A Quantitative Measure of the Richness of Galaxy Clusters

Using photometric catalogs from wide-field CCD images, we derive the cluster-galaxy correlation amplitudes, Bgc, for 47 low-redshift Abell clusters. We apply a number of tests to establish the robustness of the Bgc parameter as a quantitative measure of richness of galaxy clusters. These include using different galaxy luminosity functions to normalize the excess galaxy counts, counting galaxies to different absolute magnitude limits, and counting galaxies to different cluster-centric radii. These tests show that with a properly normalized luminosity function, the Bgc parameter is relatively insensitive (at better than 1/2 of an Abell Richness class) to magnitude limit, areal coverage, and photometric errors of up to about 0.25 mag. We compare the Bgc values to both the Abell Richness Class (ARC) and Abell count numbers (N_A). It is found that there is a good correlation between Bgc and N_A for Abell clusters with z<~0.1, with a dispersion of about one ARC; whereas the Abell Richness classifications for Abell clusters at z>~0.1 are much less well correlated with the true cluster richness. We also find evidence that the richness of ARC>=3 clusters has the tendency of being over estimated in the Abell catalog.