4.2. Masses of Clusters

If Newton's laws are valid over intracluster distances, then the virial theorem,

 (8)

should apply. If, further, a cluster is in a steady state, the polar moment of inertia should be constant, so = 0, in which case

 (9)

Masses computed for clusters of galaxies are usually based on equation (9). The requirement that strictly vanish does not have to be rigorously fulfilled; if only the cluster is on the verge of stability, so that the total energy is zero, we should have

 (10)

and the mass derived from equation 10 is only a factor of 2 smaller than that obtained from equation (9).

The kinetic energy T can be approximated by 1/2cl <V2> , where cl is the total cluster mass and <V2> is the velocity dispersion weighted by mass. Radial velocities are usually observed for the brighter and presumably the relatively massive cluster members; thus little error results from estimating <V2> from the observed dispersion in radial velocities in a cluster, <Vr2> . The principal uncertainty is that of projection effects. If the velocity field is isotropic at all points in a cluster, then <V2> = <3Vr2> ; if most galaxies move radially through the cluster, then <V2> may only slightly exceed <Vr2> .

The potential energy, , is

where the summation must be carried out over all pairs of galaxies of masses mi and mj separated by distance rij. If the cluster has n members, there are n(n - 1) / 2 n2/2 such pairs. It is customary to write the potential as - G cl2 / 2R', where 1/R' is a weighted mean of the (1 / rij)'s. Attempts have been made to evaluate 1 / R' directly for a few groups, and even for the Hercules cluster (A2151) by the Burbidges (1959b). More often, R' is associated with the apparent radius of a cluster. For rich symmetrical regular clusters, the potential can be expressed more conveniently:

 (11)

where (r) is the mass contained within a radial distance r of the cluster center. For a constant mass-to-light ratio, we can write (r) = fL(r), which leads to

 (12)

The integral in equation (12) can be evaluated numerically from the density distribution of luminosity in the cluster derived from observations, leading finally to

 (13)

where R is the outer cluster radius and q is a numerical factor that is near unity. For the Coma cluster, for example, Rood (1965), using the writer's observations, finds that q = 0.92. The only unknown in equation (9) (or eq. [10]) is thus the total cluster mass. Setting q = 1, we find

 (14)

where the factor allows for the factor of 2 uncertainty in whether equation (9) or equation (10) is more appropriate, and for the projection effects between <Vr2> and <V2>; thus lies in the range 1/2 to 3.

Although the dynamical method of estimating masses of galaxian clusters is well known, the assumptions involved have been reviewed here because the masses found often seem discordant with those estimated from the luminosities of the cluster members. Ratios of mass to light (in solar units) found for individual galaxies (e.g., Burbidge and Burbidge 1959a) or for double galaxies (Page 1962) range from 1 to 15 for spirals and from 10 to 70 for ellipticals and S0's. On the other hand, to reconcile the masses estimated for a number of groups and clusters of galaxies from dynamical and luminosity methods, mass-to-light ratios for some of those systems would have to lie in the range from 100 to 1000 (e.g., Limber 1962). Years ago, Zwicky (1933), and Smith (1936) called attention to the unexpectedly high mass obtained for the Virgo cluster from its internal kinematics. The Burbidges (1959b) have called attention to the same situation in the Hercules cluster (A 2151). Other discussions of the problem of the high dynamical masses found for groups and clusters include those of de Vaucouleurs (1960), Burbidge and Burbidge (1961), van den Bergh (1961c), Limber (1961), and Rood (1965). The problem, then, is to understand why mass-to-light ratios found for clusters and groups of galaxies are often higher - by even a factor of 10 or more - than those found for individual or double galaxies. Among the possibilities proposed are the following: 1. Ambartsumian (1958) has suggested that clusters (at least some of the smaller groups) have positive energy and are expanding. On the other hand, to reduce the dynamical mass of a cluster even by just a factor of 4 from that given by the virial theorem, we would need to have T = - 2; in this case, the observed speeds of galaxies are, in the mean, only 2 times greater than the speeds they will have when the cluster has expanded to infinity. In a period of at most a few times 109 years, all clusters would dissipate enough to have lost their identity, and it would be difficult to understand why most galaxies are still in clusters.

