On the orbital evolution and growth of protoplanets embedded in a gaseous disc

Abstract

We present a new computation of the linear tidal interaction of a protoplanetary core with a thin gaseous disc in which it is fully embedded. For the first time a discussion of the orbital evolution of cores on eccentric orbits with eccentricity (e) significantly larger than the gas-disc scaleheight-to-radius ratio (Hr) is given. We find that the direction of orbital migration reverses for e>1.1Hr. This occurs as a result of the orbital crossing of resonances in the disc that do not overlap the orbit when the eccentricity is very small. In that case, resonances always give a net torque corresponding to inward migration. Simple expressions giving approximate fits to the eccentricity damping rate and the orbital migration rate are presented. We go on to calculate the rate of increase of the mean eccentricity for a system of protoplanetary cores caused by dynamical relaxation. By equating the eccentricity damping time-scale with the dynamical relaxation time-scale we deduce that, for parameters thought to be applicable to protoplanetary discs, an equilibrium between eccentricity damping and excitation through scattering is attained on a 103-104 yr time-scale, at 1 au. This equilibrium is maintained during the further migrational and collisional evolution of the system, which occurs on much longer time-scales. The equilibrium thickness of the protoplanet distribution is related to the equilibrium eccentricity, and is such that it is generally well confined within the gas disc. By use of a three-dimensional direct summation N-body code we simulate the evolution of a system of protoplanetary cores, initialized with a uniform isolation mass of 0.1 M⊕, incorporating our eccentricity damping and migration rates. Assuming that collisions lead to agglomeration, we find that the vertical confinement of the protoplanet distribution permits cores to build up in mass by a factor of ∼10 in only ∼104 yr, within 1 au. The time-scale required to achieve this is comparable to the migration time-scale. In the context of our model and its particular initial conditions, we deduce that it is not possible to build up a massive enough core to form a gas giant planet, before orbital migration ultimately results in the preferential delivery of all such bodies to the neighbourhood of the central star. This problem could be overcome by allowing for the formation of massive cores at much larger radii than are usually considered. It remains to be investigated whether different disc models or initial planetesimal distributions might be more favourable for slowing or halting the migration, leading to possible giant planet formation at intermediate radii.