| © CAMBRIDGE UNIVERSITY PRESS 1999 |

**3.4. The Intracluster Gas: Hydrostatic Equilibrium?**

The standard model of clusters assumes that both the gas and the galaxies are in approximate hydrostatic equilibrium with the binding cluster potential (Bahcall and Sarazin 1977; Forman and Jones 1984; Sarazin 1986; Evrard 1990; Bahcall and Lubin 1994). In this model the gas distribution obeys

(32) |

where *P*_{gas} and
_{gas}
are the gas pressure and density, and
*M*_{cl}(
*r*) is the total cluster binding mass
within a radius *r*. The cluster mass can thus be represented as

(33) |

where *T* is the gas temperature and
*µ**m*_{p} is the mean particle mass of the gas.

The galaxies in the cluster respond to the same gravitational field, and they satisfy

(34) |

where _{r} is
the radial velocity dispersion of galaxies in the
cluster, _{gal}(*r*) is the galaxy density profile, and
*A*
represents a possible anisotropy in the galaxy velocity distribution
[*A* = 1 -
(_{t} /
_{r})^{2},
where *t* and *r* represent the
tangential and radial velocity components].

The above two relations yield

(35) |

where the
_{spec}
parameter, defined by the left side of
the above relation, can be determined directly from observations of
cluster velocity dispersions and gas temperatures. The
_{spec}
parameter represents the ratio of energy per unit mass in the
galaxies to that in the gas. Observations of a large sample of
clusters yield a mean best-fit value of
_{spec}
1 ± 0.1
(Lubin and Bahcall 1993;
see also Section 3.5). This
suggests that, on average, the gas and galaxies follow each other with
comparable energies
(^{2}_{r}
*kT*/*µ**m*_{p}). The observed mean value
_{spec}
1 ± 0.1 is
consistent with the
value of
determined from the right-hand side of the
relation (referred to as
_{fit},
and determined from the gas and galaxy density profile fits). Using
_{gas}(*r*)
*r*^{-2}
(Section 3.2) and
_{gal}(*r*)
*r*^{-2.4 ± 0.2}
(Section 2.6), one finds
_{fit}
0.85 ± 0.1 (for an
isothermal distribution)
(Bahcall and Lubin 1994).
The above consistency supports the assumption that the gas is
in an approximate hydrostatic equilibrium with the cluster potential,
and suggests that the galaxies and gas approximately trace each other
in the clusters.