For a two-body system (sun and one companion), there is a simple, elegant solution: a conic section (circle or ellipse, parabola or hyperbola). However, the presence of a third body (or in the case of the solar system, the sun, the nine major planets, and myriad minor bodies) allows no simple solution to these simple equations!
The orbital evolution of planetary orbits on giga-year time scales has been investigated recently via several numerical simulations. These have led to a most interesting conclusion that the orbits of the planets themselves evolve chaotically. The characteristic Lyapunov time is 5–10 million years. A new analytic theory (
2) shows that chaos among the Jovian planets results from a delicate interaction—also a three-body resonance—among Jupiter, Saturn, and Uranus. The theory also confirms the numerical estimate of the Lyapunov time associated with this chaos and shows that the escape time of Uranus is long (1018 years), substantially longer than the lifetime of our sun.
Although the numerical simulations all indicate chaos in planetary orbits, in a qualitative sense the planetary orbits are stable—because the planets remain near their present orbits—over the lifetime of the sun. However, the presence of chaos implies that there is a finite limit to how accurately the positions of the planets can be predicted over long times. Of all of the planets, Mercury's orbit appears to exhibit changes of the largest magnitude in orbital eccentricity and inclination. Fortunately, this is not fatal to the global stability of the whole planetary system owing to the small mass of Mercury. Changes in the orbit of the Earth, which can have potentially large effects on its surface climate system through solar insolation variation, are found also to be small.
Takashi Ito discussed several properties that may be responsible for the long term stability of our solar system. Among these, the difference in
dynamical separation¶ between terrestrial and Jovian planetary subsystems seems to be quite interesting and important. The terrestrial planets have smaller masses, shorter orbital periods, and wider dynamical separation. They are strongly perturbed by the Jovian planets, which have larger masses, longer orbital periods, and narrower dynamical separation. As a subsystem, the Jovian planets are not perturbed by any other massive bodies.
Ito and Tanikawa (
3) have performed a set of numerical experiments to understand how these differences between terrestrial and Jovian planets affect their long term stability. They have considered various kinds of terrestrial planetary subsystems with equal dynamical separations and determined their typical instability time scales under the disturbance from the massive Jovian planets. They find that the terrestrial planetary subsystems with smaller dynamical distances (<18
RH) are likely to become unstable in a short time scale (<107 years). This rapid instability is caused by the strong gravitational perturbation from massive Jovian planets. Thus it seems that the present wide dynamical separation among terrestrial planets (>26
RH) is possibly one of the significant conditions to maintain the stability of the planetary orbits in giga-year time spans.
