  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 timescales 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 that 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.
