4.1 Implications of the Correlation of M_gas / M_stellar with T_gas

Although the correlation between M_gas / M_stellar and T_gas is based on only a few systems and requires further confirmation, we briefly explore the possible implications of the correlation.

4.1.1 Efficiency of Galaxy Formation

The ratio of the gas mass to the stellar mass, M_gas / M_stellar, shown in Figure 8, can be related to the efficiency of star formation. We assume a scenario in which the luminous matter (stars and the ICM) form the bulk of the baryonic material and the remainder of the virial mass is in the form of hot or cold dark matter. Then, as long as groups and clusters are closed systems which do not lose their intracluster material, the efficiency of galaxy formation, the conversion of baryons from gas to stars in galaxies, can be written as

(2)

where M_lum = M_stellar + M_gas, or equivalently as

(3)

(assuming the expelled gas from galaxies is small and can be neglected). Thus by measuring M_gas / M_stellar, we can study the efficiency of star formation in systems ranging from groups to rich clusters. Our analysis shows that the star (and galaxy) formation efficiency ranges from 50% for groups to as little as 15% for rich clusters. If all the ICM in groups is gas ejected from galaxies, and we use this injection rate for all clusters, then the galaxy formation efficiency would be 100% for groups but lower for rich clusters (as low as 17% for M_gas / M_stellar = 6 to as high as 50% for M_gas / M_stellar = 3). Although the amount of luminous material (gas+stars) remains relatively constant for all clusters (Blumenthal et al. 1984), the efficiency of galaxy formation decreases as one moves to richer systems. In other words although the richest systems obviously produced more galaxies, their efficiency of galaxy formation was lower.

Interpreting the ratio of gas mass to stellar mass as a measure of galaxy formation efficiency requires that clusters be "closed" systems, that is, no material may be added or lost. The gas in the ICM is enriched both during an early phase of massive star supernovae (Type II) and continuing through the present with primarily Type I supernovae and mass loss from older stars. Since the gravitational potential of poor clusters is sufficient to bind the enriched material ejected in supernova winds driven from the galaxies, none of the material in the ICM should be lost from these systems. Furthermore, based on the computed enrichment rates, extensive amounts of matter could not have been expelled by the galaxies and entirely lost from poor and rich clusters if their ICM's are to have significant heavy-element abundances. Therefore the change in the ratio of gas mass to stellar mass with cluster richness cannot be explained by a loss of hot intracluster material from the groups and poor clusters. The relative constancy over rich and poor clusters of the fraction of the cluster virial mass made up by luminous material (stars and gas) also supports the notion of a "closed" system.

4.1.2 Correlation of Iron Abundance with T_gas

As described above (see also Jones and Forman 1990 and David et al. 1989a) the ratio of the gas to stellar mass, M_gas / M_stellar, increases from unity in groups and poor clusters with low temperatures (~ 2 keV) to values of 3-6 in systems with high gas temperatures (6-10 keV). This correlation, combined with an understanding of the production of heavy elements, predicts a correlation of heavy element abundance with gas temperature.

The groups which are luminous X-ray sources are dense systems and have stellar populations comparable to rich clusters (Morgan, Kayser, and White 1975). Also, the correlation of galaxy population with local density (Dressler 1980, and Postman and Geller 1984) supports the similarity of the galaxy populations in the groups and clusters. Therefore, the production of heavy elements should be directly proportional to the stellar light, or equivalently stellar mass, since comparable populations will have similar mass-to-light ratios. Thus, the larger the ratio of gas mass to stellar mass, the more dilute the stellar products like iron. Since M_gas / M_stellar increases with increasing T_gas, we predict that hotter clusters (those with larger M_gas / M_stellar) will have lower iron abundances than cooler clusters. This prediction assumes that the clusters and groups are closed systems, i.e. no gas is expelled or accreted.

Figure 9 shows quantitative predictions for the correlation of iron abundance with gas temperature. The two solid curves are derived by taking a simple parameterization for the dependence of M_gas / M_{stellar on Tgas and assuming that enriched material is expelled from galaxies only during an early wind phase during which Type II supernovae can readily drive a galactic wind (see David, Forman, and Jones, 1989b). The two curves assume different initial mass functions (the upper curve has a power law exponent = 2 and the lower curve has = 2.5). Note that an amount of enriched material equal to that expelled in the wind is produced by stellar evolution and could be liberated by ram pressure stripping. The present estimates of supernova yields can explain the observed heavy element abundances in the intracluster gas as Figure 9 shows. The ejected gas is extremely enriched and is diluted to the observed values by mixing with the predominantly primordial component of the intracluster medium.}

Figure 9. The iron abundance (as a fraction of the solar value) is plotted against gas temperature. The data are taken from Henriksen (1985), Hughes et al. (1988) and Arnaud et al. (1987). The smooth curves are predictions based on a parameterization of the relation between Mgas / Mstellar and Tgas as well as a model for the evolution of stars with two different initial mass functions.

The present measurements of iron abundances are too inaccurate to verify the above model or test possibilities for the origin of the ICM. Mushotzky (1984) and Henriksen (1985) summarize present results. For rich clusters, Henriksen reports a possible correlation of decreasing iron abundance with increasing gas temperature, as predicted, but the data are not sufficiently precise to yield quantitative results. Those observations were for only quite luminous clusters (L_x > 2 x 10⁴⁴ ergs sec^-1) while in general we expect the abundances to be highest in the low luminosity clusters. Hughes et al. (1988) have measured a precise iron abundance of 22% of the solar value for the rich Coma cluster. To adequately test for differences in abundances, it is particularly important to obtain comparable measurements for low temperature (high galaxy formation efficiency) systems. By determining accurate values of the heavy-element abundances of the ICM in both poor and rich clusters, one could better investigate the properties of the IMF (e.g., exponent), the efficiency of galaxy formation, and the origin and enrichment of the ICM. A precise determination of the heavy element abundance of the intracluster medium for a sample of clusters ranging from groups to rich clusters has implications for the amount of material in the ICM that must be primordial. In particular, determining a high solar abundance for the ICM in Morgan groups, as suggested by the arguments above, would confirm that the origin of most of the hot gas in rich clusters must be primordial.

4.1.3 The Energy Content of the ICM

The changing ratio of gas mass to stellar mass also will affect the energy (or temperature) of the ICM. By measuring the surface brightness profiles and independently by measuring the ratio of the velocity dispersion to the gas temperature, one can estimate the energy per unit mass of the galaxies compared to that of the gas. From the surface brightness profiles, this value for rich clusters is generally ~ 2/3. The values calculated from the measured velocity dispersions and gas temperatures have a wider range (but see Flanagan (1988) who suggests a resolution for the Perseus discrepancy). By comparison to rich clusters, the surface brightness profiles for hot gas around single dominant cluster galaxies such as M87 and the cD groups such as AWM7 yield a value ~ 1/2 (Kriss, Cioffi, and Canizares 1983; their parameter = -3). This implies that the groups and individual central galaxies have more energy per unit mass in gas compared to the constituent galaxies than do rich clusters. For the groups and poor clusters where the stellar mass is comparable to the gas mass, there may be significant heating of the ICM by the ejected material which may account for the observed difference between the groups and the clusters.