Using a large and well-controlled sample of clusters of galaxies, we investigate the relation between cluster velocity dispersions and X-ray temperatures of intracluster gas. The cluster selection is based on nonparametric methods. In particular, we present the two-dimensional optical maps of our sample clusters, obtained via the kernel adaptive technique, using an optimized smoothing parameter. In order to obtain a reliable estimate of the total velocity dispersion of a cluster, independent of the level of anisotropies in galaxy orbits, we analyze the integrated velocity dispersion profiles over increasing distances from the cluster centers. Both increasing and decreasing integrated profiles are found, but the general trend is a flattening of the integrated velocity dispersion profile at the largest radii, thus enabling us to take the asymptotic value of the integrated profile as an estimate of the total velocity dispersion, which is independent of possible anisotropies. Distortions in the velocity fields, the effect of close clusters, the presence of substructures, and the presence of a population of (spiral) galaxies not in virial equilibrium with the cluster potential are taken into account for reducing the errors in the estimate of the cluster velocity dispersions. Using our final sample of 37 clusters for which a reliable estimate of velocity dispersion could be obtained, we derive a relation between the velocity dispersions and the X-ray temperatures, with a scatter reduced by more than 30% with respect to previous works. A χ~2 fit to the temperature-velocity dispersion relation does not exclude the hypothesis that the ratio between galaxy and gas energy density (the so-called β_(spec)) is a constant for all clusters. In particular, the value of β_(spec) = 1, corresponding to energy equipartition, is acceptable. However, the large data scatter in the σ-T relation may suggest the presence of intrinsic dispersion. This intrinsic dispersion may be due to spurious effects (we consider the effect of cluster ellipticity) as well as to physical reasons, different values of β_(spec) pertaining to clusters with different properties.
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