Three-dimensional simulations of multiple protoplanets embedded in a protostellar disc

Context. Theory predicts that low-mass protoplanets in a protostellar disc migrate into the central star on a time scale that is short compared with the disc lifetime or the giant planet formation time scale. Protoplanet eccentricities of e greater than or similar to H/r can slow or reverse migration, but previous 2D studies of multiple protoplanets embedded in a protoplanetary disc have shown that gravitational scattering cannot maintain significant planet eccentricities against disc-induced damping. The eventual fate of these systems was migration into the central star. Aims. Here we simulate the evolution of low-mass protoplanetary swarms in three dimensions. The aim is to examine both protoplanet survival rates and the dynamical structure of the resulting planetary systems, and to compare them with 2D simulations. Methods. We present results from a 3D hydrodynamic simulation of eight protoplanets embedded in a protoplanetary disc. We also present a suite of simulations performed using an N-body code, modified to include prescriptions for planetary migration and for eccentricity and inclination damping. These prescriptions were obtained by fitting analytic formulae to hydrodynamic simulations of planets embedded in discs with initially eccentric and/or inclined orbits. Results. As was found in two dimensions, differential migration produces groups of protoplanets in stable, multiple mean-motion resonances that migrate in lockstep, preventing prolonged periods of gravitational scattering. In almost all simulations, this leads to large-scale migration of the protoplanet swarm into the central star in the absence of a viable stopping mechanism. The evolution involves mutual collisions, occasional instances of large-scale scattering, and the frequent formation of the long-lived, co-orbital planet systems that arise in > 30% of all runs. Conclusions. Disc-induced damping overwhelms eccentricity and inclination growth due to planet-planet interactions, leading to large-scale migration of protoplanet swarms. Co-orbital planets are a natural outcome of dynamical relaxation in a strongly dissipative environment, and if observed in nature would imply that such a period of evolution commonly arises during planetary formation.