  We present a new model for the X-ray properties of the intracluster medium
that explicitly includes heating of the gas by the energy released during the
evolution of cluster galaxies. We calculate the evolution of clusters by
combining the semi-analytic model of galaxy formation of Cole et al. with a
simple model for the radial profile of the intracluster gas. We focus on the
cluster X-ray luminosity function and on the relation between X-ray temperature
and luminosity. We show that cooling alone is not enough to account for the
flatness of the observed T-L relation or for the lack of strong redshift
evolution in the observed X-ray luminosity function. Gas heating, on the other
hand, can solve these two problems: in the \Lambda-CDM cosmology, our model
reproduces fairly well the T-L relation and the X-ray luminosity function.
Furthermore, it predicts only weak evolution in these two properties out to
z=0.5, in agreement with recent observational data. A successful model requires
an energy input of 1--2 x 10^49 ergs per solar mass of stars formed. This is
comparable to the total energy released by the supernovae associated with the
formation of the cluster galaxies. Thus, unless the transfer of supernovae
energy to the intracluster gas is very (perhaps unrealistically) efficient,
additional sources of energy, such as mechanical energy from AGN winds are
required. However, the amplification of an initial energy input by the response
of the intracluster medium to protocluster mergers might ease the energy
requirements. Our model makes definite predictions for the X-ray properties of
groups and clusters at high redshift which may soon be tested with data from
the Chandra and Newton satellites.
