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Long-term integrations and stability of planetary orbits in our Solar system

We present the results of very long-term numerical integrations of planetary orbital motions over 10 9 -yr time-spans including all nine planets. A quick inspection of our numerical data shows that the planetary motion, atleast in our simple dynamical model, seems to be quite stable even over this very long time-span. A closer look at the lowest-frequency oscillations using a low-pass filter shows us the potentially diffusive character of terrestrial planetary motion, especially that of Mercury. The behaviour of the eccentricity of Mercury in our integrations is qualitatively similar to the results from Jacques Laskar's secular perturbation theory (e.g. e m a x ∼ 0.35 over ∼′4 Gyr). However, there are no apparent secular increases of eccentricity or inclination in any orbital elements of the planets, which may be revealed by still longer-term numerical integrations. We have also performed a couple of trial integrations including motions of the outer five planets over the duration of ′5 x 101° yr. The result indicates that the three major resonances in the Neptune-Pluto system have been maintained over the 10 1 1 -yr time-span.