Testing the accuracy of radiative cooling approximations in smoothed particle hydrodynamical simulations

Hydrodynamical simulations of star formation have stimulated a need to develop fast and robust algorithms for evaluating radiative cooling. Here we undertake a critical evaluation of what is currently a popular method for prescribing cooling in smoothed particle hydrodynamical simulations, i.e. the polytropic cooling due originally to Stamatellos et al. This method uses the local density and potential to estimate the column density and optical depth to each particle and then uses these quantities to evaluate an approximate expression for the net radiative cooling. We evaluate the algorithm by considering both spherical and disc-like systems with analytic density and temperature structures. In spherical systems, the total cooling rate computed by the method is within around 20 per cent for the astrophysically relevant case of opacity dominated by ice grains and is correct to within a factor of order unity for a range of opacity laws. In disc geometry, however, the method systematically underestimates the cooling by a large factor at all heights in the disc. For the self-gravitating disc studied, we find that the method underestimates the total cooling rate by a factor of similar to 200. This discrepancy may be readily traced to the methods systematic overestimate of the disc column density and optical depth, since (being based only on the local density and potential) it does not take into account the low column density route for photon escape normal to the disc plane. We note that the discrepancy quoted above applies in the case that the stars potential is not included in the column density estimate and that even worse agreement is obtained if the full (star plus disc) potential is employed. These results raise an obvious caution about the methods use in disc geometry whenever an accurate cooling rate is required, although we note that there are situations where the discrepancies highlighted above may not significantly affect the global outcome of simulations. Finally, we draw attention to our introduction of an analytic self-gravitating disc structure that may be of use in the calibration of future cooling algorithms.