Using a modified leapfrog method as a basic mapping, we produce a new numerical integrator for the stellar dynamical few‐body problem. We do not use coordinate transformation and the differential equations are not regularized, but the leapfrog algorithm gives regular results even for collision orbits. For this reason, application of extrapolation methods gives high precision. We compare the new integrator with several others and find it promising. Especially interesting is its efficiency for some potentials that differ from the Newtonian one at small distances.
We present the results of very long‐term numerical integrations of planetary orbital motions over 109 ‐yr time‐spans including all nine planets. A quick inspection of our numerical data shows that the planetary motion, at least 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. emax∼ 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 × 1010 yr. The result indicates that the three major resonances in the Neptune–Pluto system have been maintained over the 1011‐yr time‐span.
We conducted extensive numerical experiments of equal mass three‐body systems until they became disrupted. The system lifetimes, as a bound triple, and the Lyapunov times show a correlation similar to what has been earlier obtained for small bodies in the Solar system. Numerical integrations of several sets of differently randomized initial conditions produced the same relationship of the instability time and Lyapunov time. Marginal probability densities of the various times in the three‐body experiments are also discussed. Our high accuracy numerical method for three‐body orbit computations and Lyapunov time determinations is concisely described.
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