The ESO Nearby Abell Cluster Survey .2. The distribution of velocity dispersions of rich galaxy clusters

The ESO Nearby Abell Cluster Survey (the ENACS) has yielded 5634 redshifts for galaxies in the directions of 107 rich, Southern clusters selected from the ACO catalogue (Abell et al. 1989). By combining these data with another 1000 red-shifts from the literature, of galaxies in 37 clusters, we construct a volume-limited sample of 128 R(ACO) greater than or equal to 1 clusters in a solid angle of 2.55 sr centered on the South Galactic Pole, out to a redshift z = 0.1. For a subset of 80 of these clusters we can calculate a reliable velocity dispersion, based on at least 10 (but very often between 30 and 150) redshifts. We deal with the main observational problem that hampers an unambiguous interpretation of the distribution of cluster velocity dispersions, namely the contamination by fore- and back-ground galaxies. We also discuss in detail the completeness of the cluster samples for which we derive the distribution of cluster velocity dispersions. We find that a cluster sample which is complete in terms of the field-corrected richness count given in the ACO catalogue gives a result that is essentially identical to that based on a smaller and more conservative sample which is complete in terms of an intrinsic richness count that has been corrected for superposition effects. We find that the large apparent spread in the relation between velocity dispersion and richness count (based either on visual inspection or on machine counts) must be largely intrinsic; i.e. this spread is not primarily due to measurement uncertainties. One of the consequences of the (very) broad relation between cluster richness and velocity dispersion is that all samples of clusters that are defined complete with respect to richness count are unavoidably biased against low-sigma(nu) clusters. For the richness limit of our sample this bias operates only for velocity dispersions less than approximate to 800 km/sec. We obtain a statistically reliable distribution of global velocity dispersions which, for velocity dispersions sigma(nu) greater than or similar to 800 km/s, is free from systematic errors and biases. Above this value of sigma(nu) our distribution agrees very well with the most recent deter mination of the distribution of cluster X-ray temperatures, from which we conclude that beta = sigma(nu)(2) mu m(H)/kT(X) approximate to 1. The observed distribution n(> sigma(nu)), and especially its high-sigma(nu) tail above approximate to 800 km/s, provides a reliable and discriminative constraint on cosmological scenarios for the formation of structure. We stress the need for model predictions that produce exactly the same information as do the observations, namely dispersions of line-of-sight velocity of galaxies within the turn-around radius and inside a cylinder rather than a sphere, for a sample of model clusters with a richness limit that mimics that of the sample of observed clusters.