Catching drifting pebbles I. Enhanced pebble accretion efficiencies for eccentric planets

Context. Coagulation theory predicts that micron-sized dust grains grow into pebbles, which drift inward towards the star when they reach sizes of mm cm. When they cross the orbit of a planet, a fraction of these drifting pebbles will be accreted. In the pebble accretion mechanism, the combined effects of the planet's gravitational attraction and gas drag greatly increase the accretion rate. Aims. We calculate the pebble accretion efficiency epsilon(2D) - the probability that a pebble is accreted by the planet - in the 2D limit (pebbles reside in the midplane). In particular, we investigate the dependence of epsilon(2D) on the planet eccentricity and its implications for planet formation models. Methods. We conduct N-body simulations to calculate the pebble accretion efficiency in both the local frame and the global frame. With the global method we investigate the pebble accretion efficiency when the planet is on an eccentric orbit. Results. We find that the local and the global methods generally give consistent results. However, the global method becomes more accurate when the planet is more massive than a few Earth masses or when the aerodynamic size (Stokes number) of the pebble is larger than 1. The efficiency increases with the planet's eccentricity once the relative velocity between the pebble and the planet is determined by the planet's eccentric velocity. At high eccentricities, however, the relative velocity becomes too high for pebble accretion. The efficiency then drops significantly and the accretion enters the ballistic regime. We present general expressions for epsilon(2D). Applying the obtained formula to the formation of a secondary planet, in resonance with an already-formed giant planet, we find that the embryo grows quickly due to its higher eccentricity. Conclusions. The maximum epsilon(2D) for a planet on an eccentric orbit is several times higher than for a planet on a circular orbit, but this increase gives the planet an important headstart and boosts its following mass growth. The recipe for epsilon(2D) that we have obtained is designed to be implemented into N-body codes to simulate the growth and evolution of planetary systems.