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We present galaxy luminosity functions in the K band for a sample of 10 clusters selected from the Einstein EMSS cluster catalog with $0.3 < z < 0.4$. X-ray selection of clusters requires the presence of a deep potential well to contain the $10^7K$ plasma emitting the X-rays, and thus ensures a sample free from the projection effects that are inherent in optically selecting clusters by galaxy overdensity. Similarly, selection in the K-band ensures a well-defined sample of galaxies. The near-infrared $K$--corrections and expected luminosity evolution are smooth, independent of Hubble type, and well understood. While the optical flux from a galaxy is dominated by short-lived massive stars and galaxies change on short timescales, the K-band flux is dominated by long-lived near-solar mass stars which are comparable in lifetime and timescales of evolution to the age of the galaxy. Because near-solar mass stars make up the bulk of the galaxy, the absolute K magnitude is a measure of the mass of light matter in a galaxy, and the K-band luminosity function can be considered a mass function.
We have obtained $1.2 Mpc \times 1.2 Mpc$ images of 10 clusters to $K=18.5$, and have measured the luminosity function to $M_K=-22.5$, ($H_0=50km/s/Mpc$.) We compare each cluster luminosity function to a composite of all 10 clusters and to measured field galaxy luminosity functions using nonparametric statistical methods. We measure the dependence of the characteristic luminosity ($M_K^*$) upon X-ray luminosity and the blue galaxy fraction of the Butcher-Oemler effect.
