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