  In order to study the process of cooling in dark-matter (DM) halos and assess
how well simple models can represent it, we run a set of radiative SPH
hydrodynamical simulations of isolated halos, with gas sitting initially in
hydrostatic equilibrium within Navarro-Frenk-White (NFW) potential wells. [...]
After having assessed the numerical stability of the simulations, we compare
the resulting evolution of the cooled mass with the predictions of the
classical cooling model of White & Frenk and of the cooling model proposed in
the MORGANA code of galaxy formation. We find that the classical model predicts
fractions of cooled mass which, after about two central cooling times, are
about one order of magnitude smaller than those found in simulations. Although
this difference decreases with time, after 8 central cooling times, when
simulations are stopped, the difference still amounts to a factor of 2-3. We
ascribe this difference to the lack of validity of the assumption that a mass
shell takes one cooling time, as computed on the initial conditions, to cool to
very low temperature. [...] The MORGANA model [...] better agrees with the
cooled mass fraction found in the simulations, especially at early times, when
the density profile of the cooling gas is shallow. With the addition of the
simple assumption that the increase of the radius of the cooling region is
counteracted by a shrinking at the sound speed, the MORGANA model is also able
to reproduce for all simulations the evolution of the cooled mass fraction to
within 20-50 per cent, thereby providing a substantial improvement with respect
to the classical model. Finally, we provide a very simple fitting function
which accurately reproduces the cooling flow for the first ~10 central cooling
times. [Abridged]
