Annu. Rev. Astron. Astrophys. 1979. 17: 135-87 Copyright © 1979 by Annual Reviews. All rights reserved

7.1 The Virial Theorem

In applying the virial theorem to large clusters, it is assumed that the cluster is in a stationary state. The kinetic energy is obtained from an appropriately weighted determination of the space velocity dispersion V² (Rood et al. 1970; see also Rood & Dickel 1978a). Choosing a proper weighting scheme is not a trivial point; for example, Chincarini & Rood (1977) show that the calculated mass of Abell 194 (hereafter, A194) varies by almost a factor of three among plausible weighting schemes. The issue is whether to weight by observed luminosities or by masses inferred from mean mass-to-light ratios. The latter method introduces great uncertainty since mean M / L's are poorly known for early-type galaxies, which are a significant fraction of the membership in large clusters.

As with small groups, a correction must also be applied to convert the observed projected velocity dispersion ² to V², which will in general depend on the types of orbits in the cluster. If the orbits are primarily radial and we determine ² from the cluster core, then V² may only slightly exceed ², while if the orbits are nearly circular with an isotropic distribution V² 3² (Abell 1977). Thus the kinetic energy term may be uncertain by as much as a factor of 3, although Rood (1970) has suggested that usually 2 < V² / ² < 3 and that a ratio of 2.1 is most appropriate for the Coma cluster.

In calculating the potential energy, the mass distribution of the cluster is usually taken to be the same as that of the galaxies. Schwarzschild (1954) showed that the number of galaxies per unit area, S, along a strip passing a minimum distance q from the cluster core could be used to find the effective virial radius,

(10)

Unfortunately, S is difficult to measure precisely, as it depends on the statistical correction for background galaxies, which is most important near the outer parts of the cluster and can introduce significant errors in R_VT (Rood et al. 1972). As an alternative to Equation (10), the observed surface density profile may be inverted to give the space density distribution (r), and the potential integral then can be explicitly evaluated (Oemler 1974, Abell 1977). Once R_VT has been found, the virial mass of the cluster can be calculated in the usual way as M_VT = V² R_VT / G.

In this section we use luminosities on the V system since virtually all cluster work has employed this convention. Furthermore, most cluster photometry is necessarily somewhat less precise than that for nearby, bright galaxies, and thus the derived M / L are more uncertain than those found by other methods. We convert to M / L_B at the end of the section.

We have recomputed virial masses and M / L's in a homogeneous way for seven large clusters. We have taken ² from Yahil & Vidal (1977) and cluster luminosities and virial radii from Oemler's study of galaxy populations in 15 clusters. In order to illustrate a standard virial mass, we have assumed V² / ² = 3. The dispersions have been multiplied by (1 + z)^-1 to correct for redshift (Harrison 1974, Faber & Dressler 1977), and cluster distances are based on mean radial velocities. We find the median M / L_V 290, with a range of 165-800. This result is typical of values quoted in the literature (Bahcall 1975) and is substantially larger than M / L's of individual galaxies found in Table 2.

However, there is considerable room to maneuver within the framework of the virial theorem. The ratio M / L could be reduced by a factor of 2 or 3 if our guess concerning velocity isotropy is incorrect. An additional decrease might be obtained by mass-weighting the velocity dispersion and by correcting the luminosities for halos of galaxies and faint cluster members. Chincarini & Rood (1977) show that these considerations can reduce M / L_pg for A194 to 36 and that of Coma to 170. It is also possible that irregular clusters such as Hercules are not yet in virial equilibrium. The simple requirement for boundedness would reduce the mass by a factor of two. Hercules would then have a visual mass-to-light ratio of 270, rather typical of other large clusters.