对火星轨道变化问题的最后解释 (3 / 5)
        2.4.2 Error in planetary longitudes

        Since the symplectic maps preserve total energy and total angular momentum of N-body dynamical systems inherently well， the degree of their preservation may not be a good measure of the accuracy of numerical integrations， especially as a measure of the positional error of planets， i.e. the error in planetary longitudes. To estimate the numerical error in the planetary longitudes， we performed the following procedures. We pared the result of our main long-term integrations with some test integrations， which span much shorter periods but with much higher accuracy than the main 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 yr， starting with the same initial conditions as in the N?1 integration. We consider that this test integration provides us with a ‘pseudo-true’ solution of planetary orbital evolution. Next， we pare the test integration with the main integration， N?1. For the period of 3 × 105 yr， we see a difference in mean 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 Gyr， since the error of longitudes increases linearly with time in the symplectic map. Similarly， the longitude error of Pluto can be estimated as ～12°. This value for Pluto is much better than the result in Kino**a & Nakai (1996) where the difference is estimated as ～60°.

        3 Numerical results – I. Glance at the raw data

        In this section we briefly review the long-term stability of planetary orbital motion through some snapshots of raw numerical data. The orbital motion of planets indicates long-term stability in all of our numerical integrations: no orbital crossings nor close encounters between any pair of planets took place.

        3.1 General description of the stability of planetary orbits

        First， we briefly look at the general character of the long-term stability of planetary orbits. Our interest here focuses particularly on the inner four terrestrial planets for which the orbital time-scales are much shorter than those of the outer five planets. As we can see clearly from 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 Gyr. The solid lines denoting the present orbits of the planets lie almost 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 planetary orbital motion remain nearly the same as they are at present.

        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.

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