2.3. Baryons in Clusters of Galaxies

The cluster mass function is approximated as (Bahcall & Cen 1993)

(15)

where M^* = (1.8 ± 0.3) x 10¹⁴ h^-1 M, and M is the total gravitational mass within a sphere of a radius of 1.5h^-1 Mpc (the Abell radius) centered on the cluster. At the Abell radius the mass distribution is close to dynamical equilibrium. The envelope of matter around the cluster at greater distances merges into the ``large-scale structure'' and we take this matter to be included in the field (as discussed in Section 2.4). We define clusters as objects with mass M > 10¹⁴ h M. The integral dM M dn_cl / dM gives the mean mass density in clusters,

(16)

The contribution of this gravitational mass to the density parameter is

(17)

Intracluster plasma masses are well determined from X-ray observations (Fabricant et al. 1986; Hughes 1989; White et al. 1993). The ratio of X-ray emitting gas mass to gravitational mass within the Abell radius from the survey by White & Fabian (1995) is

(18)

In the White & Fabian (1995) sample this ratio shows no correlation with cluster mass. The value (18) is independently verified by Myers et al. (1997), (M_HII / M_grav)_cl = (0.061 ± 0.011) h^-1 from the measurement of the Sunyaev-Zeldovich effect in three clusters. The product of equations (17) and (18) is

(19)

This is the contribution to the baryon budget by intracluster plasma.

Let us estimate the baryons in stars in galaxies in clusters. Although these stars are included in the estimate for stars in Section 2.1, it is useful to compare their mass with the plasma mass, which we can derive from the cluster mass-to-light ratio. In the discussion of the Coma cluster by White et al. (1993), straightforward applications of galaxy velocity dispersions or the X-ray pressure gradient give (M_grav / L_B)_cl ~ 370h, or values as large as 500h if the analysis is constrained by models from numerical simulations of cluster formation. The CNOC value (Carlberg et al. 1996) transformed to the B band by (B - r) = 0.65 and < B - r > = 1.03 ± 0.1 for the average color of S0 galaxies, after a passive evolutionary correction for early type galaxies, is somwhat larger, 560h. However, the CNOC luminosity density is correspondingly smaller than our adopted value (equation (4)), and it is the product of this with M/L that matters in deriving most global quantities; moreover we suspect that the product is more reliably estimated since in the end we are estimating the luminosity density in cluster galaxy stars, which does not depend on a mass estimate. We therefore adopt a central value,

(20)

For cluster galaxies (Dressler 1980) we adopt the composition within the Abell radius

(21)

(This is somewhat richer in early-type galaxies than the estimate given by Schechter & Dressler (1987), but their sample extends beyond the Abell radius.) The mass-to-light ratios in equations (7) and (8) and the bulge-to-disk ratios in Table 1 give (M/L)_star = 4.5 ± 1. The ratio to equation (20) gives h M_stars / M_grav = 0.010^+0.005_-0.004. The product with equation (17) is the contribution to the density parameter by stars in clusters,

(22)

about ten percent of the total in equations (10)-(12).