The brightest galaxy in a cluster as a distance indicator

. The customary use of the brightest (or fifth brightest) galaxy in a cluster as a distance indicator is studied. Estimates of the absolute magnitude of a distant galaxy will tend to be too faint and its distance will be underestimated due to the unavoidable selection of available clusters for observation. The present study is limited to the effects of only two conditions, which are certainly involved in the selection of clusters: (a) The cluster must have at least a certain number n galaxies brighter than a limit m1, (b) the rth brightest galaxy must be brighter than m2 =< m1. An exact theory of the effects of conditions (a) and (b) is developed and applied to the problem of estimating distances to clusters and to the interpretation of the magnitude-redshift relation. It is found that near the threshold of the instruments used, the effects of selection bias are strong.