4.1. General Considerations
The incomplete, and even conflicting, observational data do not permit us to draw a very complete picture of the dynamics of even a single cluster. At best, from observational data alone, we can make only rough estimates of clusters' probable dynamical structures.
The time of relaxation in the central regions of a typical regular cluster, like the Coma cluster, is almost certainly greater than 109 years, and probably lies within a small factor of 1010 years (Oort 1958; van Albada 1960). It is doubtful, therefore, that in the presumed age of the Universe (~ 1010 years) gravitational encounters between galaxies can have set up anything approaching a state of statistical equilibrium in a regular cluster. If, however, such a state could have been reached, we would expect: (1) an equipartition of energy with a consequent segregation of velocities of galaxies according to their masses; (2) a radial segregation of galaxies within the cluster according to mass, with the less massive, faster-moving galaxies occupying a larger spatial distribution; and (3) a mass distribution within at least the central regions of the cluster resembling that of the isothermal polytrope. Observational verification of all of these effects has been reported by various investigators, but these observations are inconclusive.
In cluster 194, Zwicky and Humason (1964a) report radial velocities for 41 galaxies that they presume to be cluster members. The mean velocity is 5321 km s-1, with an rms dispersion of 406 km s-1. The brightest 21 galaxies, however, have a mean velocity of 5254 km s-1, with a dispersion of 360 km s-1, while the corresponding figures for the faintest 21 galaxies are 5392 and 439 km s-1. Zwicky and Humason interpret the slightly higher dispersion in velocity of the fainter members to be indicative of a degree of equipartition of energy. The brightest six galaxies do have velocities very near the mean for the cluster, but for the remaining galaxies the effect vanishes; if anything, the dispersion in velocity decreases with increasing magnitude, as is seen at once on a plot of Zwicky and Humason's data. Indeed, as those authors themselves point out, there is a larger difference between the velocity dispersion of galaxies in the southwest and north-east halves of the cluster, than between the bright and faint members. In short, the case for a velocity-dispersion dependence on magnitude in cluster 194 is not convincing. Negative results for such segregation have been found for five other groups and clusters (Lovasich, Mayall, Neyman, and Scott 1961; Neyman and Scott 1961; Rood 1965). In their analysis of the Coma cluster, Rood et al. (1972) find no evidence for a luminosity dependence of velocity dispersion, with the possible exception that the very brightest galaxies in the central core may have a smaller rms dispersion; they regard these results, however, as statistically inconclusive.
Zwicky (1957) reports a very pronounced radial segregation of bright and faint members in the Coma cluster; but as was pointed out in the last section, this result depends on what assumption is made about the field. The writer's own investigation indicates that a slight segregation possibly exists (fig. 6), but not necessarily so. Rudnicki's (1963) finding of a segregation of bright and faint galaxies in cluster 426 may result from irregular interstellar absorption. The case for the Virgo cluster is unclear because it may be more than one cluster. The slight segregation found by Hodge, Pyper, and Webb (1965) in the Fornax cluster appears to be real, but Fornax is a poor cluster, and the segregation is at best small. Other investigations have also suggested slight radial segregation of bright and faint galaxies in some clusters (N. Bahcall 1972; Kwast 1966; Rudnicki and Baranowska 1966b; Rood 1969; Rood and Abell 1973; Noonan 1971), but the evidence is sometimes contradictory, and in no case is there segregation present in the amount required by complete statistical equilibrium.
Figure 6. The radial density distribution of bright and faint galaxies in the Coma cluster (No. 1656) derived from Abell's counts. The zero points of the ordinates are arbitrary and have been shifted to agree for bright and faint galaxies at the cluster center.
As we have already seen, both Zwicky and N. Bahcall have shown that observations of the density distributions in several regular clusters are compatible with isothermal distributions, but their data can also be represented about as well by various other distributions that have nothing to do with statistical equilibrium.
In summary, there is no convincing evidence that equipartition of energy exists in any cluster, although there is some evidence for partial radial segregation by mass in a few clusters. Nor would we expect equipartition of energy, because of the long times of relaxation in clusters. Ages of clusters are probably at most only a few relaxation times; some dynamical evolution has probably taken place, but it is unlikely that a final state has been reached.