A new and simple prescription for planet orbital migration and eccentricity damping by planet-disc interactions based on dynamical friction

During planet formation, gravitational interaction between a planetary embryo and the protoplanetary gas disc causes orbital migration of the planetary embryo, which plays an important role in shaping the final planetary system. While migration sometimes occurs in the supersonic regime, wherein the relative velocity between the planetary embryo and the gas is higher than the sound speed, migration prescriptions proposed thus far describing the planet-disc interaction force and the time-scales of orbital change in the supersonic regime are inconsistent with one another. Here we discuss the details of existing prescriptions in the literature and derive a new simple and intuitive formulation for planet-disc interactions based on dynamical friction, which can be applied in both supersonic and subsonic cases. While the existing prescriptions assume particular disc models, ours include the explicit dependence on the disc parameters; hence, it can be applied to discs with any radial surface density and temperature dependence (except for the local variations with radial scales less than the disc scale height). Our prescription will reduce the uncertainty originating from different literature formulations of planet migration and will be an important tool to study planet accretion processes, especially when studying the formation of close-in low-mass planets that are commonly found in exoplanetary systems.