Simple models of cooling flows

A semi-analytic model of cluster cooling flows is presented. The model assumes that episodic nuclear activity followed by radiative cooling without mass-dropout cycles the cluster gas between a relatively homogeneous, nearly isothermal post-outburst state and a cuspy configuration in which a cooling catastrophe initiates the next nuclear outburst. Fitting the model to Chandra data for the Hydra cluster, a lower limit of 284 Myr until the next outburst of Hydra A is derived. Density, temperature and emission-measure profiles at several times prior to the cooling catastrophe are presented. It proves possible to fit the mass M (sigma) with entropy index P rho(-gamma) less than sigma to a simple power-law form, which is almost invariant as the cluster cools. We show that radiative cooling automatically establishes this power-law form if the entropy index was constant throughout the cluster gas at some early epoch or after an active galactic nucleus (AGN) activity cycle. To high precision, the central value of sigma decreases linearly with time. The fraction of clusters in a magnitude-limited sample that have gas cooler than T is calculated, and is shown to be small for T = 2 keV. Similarly, only 1 per cent of clusters in such a sample contain gas with P rho(-gamma) < 2 keV cm(2) . Entropy production in shocks is shown to be small. The entropy that is radiated from the cluster can be replaced if a few per cent of the cluster gas passes through bubbles heated during an outburst of the AGN.