**5.5.3. More complicated distributions**

The models discussed above assume that clusters are spherical, that the gas is hydrostatic, that the cluster potential is known in advance, and that the entropy distribution in the gas is given by a very simple polytropic distribution. Intracluster gas models have been calculated which attempt to generalize each of these assumptions. Here, some more complicated hydrostatic models are reviewed.

Stimpel and Binney (1979) (see also Binney and Stimpel, 1978) showed how spheroidal models for the intracluster gas distribution could be derived, using the observed galaxy distribution to determine the shape of the cluster potential. These models were fit to the galaxy counts in the Coma cluster, and models for the distribution of X-ray emission and microwave diminution (Section 3.5) were derived. There is an uncertainty in the shape of the cluster potential because the galaxy counts themselves cannot determine whether the cluster shape is more nearly prolate or oblate. However, Stimpel and Binney show that the resulting X-ray distributions in the two cases are considerably different, and thus X-ray observations can be used to determine the true shapes of elongated clusters (Chanan and Abramopoulos, 1984).

In general, the entropy distribution in a cluster will depend on the origin of the intracluster gas and the history of cluster. A number of authors have solved the hydrodynamic equations for simple models for the origin of the intracluster gas and of the subsequent history of the cluster; these calculations will be reviewed in Section 5.10. It is perhaps not surprising that these calculations do not generally lead to entropy distributions of the very simple sort (isothermal, adiabatic, or polytropic) considered above.

A number of authors have attempted to model these more detailed entropy distributions by allowing the polytropic index or the isothermal parameter to vary with radius. One example is to allow the gas to be isothermal in the cluster core, where conduction is effective (Section 5.4.2), but adiabatic in the outer parts (Cavaliere and Fusco-Femiano, 1978; Cavaliere, 1980). Cavaliere and Fusco-Femiano (1978) also included the effect of the gas density on the cluster potential.