Indirect forces in disc-planet interaction

Abstract

Gravitational coupling between a protoplanetary disc and an embedded planet is often studied in a frame attached to a central star. This frame is non-inertial because of the stellar reflex motion, leading to indirect forces arising in the star-planet-disc system. Here we examine the impact produced by these forces on several aspects of disc-planet coupling using analytical and numerical means. We explore how neglecting indirect forces changes (1) the spatial pattern of the surface density perturbation in the disc, (2) the calculation of the torque exerted on the disc by the planet, and (3) the torque on the planet exerted by the disc. For low-mass planets, in the linear regime, the differences in the perturbation pattern are only in its $m=1$ azimuthal harmonic, with an amplitude increasing with the distance from the star. In this regime both the torque on the planet and the deposition torque density in the disc are only weakly affected by non-inclusion of indirect forces, corroborating some results of studies neglecting indirect forces altogether. For higher mass planets, a broader range of azimuthal harmonics of the perturbation are affected. Also, indirect forces have a stronger effect on the planetary torque and on planet migration in the Type II regime. We highlight the importance of including the planetary indirect force in the calculation of the torque on the disc (if disc evolution accounts for indirect force) to ensure conservation of angular momentum carried by the planet-driven density waves. The corresponding indirect torque has an oscillatory, radially-diverging character.