  The aim of the present work is to show that, contrary to popular belief,
galaxy clusters are **not** expected to be self-similar, even when the only
energy sources available are gravity and shock-wave heating. In particular, we
investigate the scaling relations between mass, luminosity and temperature of
galaxy groups and clusters in the absence of radiative processes. Theoretical
expectations are derived from a polytropic model of the intracluster medium and
compared with the results of high-resolution adiabatic gasdynamical
simulations. It is shown that, in addition to the well-known relation between
the mass and concentration of the dark matter halo, the effective polytropic
index of the gas also varies systematically with cluster mass, and therefore
neither the dark matter nor the gas profiles are exactly self-similar. It is
remarkable, though, that the effects of concentration and polytropic index tend
to cancel each other, leading to scaling relations whose logarithmic slopes
roughly match the predictions of the most basic self-similar models. We provide
a phenomenological fit to the relation between polytropic index and
concentration, as well as a self-consistent scheme to derive the non-linear
scaling relations expected for any cosmology and the best-fit normalizations of
the M-T, L-T and F-T relations appropriate for a Lambda-CDM universe. The
predicted scaling relations reproduce observational data reasonably well for
massive clusters, where the effects of cooling and star formation are expected
to play a minor role.
