Planetesimal disk evolution driven by embryo-planetesimal gravitational scattering

The process of gravitational scattering of planetesimals by a massive protoplanetary embryo is explored theoretically. We propose a method to describe the evolution of the disk surface density, eccentricity, and inclination caused by the embryo-planetesimal interaction. It relies on an analytical treatment of the scattering in two extreme regimes of the planetesimal epicyclic velocities: shear dominated (dynamically "cold") and dispersion dominated (dynamically "hot"). In the former, planetesimal scattering can be treated as a deterministic process. In the latter, scattering is mostly weak because of the large relative velocities of interacting bodies. This allows one to use the Fokker-Planck approximation and the two-body approximation to explore the disk's evolution. We compare the results obtained by this method with the outcomes of direct numerical integrations of planetesimal orbits, and they agree quite well. In the intermediate-velocity regime, an approximate treatment of the disk evolution is proposed based on interpolation between the two extreme regimes. We also calculate the rate of the embryo's mass growth in an inhomogeneous planetesimal disk and demonstrate that it is in agreement with both the simulations and earlier calculations. Finally, we discuss the question of the direction of the embryo-planetesimal interaction in the dispersion-dominated regime and demonstrate that it is repulsive. This means that the embryo always forms a gap in the disk around it. The machinery developed here will be applied to realistic protoplanetary systems in future papers.