Testing the accuracy of radiative cooling approximations in SPH
  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 SPH 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 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 under-estimates the
cooling by a large factor at all heights in the disc. For the self-gravitating
disc studied, we find that the method under-estimates the total cooling rate by
a factor of 200. This discrepancy may be readily traced to the method's
systematic over-estimate 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. These results raise an obvious caution about the method's 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.
