Radiative transfer is an important component of hydrodynamic simulations as it determines the thermal properties of a physical system. It is especially important in cases where heating and cooling regulate significant processes, such as in the collapse of molecular clouds, the development of gravitational instabilities in protostellar discs, disc-planet interactions, and planet migration. We compare two approximate radiative transfer methods which indirectly estimate optical depths within hydrodynamic simulations using two different metrics: (i) the gravitational potential and density of the gas (Stamatellos et al.), and (ii) the pressure scale-height (Lombardi et al.). We find that both methods are accurate for spherical configurations e.g. in collapsing molecular clouds and within clumps that form in protostellar discs. However, the pressure scale-height approach is more accurate in protostellar discs (low and high-mass discs, discs with spiral features, discs with embedded planets). We also investigate the $β$-cooling approximation which is commonly used when simulating protostellar discs, and in which the cooling time is proportional to the orbital period of the gas. We demonstrate that the use of a constant $β$ cannot capture the wide range of spatial and temporal variations of cooling in protostellar discs, which may affect the development of gravitational instabilities, planet migration, planet mass growth, and the orbital properties of planets.

16 pages, 12 figures, accepted by MNRAS