How does gas cool in dark matter haloes?

In order to study the process of cooling in dark matter haloes and assess how well simple models can represent it, we run a set of radiative smoothed particle hydrodynamics (SPH) simulations of isolated haloes, with gas sitting initially in hydrostatic equilibrium within Navarro-Frenk-White potential wells. Simulations include radiative cooling and a scheme to convert high-density cold gas particles into collisionless stars, neglecting any astrophysical source of energy feedback. 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 2 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. Indeed, we find from simulations that cooling SPH particles take most time in travelling, at roughly constant temperature and increasing density, from their initial position to a central cooling region, where they quickly cool down to similar to 10(4) K. We show that in this case the total cooling time is shorter than that computed on the initial conditions, as a consequence of the stronger radiative losses associated to the higher density experienced by these particles. As a consequence the mass cooling flow is stronger than that predicted by the classical model. The MORGANA model, which computes the cooling rate as an integral over the contribution of cooling shells and does not make assumptions on the time needed by shells to reach very low temperature, 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 in 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 similar to 10 central cooling times.