THE ESTIMATION OF ABSOLUTE MAGNITUDE AND DISTANCE FROM THE TULLY-FISHER RELATION

Several aspects of using the Tully-Fisher relation to estimate magnitudes and distances are examined, in particular the effect of the luminosity function on the theoretical estimation of regression parameters, unbiased estimates of absolute magnitude, and the Malmquist bias of field galaxies. With respect to observations of clusters of galaxies, if the limiting absolute magnitude is close to or brighter than the Schecter magnitude M* there is a significant difference in the slopes and offsets of the regression of magnitude upon log (line width) as calculated from Gaussian and Schecter functions. The linear regression estimate of absolute magnitude is biased in the case of the Schecter luminosity function, and this is especially important for absolute magnitudes which are brighter than M*. With respect to field galaxies and Malmquist bias (defined as the difference between the estimated absolute magnitudes for field and cluster galaxies for the same line width) is constant for a Gaussian luminosity function but depends upon line width for a Schecter luminosity function. The difference between the Gaussian and Schecter values is insignificant for galaxies fainter than M. but significant for galaxies brighter than M*. Useful approximate relations for estimating the absolute magnitude and distance of field galaxies for arbitrary luminosity functions and arbitrary density distributions are given.