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 \pm 10 km s ^ { -1 } for small groups to more than 854 \pm 102 km s ^ { -1 } for large clusters .
We show the scatter to be at most 40.5 \pm 3.5 % , declining to 14.9 \pm 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 one-sigma 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 .