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

7.2 Other Methods of Mass Determination

Unfortunately, the use of the virial theorem to measure cluster masses is sensitive to the structure of the outer parts of the cluster (Rood et al. 1972). An alternative is to consider only the well observed core regions. Zwicky (1937, 1957) found that the number-density profile of the Coma cluster could be fit by a bounded isothermal sphere. Using such a model, Bahcall (1975) derived a mean core radius, r_c, of 0.25±0.04 Mpc for 15 clusters. Dressler (1978a) also found a small dispersion in core radius but his mean value was 0.5 Mpc. Avni & Bahcall (1976) have tested the measuring techniques for r_c on simulated clusters. They found that the existence of cluster cores is real, not an artifact of the data analysis, but that the close fit of Bahcall's clusters to a common isothermal model is due to the fitting procedure used.

If r_c and the core velocity dispersion, _c, are known for a cluster, a dynamical model can be used to calculate the central mass density. The King (1966, 1972) models were shown by Rood et al. (1972) to provide a reasonable representation of both the run of surface density and of velocity dispersion with radius in the Coma cluster. With this method, M / L_V values of 280, 350, and 250 have been derived for the cores of Coma (Rood et al. 1972), Perseus (Bahcall 1974), and Virgo (van den Bergh 1977). The value for Coma has been rescaled using a revised core velocity dispersion of 1260 km s^-1 (Gregory & Tifft 1976).

Core-fitting procedures are subject to operational difficulties which Rood et al. clearly describe: first, the core radius is difficult to determine (e.g. the factor of two difference in mean r_c between Bahcall and Dressler); second, due to small-number statistics, the core velocity dispersion is uncertain and the core luminosity distribution is grainy. The situation is often further complicated by the presence of centrally located luminous galaxies, which make it difficult to measure the core luminosity accurately. To avoid the problem of graininess in the number density, Dressler (1978b) used the smoothed central number density from the model fit, assigned a mean luminosity per galaxy using a Schechter (1976) luminosity function, and applied a 20% correction to convert from isophotal to total luminosities. From a sample of nine rich clusters with velocity dispersions primarily taken from Faber & Dressler (1977), he found a median M / L_V equal to 270. For comparison. he computed a global M / L_V for each cluster using a fit to a de Vaucouleurs (1948) law (see Young 1976). This approach yielded a median M / L_V equal to 280. The global and core-fitting methods are therefore in good agreement for these clusters. Rood et al. also analyzed the Coma cluster with a global de Vaucouleurs law and found M / L_V = 200.

With the increasing sophistication of computer N-body models, it has become possible to custom build a model to represent a specific cluster. White (1976b) extended the N-body program of Aarseth (1969) to produce a realistic dynamical model of the Coma cluster using 700 particles. The model was scaled to Oemler's gravitational radius and the radial velocities of Rood et al. and Gregory (1975). From the model, White (1976b) found M / L_V = 258±36. Melnick et al. (1977) have shown that the value of M / L_V is reduced slightly to 207±47 if L_V is corrected to the total light in the central regions of Coma.

All of these analyses of the Coma cluster yield values of M / L_V close to 250. The exception is Abell's (1977) study, which concludes that M / L_V = 120. This result is due to two factors. First, the derived mass is small because Abell assumed V² / ² = 2 rather than 3. More important, however, the form of the luminosity function chosen by Abell results in a luminosity twice as large as Oemler's. Relative to Oemler, Abell counts more galaxies with 13.5 < m_v < 14.5. Furthermore, Oemler's faint-end extrapolation adds fewer galaxies than actually counted by Abell. Both effects contribute roughly equally to the net difference, with the excess at the bright end being somewhat more significant. Godwin & Peach (1977) also measured the luminosity function in Coma and found more galaxies with 15 m_v 17.5 than either Abell or Oemler.

It suffices to note that luminosity functions of clusters are still not well known. This problem is especially severe in poorly studied clusters, where one must assume a universal luminosity function to estimate the contribution of faint galaxies. Furthermore, there is good evidence for significant cluster-to-cluster variations in the luminosity function, and luminosity functions may not be amenable to simple analytic forms (Dressler 1978a).

Ignoring uncertainties in the total luminosity for the moment, we emphasize the good agreement between the mass-to-light ratios for Coma found by many different authors using various techniques. Furthermore, these various approaches weight subsets of the data in very different fashions, some emphasizing the core, others the outer regions. It is indeed remarkable that these techniques yield such consistent values.