2. Perhaps the high masses of some clusters are due to, a few extremely massive galaxies. There does appear to be a tendency for the mass-to-light ratio to be an increasing function of mass for elliptical galaxies (Rood 1965). However, it is very hard to invent convincing stellar populations that can give the extremely high mass-to-light ratios that would be required. Moreover, many of the smaller clusters and groups for which the mass-to-light ratio appears to be high have only spirals among their massive members.

3. The derived mass-to-light ratios are proportional to the assumed value of the Hubble constant. A greatly expanded distance scale would therefore reduce them. But it would also reduce the corresponding ratios found for most individual galaxies and for double galaxies, so the discrepancy would remain.

4. There is a probability that metagalactic structure of larger size than clusters exists i.e., second-order clusters (see next section). If so, and if gravitational forces exist between clusters in such a system, then two or more clusters seen overlapping in projection may occasionally be mistaken for an isolated system, and their relative motions can lead to a spuriously high observed velocity dispersion. In some cases, such an effect may partially be responsible for the mass-to-light discrepancy. The Hercules cluster, in particular, is in a region rich in groups and clusters of about the same distance and may not be a single dynamic unit. De Vaucouleurs's suggestion that the Virgo cluster is at least two clusters has already been mentioned, as has the possibility of inappropriately assuming cluster membership for outlying galaxies in the Coma cluster. In fact, Gott, Wrixon, and Wannier (1973) have shown that some systems that have been identified as groups may actually be optical alignments of field galaxies that are not gravitationally bound at all, and thus that high mass-to-light ratios reported for these systems may be entirely spurious.

5. Subclustering, or incidence of duplicity within a cluster, results in lowering the average separation of galaxies, thereby increasing the potential energy for a given mass. Van den Bergh (1960) has pointed out that binary galaxies in the Virgo cluster can possibly affect significantly the derived cluster mass. Studies of subclustering in several clusters, however (Abell, Neyman, and Scott 1964), show that the phenomenon can not be important generally.

6. A popular explanation is that clusters contain a large amount of invisible matter which adds to their masses but not to their visual luminosities. Zwicky (1957) believes that intergalactic obscuring matter in clusters hides more remote clusters, although the existence of such absorption is not universally accepted. For the Coma cluster, he estimates an obscuration of 0.6 mag (Zwicky 1959). The dust required to produce this absorption, however, corresponds to a density of only 10-30 g cm-3, if the dust has the same obscuring properties as interstellar grains in the Galaxy (de Vaucouleurs 1960); this would be a negligible contribution to the cluster mass. Evidence for luminous haze in the centers of clusters is inconclusive and controversial; if visible luminous matter does exist, it can in no case contribute substantially to the mass of a cluster. Field (1959) has shown that neutral hydrogen cannot be important enough in intergalactic space to affect cluster masses measurably. Observations of the 21-cm line in absorption reported by Robinson, van Damme, and Koehler (1963) suggest the presence of neutral hydrogen in the Virgo cluster, but the total mass indicated is more than two orders of magnitude less than that of the cluster itself. Recent 21-cm observations of three groups by Gott, Wrixon, and Wannier (1973) indicate that neutral hydrogen falls short by at least a factor of 25 of having enough mass to gravitationally bind those systems. Nor is hot plasma likely to be able to contribute appreciably to the potential energies of clusters; an analysis by Davidson, Bowyer, and Welch (1973) of the radio, soft X-ray, and far-ultraviolet observations of the Coma cluster, for example, seems to rule out the possibility that ionized gas contributes enough mass to bind that cluster. Still, intracluster material cannot yet be ruled out in all cases.