2.3. Richness - the number of galaxies in a cluster
The richness of a cluster is a measure of the number of galaxies associated with that cluster. Because of the presence of background galaxies, it is not possible to state with absolute confidence that any given galaxy belongs to a given cluster. Thus one cannot give an exact tally of the number of galaxies in a cluster. Richness is a statistical measure of the population of a cluster, based on some operational definition of cluster membership. The richness will be more useful if it is defined in such a way as to be reasonably independent of the distance to and morphology of a cluster.
Zwicky et al. (1961-1968) define the richness of their clusters as the total number of galaxies visible on the red Sky Survey plates within the cluster isopleth (Section 2.1); the number of background galaxies expected is subtracted from the richness. These richnesses are clearly very dependent on a cluster redshift (Abell, 1962; Scott, 1962). First, a wider magnitude range is counted for nearby clusters, because the magnitude of the first brightest galaxy is further from the plate limit. Second, a larger area of the cluster is counted for nearby clusters, because the point at which the surface density is twice that of the background will be farther out in the cluster.
Abell (1958) has divided his clusters into richness groups using criteria that are nearly independent of distance (see Section 2.1); that is, the magnitude range and area considered do not vary with redshift. Just (1959) has found a slight richness-distance correlation in Abell's catalog; however, the effect is small and is probably explained by a slight incompleteness (10%) of the catalog for distant clusters (Paal, 1964). When more accurate determinations have been made, it has been found that the Abell richness generally correlated well with the number of galaxies, but that the Abell richness may be significantly in error in some individual cases (Mottmann and Abell, 1977; Dressler, 1980a). Thus the Abell richnesses are very useful for statistical studies, but must be used with caution in studies of individual clusters. Note also that it is generally preferable to use the actual Abell counts rather than just the richness group (Abell, 1982).