The luminosity function for galaxies and the clustering of galaxies

An analytic representation for the luminosity function for galaxies is proposed. Best fits of this function to counts of nearby bright galaxies and to counts of galaxies in rich clusters have been obtained. The results are marginally consistent with a single luminosity function valid for both samples. The proposed representation contains a characteristic magnitude M* which exhibits an equivalent dispersion of only .24 magnitudes from cluster to cluster. The narrow dispersion in absolute magnitude observed for the brightest members of clusters is understood in large part as statistical fluctuation about a universal luminosity function, but the correlation of absolute magnitude with richness expected from the proposed representation is not observed. It is shown that galaxies will condense into clusters of the sizes presently observed if the perturbations giving rise to galaxies were randomly distributed at recombination. A model for the origin of clusters is proposed which assumes (a) that galaxies collapse without dissipation, (b) that the perturbations giving rise to galaxies are centrally condensed, and (c) that most of the matter density in the universe is in galaxies. The problem of the distribution of cluster sizes for Poisson distributed points is discussed and an analytic approach to a solution is developed. Numerical experiments show factor of two agreement with the solution obtained. It is shown that if galaxies were randomly distributed at some early epoch massive galaxies are less likely to be isolated than less massive galaxies. An observational definition of an "isolated galaxy" is offered and an observational test of the model is proposed