Temporary Capture of Asteroids by an Eccentric Planet

We have investigated the probability of temporary capture of asteroids in eccentric orbits by a planet in a circular or eccentric orbit through analytical and numerical calculations. We found that, in the limit of the circular orbit, the capture probability is ∼0.1% of encounters to the planet’s Hill sphere, independent of planetary mass and semimajor axis. In general, temporary capture becomes more difficult as the planet’s eccentricity ({e}_{{p}}) increases. We found that the capture probability is almost independent of {e}_{{p}} until a critical value ({e}_{{p}}^{{c}}) that is given by ≃5 times the Hill radius scaled by the planet’s semimajor axis. For {e}_{{p}}> {e}_{{p}}^{{c}}, the probability decreases approximately in proportion to {e}_{{p}}^-1. The current orbital eccentricity of Mars is several times larger than {e}_{{p}}^{{c}}. However, since the range of secular change in Martian eccentricity overlaps {e}_{{p}}^{{c}}, the capture of minor bodies by Mars in the past is not ruled out.