The Correlation Function of Flux-Limited X-Ray Clusters

We show that the spatial correlation function of a flux-limited sample of X-ray selected clusters of galaxies will exhibit a correlation scale that is smaller than the correlation scale of a volume-limited, richness-limited sample of comparable apparent spatial density. The flux-limited sample contains clusters of different richnesses at different distances: poor groups are found nearby and rich clusters at greater distances. Since the cluster correlation strength is known to increase with richness, the flux-limited sample averages over the correlations of poor and rich clusters. On the other hand, a volume-limited, richness-limited sample has a minimum richness threshold, and a constant mixture of richnesses with redshift. Using the observed correlation scale for rich ($R\ge 1$) clusters, $r_o (R\ge 1) = 21\pm 2 h^{-1}$Mpc that was determined from previous volume-limited studies, we derive for the ROSAT flux-limited X-ray cluster sample $r_o({\rm flux}{\rm -}{\rm limited}) \approx 14h^{-1}$Mpc, in agreement with the recently observed value of $13.7\pm 2.3 h^{-1}$Mpc.