Protoplanets with core masses below the critical mass fill in their Roche lobe

Abstract

We study the evolution of a protoplanet of a few earth masses embedded in a protoplanetary disc. If we assume that the atmosphere of the protoplanet, i.e. the volume of gas in hydrostatic equilibrium bound to the core, has a surface radius smaller than the Roche lobe radius, we show that it expands as it accretes both planetesimals and gas at a fixed rate from the nebula until it fills in the Roche lobe. The evolution occurs on a time-scale shorter than the formation or migration time-scales. Therefore, we conclude that protoplanets of a few earth masses have an atmosphere that extends to the Roche lobe surface, where it joins on to the nebula. This is true even when the Bondi radius is smaller than the Roche lobe radius. This is in contrast to the commonly used models in which the static atmosphere extends up to the Bondi radius and is surrounded by a cold accretion flow. As a result, any calculation of the tidal torque exerted by the disc on to the protoplanet should exclude the material present in the Roche lobe, since it is bound to the protoplanet.