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, Δ, measured in mutual Hill radii,R_H, exceeds[formula], 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 Δ < 10 are always unstable, with the time,t, of first close encounter given approximately by logt=bΔ +c, wherebandcare constants. It is likely that systems with Δ > 10 are also unstable. The slopebdepends weakly on the number of planets, but is independent of planetary mass,m, if we measure Δ in units that are proportional tom^1/4rather than the usualR_H∝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.