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Imaging data from the Sloan Digital Sky Survey is used to measure theempirical size-richness relation for a large sample of galaxy clusters. Usingpopulation subtraction methods, we determine the radius at which the clustergalaxy number density is 200/Omega_m times the mean galaxy density, withoutassuming a model for the radial distribution of galaxies in clusters. If thesegalaxies are unbiased on Mpc scales, this galaxy-density-based R_200 reflectsthe characteristic radii of clusters. We measure the scaling of thischaracteristic radius with richness over an order of magnitude in clusterrichness, from rich clusters to poor groups. We use this information to examinethe radial profiles of galaxies in clusters as a function of cluster richness,finding that the concentration of the galaxy distribution decreases withrichness and is systematically lower than the concentrations measured for darkmatter profiles in N-body simulations. Using these scaled radii, we investigatethe behavior of the cluster luminosity function, and find that it is wellmatched by a Schechter function for galaxies brighter than M_r = -18 only afterthe central galaxy has been removed. We find that the luminosity functionvaries with richness and with distance from the cluster center, underscoringthe importance of using an aperture that scales with cluster mass to comparephysically equivalent regions of these different systems. We note that thelowest richness systems in our catalog have properties consistent with thoseexpected of the earliest-forming halos; our cluster-finding algorithm, inaddition to reliably finding clusters, may be efficient at finding fossilgroups.Comment: 17 pages, 12 figures. Accepted for publication in Ap

Measurement of Galaxy Cluster Sizes, Radial Profiles, and Luminosity Functions from SDSS Photometric Data