Appendix A: Relationship between N and Abell richness

We adopt N, the field-corrected number of all galaxies brighter than m^* + 5, as the indicator of cluster richness in this paper. In most previous studies, however, Abell's richness parameter c, which is the field-corrected number of galaxies inside the Abell radius with magnitude between m₃ and m₃+2, is conventionally used. For convenience, we estimate the relationship between N and c by simply using random values whose probability density function obeys a Schechter function.

For a given value of N, we generate 20 "clusters'' and count numbers of galaxies with magnitudes between m₃ and m₃+2.

  

Figure 12: Relationship between our richness parameter N and Abell richness c. Solid line shows the median value of c of 20 artificial clusters with a fixed N. Dashed lines show $\pm 1\sigma$ range

Figure 12 shows the relationship. The median of 20 c values corresponding to a given N is shown by solid line. The region between the two dashed lines represents the central $\pm 1\sigma$ area. The median value of c is well approximated by the following power-law relations

$$\begin{eqnarray} c & = & 0.46\; N^{0.67} \hspace*{0.5cm} (N\mathrel{\mathchoice ... ...halign{\hfil$\scriptscriptstyle ...$$ (8) (9)

For the Coma cluster c equals 106, which corresponds to $N\sim$ 3000.

The magnitude of the third brightest galaxy is often significantly affected by the overlapping galaxies or variations in cluster morphology; if a cluster's morphology is cD or B in Rood-Sastry classification (Rood & Sastry 1971), m₃ would be often fainter because the original third brightest galaxy may have been merged by the first or second brightest galaxy and a fainter galaxy becomes the new third brightest galaxy. On the other hand, m₃ would be brighter for L or C clusters because the third brightest galaxy may capture smaller ones. The error in estimating m₃ directly affects c and leads eventually to some serious systematic error in the evaluation of cluster richness. On the other hand, N includes all galaxies brighter than m^*+5 and is less affected by those factors. We can use N as a more robust richness indicator than c.