 planets. The fluctuation shown in the lower panels is virtually entirely a result of the massive jovian planets.

Comparing the variations of energy and angular momentum 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 planetary subsystem is more stable than the outer one: this is simply a result of the relative smallness of the masses of the four terrestrial planets compared with those of the outer jovian planets. Another thing we notice is that the inner planetary subsystem may become 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-periodic and irregular oscillations are more apparent than in the panels denoted astotal 9. Actually， the fluctuations in theinner 4 panels are to a large extent as a result of the orbital variation of the Mercury. However， we cannot neglect the contribution from 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 planetary orbital energy and angular momentum expressed by the low-pass filtered Delaunay elements. Figs 10 and 11 show long-term evolution of the orbital energy of each planet and the angular momentum in N+1 and N?2 integrations. We notice that some planets form apparent pairs in terms of orbital energy and angular momentum exchange. In particular， Venus and Earth make a typical pair. In the figures， they show negative correlations in exchange of energy and positive correlations in exchange of angular momentum. The negative correlation in exchange of orbital energy means that the two planets form a closed dynamical system in terms of the orbital energy. The positive correlation in exchange of angular momentum means that the two planets are simultaneously under certain long-term 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 system. Mercury exhibits certain negative correlations in the angular momentum versus the Venus–Earth system， which seems to be a reaction caused by the conservation of angular momentum in the terrestrial planetary subsystem.

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

As for the outer jovian planetary 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 integrations of outer planetary orbits

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

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

5.1 Resonances in the Neptune–Pluto system

Kinoshita & Nakai (1996) integrated the outer five planetary orbits over ± 5.5 × 109 yr . They found that four major resonances between Neptune and Pluto are maintained during the whole integration period， and that the resonances may be the main causes of the stability of the orbit of Pluto. The major 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 yr.

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

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

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

In our numerical 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-yr time-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 dynamical mechanism maintains this long-term stability of the planetary 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 motion and secular) between any pair among the nine planets. Jupiter and Saturn are close to a 5:2 mean motion resonance (the famous ‘great inequality’)， but not just in the resonance zone. Higher-order resonances may cause the chaotic nature of the planetary dynamical motion， but they are not so strong as to destroy the stable planetary motion within the lifetime of the real Solar system. The second feature， which we think is more important for the long-term stability of our planetary system， is the difference in dynamical distance between terrestrial and jovian planetary subsystems (Ito & Tanikawa 1999， 2001). When we measure planetary separations by the mutual Hill radii (R_)， separations among terrestrial planets are greater than 26RH， whereas those among jovian planets are less than 14RH. This difference is directly related to the difference between dynamical features of terrestrial and jovian planets. Terrestrial planets have smaller masses， shorter orbital periods and wider dynamical separation. They are strongly perturbed by jovian planets that have larger masses， longer orbital periods and narrower dynamical separation. Jovian planets are not perturbed by any other massive bodies.

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

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

——以上文段引自 Ito， T.& Tanikawa， K. Long-term integrations and stability of planetary orbits in our Solar System. 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。