第一百日（1）作屎的老猫（第4/6页）

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.