  The evolution of many astrophysical systems depends strongly on the balance
between heating and cooling, in particular star formation in giant molecular
clouds and the evolution of young protostellar systems. Protostellar discs are
susceptible to the gravitational instability, which can play a key role in
their evolution and in planet formation. The strength of the instability
depends on the rate at which the system loses thermal energy. To study the
evolution of these systems, we require radiative cooling approximations because
full radiative transfer is generally too expensive to be coupled to
hydrodynamical models. Here we present two new approximate methods for
computing radiative cooling that make use of the polytropic cooling
approximation. This approach invokes the assumption that each parcel of gas is
located within a spherical pseudo-cloud which can then be used to approximate
the optical depth. The first method combines the methods introduced by
Stamatellos et al. and Lombardi et al. to overcome the limitations of each
method at low and high optical depths respectively. The second, the "Modified
Lombardi" method, is specifically tailored for self-gravitating discs. This
modifies the scale height estimate from the method of Lombardi et al. using the
analytical scale height for a self-gravitating disc. We show that the Modified
Lombardi method provides an excellent approximation for the column density in a
fragmenting disc, a regime in which the existing methods fail to recover the
clumps and spiral structures. We therefore recommend this improved radiative
cooling method for more realistic simulations of self-gravitating discs.
