The Mean and Scatter of the Velocity Dispersion–Optical Richness Relation for maxBCG Galaxy Clusters

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Abstract
The distribution of galaxies in position and velocity around the centers of galaxy clusters encodes important information about cluster mass and structure. Using the maxBCG galaxy cluster catalog identified from imaging data obtained in the Sloan Digital Sky Survey, we study the BCG-galaxy velocity correlation function. By modeling its non-Gaussianity, we measure the mean and scatter in velocity dispersion at fixed richness. The mean velocity dispersion increases from 202 ± 10 km s-1 for small groups to more than 854 ± 102 km s-1 for large clusters. We show the scatter to be at most 40.5% ± 3.5%, declining to 14.9% ± 9.4% in the richest bins. We test our methods in the C4 cluster catalog, a spectroscopic cluster catalog produced from the Sloan Digital Sky Survey DR2 spectroscopic sample, and in mock galaxy catalogs constructed from N-body simulations. Our methods are robust, measuring the scatter to well within 1 σ of the true value, and the mean to within 10%, in the mock catalogs. By convolving the scatter in velocity dispersion at fixed richness with the observed richness space density function, we measure the velocity dispersion function of the maxBCG galaxy clusters. Although velocity dispersion and richness do not form a true mass-observable relation, the relationship between velocity dispersion and mass is theoretically well characterized and has low scatter. Thus, our results provide a key link between theory and observations up to the velocity bias between dark matter and galaxies.