Orbital migrations in planetesimal discs:: N-body simulations and the resonant dynamical friction

We have performed N -body numerical simulations of the exchange of angular momentum between a massive planet and a 3D Keplerian disc of planetesimals. Our interest is directed at the study of the classical analytical expressions of the lineal theory of density waves, as representative of the dynamical friction in discs 'dominated by the planet' and the orbital migration of the planets with regard to this effect. By means of a numerical integration of the equations of motion, we have carried out a set of numerical experiments with a large number of particles (N greater than or equal to10 000), and planets with the mass of Jupiter, Saturn and one core mass of the giant planets in the Solar system (M-c =10 M+ ). The torque, measured in a phase in which a 'steady forcing' is clearly measurable, yields inward migration in a minimum-mass solar disc (Sigmasimilar to10 g cm(-2)), with a characteristic drift time of similar to a few 10(6) yr. The planets predate the disc, but the orbital decay rate is not sufficient to allow accretion in a time-scale relevant to the formation of giant planets. We found reductions of the measured torque on the planet, with respect to the linear theory, by a factor of 0.38 for M (c) , 0.04 for Saturn and 0.01 for Jupiter, due to the increase in the perturbation on the disc. The behaviour of planets whose mass is larger than M (c) is similar to the one of type II migrators in gaseous discs. Our results suggest that, in a minimum mass, solar planetesimals disc, type I migrations occur for masses smaller than M (c) , whereas for this mass value it could be a transition zone between the two types of migration.