Hot Gas in Clusters of Galaxies: the Punctuated Equilibria Model

We develop a model to describe the evolution of the intra-cluster X-ray emitting baryons. These are included in the potential wells of galaxy clusters evolving through subsequent merging events in the framework of hierarchical clustering. The gas is assumed to re-adjusts to a new hydrostatic equilibrium after each merging event. Before merging it is gravitationally heated at the local virial temperature when bound in subclusters; at early $z$ the gas is preheated by supernova activity following star formation. In detail, we compute analytically the following steps: the dynamic histories of dark matter halos with their merging events; the associated infall of gas into a halo, with compressions and shocks estabilishing the conditions at the cluster boundary; the updated disposition of the gas in the potential well matching such conditions; the statistical convolution of the key quantities over the merging histories. The model predicts the density and surface brightness profiles with no free parameters; the so-called $\beta$ parameter is itself an outcome of the model, and the polytropic index $\gamma$ is internally constrained to a narrow range. We obtain declining temperature profiles, and profiles for the density and for the surface brightness shallower in groups compared with clusters. Our model groups also contain a lower baryonic fraction on average, but with a scatter considerably larger than at cluster scales. Various statistics are obtained analytically upon averaging over the merging histories. In particular, we predict in different cosmologies the statistical correlation $L-T$ of luminosity with temperature; similarly we derivethe correlation $R_X-T$ for the size of the X-ray emitting region. The intrinsic scatter in both correlations is also predicted