**5.5. Distribution of the intracluster gas-hydrostatic models**

In general, the elastic collision times for ions and electrons (equations 5.32, 5.33) in the intracluster gas are much shorter than the time scales for heating or cooling (Section 5.3) or any dynamical process, and the gas can be treated as a fluid. The time required for a sound wave in the intracluster gas to cross a cluster is given by

(5.54) |

where *T*_{g} is the gas temperature and *D* is the
cluster diameter. Since this
time is short compared to the probable age of a cluster of
10^{10} yr, the gas will be
hydrostatic and the pressure will be a smoothly varying function of position
unless the cluster gravitational potential varies on a shorter time
scale or the gas
is heated or cooled more rapidly than this. The cooling time due to thermal
bremsstrahlung (equation 5.23) is much longer than the sound crossing
time, and
the same is true for the time scales for heating by any of the processes
that have been suggested (Section 5.3).
Thus the gas distribution is usually assumed to be hydrostatic:

(5.55) |

where *p* =
_{g}
*kT*_{g} / *µ*m_{p} is the gas
pressure,
_{g}
is the gas density, and
(*r*) is
the gravitational potential of the cluster. Now, the sound crossing time
condition
given above ensures that the pressure in the gas be uniform on small
scales, but the same need not be completely true of the entropy. In the
rest of this section, we will assume that the intracluster gas is locally
homogeneous.

If, in addition, the cluster is assumed to be spherically symmetric, equation (5.55) reduces to

(5.56) |

where *r* is the radius from the cluster center and
*m*(*r*) is the total cluster
mass within *r*. If the contribution of the intracluster gas to the
gravitational potential is ignored, then the distribution of the
intracluster gas is determined by the cluster potential
(*r*)
and the temperature distribution of the gas *T*_{g}(*r*).
Further, then equation (5.56) is a linear equation for the logarithm of
the gas density; the central density (or any other value) can be
specified in order to
determine the full run of densities. If self gravity is included, then the
density scale, temperature variation, and cluster potential cannot be given
independently.

In most models, the cluster potential is assumed to be that of a self-gravitating isothermal sphere (Section 2.7). For convenience, the analytic King approximation to the isothermal sphere is often assumed (equation 2.13), so that the cluster total density, mass, and potential are given by

(5.57) |

(5.58) |

(5.59) |

Here, _{0}
is the central density and *r*_{c} is the core radius of
the cluster (Section 2.7).
The central density and core radius are related to the line-of-sight
velocity dispersion
_{r}
(Section 2.6) of an isothermal cluster by

(5.60) |

which follows from equation (2.9). Thus the central value of the cluster gravitational potential is

(5.61) |

It is perhaps worth noting that although the analytic King model is a good fit to the inner portions of an isothermal sphere, the analytic King potential has a finite depth (given by equation 5.61), while the isothermal sphere potential is infinitely deep. The total mass associated with the analytic King distribution diverges as the radius increases, and it is usual to cut the distribution off in some way (equations 2.11 and 2.12). The effect of these cutoffs on the cluster potential and on the resulting hydrostatic gas models is discussed in Sarazin and Bahcall (1977) and Bahcall and Sarazin (1978).

In calculating the cluster potential, we have ignored the effect of
individual galaxies. The potential wells associated with individual
galaxies are considerably shallower than those associated with
clusters. This is shown by the fact that the velocity dispersions of
stars in galaxies
(_{*}
300 km/s;
Faber and Gallagher,
1979)
are much smaller than the velocity dispersions of galaxies in
clusters (_{r}
1000 km/s); the
gravitational potentials increase with the
square of the velocity dispersions. Thus an individual galaxy will not
significantly perturb the distribution of intracluster gas
(Gull and Northover,
1975).