The stability of multi-planet systems

A system of two small planets orbiting the Sun on low-eccentricity, low-inclination orbits is stable with respect to close encounters if the initial semi-major axis difference, a, measured in mutual Hill radii, R(H), exceeds 2 root 3, due to conservation of energy and angular momentum. We investigate the stability of systems of more than two planets using numerical integrations. We find that systems with Delta < 10 are always unstable, with the time, t, of first close encounter given approximately by log t = b Delta + c, where b and c are constants. It is likely that systems with Delta > 10 are also unstable. The slope b depends weakly on the number of planets, but is independent of planetary mass, m, if we measure Delta in units that are proportional to m(1/4) rather than the usual R(H) proportional to m(1/3). Instability in multi-planet systems arises because energy and angular momentum are no longer conserved within each two-planet subsystem due to perturbations by the additional planet(s). These results suggest that planetary embryos will not become isolated prior to the final stage of terrestrial-planet formation simply due to a failure to achieve close encounters. Other factors leading to isolation cannot be ruled out at this stage. (C) 1996 Academic Press, Inc.