  Imaging data from the Sloan Digital Sky Survey is used to measure the
empirical size-richness relation for a large sample of galaxy clusters. Using
population subtraction methods, we determine the radius at which the cluster
galaxy number density is 200/Omega_m times the mean galaxy density, without
assuming a model for the radial distribution of galaxies in clusters. If these
galaxies are unbiased on Mpc scales, this galaxy-density-based R_200 reflects
the characteristic radii of clusters. We measure the scaling of this
characteristic radius with richness over an order of magnitude in cluster
richness, from rich clusters to poor groups. We use this information to examine
the radial profiles of galaxies in clusters as a function of cluster richness,
finding that the concentration of the galaxy distribution decreases with
richness and is systematically lower than the concentrations measured for dark
matter profiles in N-body simulations. Using these scaled radii, we investigate
the behavior of the cluster luminosity function, and find that it is well
matched by a Schechter function for galaxies brighter than M_r = -18 only after
the central galaxy has been removed. We find that the luminosity function
varies with richness and with distance from the cluster center, underscoring
the importance of using an aperture that scales with cluster mass to compare
physically equivalent regions of these different systems. We note that the
lowest richness systems in our catalog have properties consistent with those
expected of the earliest-forming halos; our cluster-finding algorithm, in
addition to reliably finding clusters, may be efficient at finding fossil
groups.
