We calculate radial migration rates of protoplanets in laminar minimum mass solar nebula discs using three-dimensional self-gravitating radiation hydrodynamical (RHD) models. The protoplanets are free to migrate, whereupon their migration rates are measured. For low-mass protoplanets (10-50 M(circle plus)), we find increases in the migration time-scales of up to an order of magnitude between locally isothermal and RHD models. In the high-mass regime, the migration rates are changed very little. These results are arrived at by calculating migration rates in locally isothermal models, before sequentially introducing self-gravity, and radiative transfer, allowing us to isolate the effects of the additional physics. We find that using a locally isothermal equation of state, without self-gravity, we reproduce the migration rates obtained by previous analytic and numerical models. We explore the impact of different protoplanet models, and changes to their assumed radii, upon migration. The introduction of self-gravity gives a slight reduction of the migration rates, whilst the inertial mass problem, which has been proposed for high-mass protoplanets with circumplanetary discs, is reproduced. Upon introducing radiative transfer to models of low-mass protoplanets (approximate to 10 M(circle plus)), modelled as small radius accreting point masses, we find outward migration with a rate of approximately twice the analytic inward rate. However, when modelling such a protoplanet in a more realistic manner, with a surface which enables the formation of a deep envelope, this outward migration is not seen.
Ayliffe, B. A., & Bate, M. R. (2010). Planet migration: self-gravitating radiation hydrodynamical models of protoplanets with surfaces. Monthly Notices of the Royal Astronomical Society, 408(2), 876 - 896. https://doi.org/10.1111/j.1365-2966.2010.17221.x