对火星轨道变化问题的最后解释

ld be confirmed and investigated by fully numerical integrations.

In this paper we present preliminary results of six long-term numerical integrations on all nine planetary orbits， covering a span of several 109 yr， and of two other integrations covering a span of ± 5 × 1010 yr. The total elapsed time for all integrations is more than 5 yr， using several dedicated PCs and workstations. One of the fundamental conclusions of our long-term integrations is that Solar system planetary motion seems to be stable in terms of the Hill stability mentioned above， at least over a time-span of ± 4 Gyr. Actually， in our numerical integrations the system was far more stable than what is defined by the Hill stability criterion: not only did no close encounter happen during the integration period， but also all the planetary orbital elements have been confined in a narrow region both in time and frequency domain， though planetary motions are stochastic. Since the purpose of this paper is to exhibit and overview the results of our long-term numerical integrations， we show typical example figures as evidence of the very long-term stability of Solar system planetary motion. For readers who have more specific and deeper interests in our numerical results， we have prepared a webpage (access )， where we show raw orbital elements， their low-pass filtered results， variation of Delaunay elements and angular momentum deficit， and results of our simple time–frequency analysis on all of our integrations.

In Section 2 we briefly explain our dynamical model， numerical method and initial conditions used in our integrations. Section 3 is devoted to a description of the quick results of the numerical integrations. Very long-term stability of Solar system planetary motion is apparent both in planetary positions and orbital elements. A rough estimation of numerical errors is also given. Section 4 goes on to a discussion of the longest-term variation of planetary orbits using a low-pass filter and inc