The bivariate brightness function of galaxies and a demonstration of the impact of surface brightness selection effects on luminosity function estimations

In this paper we fit an analytic function to the bivariate brightness distribution (BBD) of galaxies. It is a combination of the classical Schechter Function convolved with a Gaussian distribution in surface brightness: thus incorporating the luminosity-surface brightness correlation as seen in many recent data sets. We fit this function to a recent measurement of the BBD based on 45 000 galaxies from the Two-Degree Field Galaxy Redshift Survey. The parameters for the best-fitting model are phi* = (0.0206+/-0.0009)h(3) Mpc(-3), M-bj(*) -5 log h = (-19.72+/-0.04) mag, alpha = -1.05+/-0.02, beta(mu) = 0.281+/-0.007, mu(e,bj)(*) = (22.45+/-0.01) mag arcsec(-2) and sigma(mu) = 0.517+/-0.006. phi*, M-bj(*) and alpha equate to the conventional Schechter parameters. beta(mu) is the slope of the luminosity-surface brightness correlation, mu(e,bj)(*), is the characteristic effective surface brightness at M-bj(*) and sigma(mu) is the width of the Gaussian. Using a BBF we explore the impact of the limiting detection isophote on classical measures of the galaxy luminosity distribution. We demonstrate that if isophotal magnitudes are used then errors of DeltaM(bj)(*) similar to 0.62 mag, Deltaphi* similar to 26 per cent and Deltaalpha similar to 0.04 are likely for mu(lim,bj) = 24.0 mag arcsec(-2). If Gaussian corrected magnitudes are used these change to DeltaM(bj)(*) similar to 0.38 mag, Deltaphi* similar to 11 per cent and Deltaalpha < 0.01 for μ(lim,bj) = 24.0 mag arcsec(-2). Hence while the faint-end slope, a, appears fairly robust to surface brightness issues, both the M* and φ* values are highly dependent. The range over which these parameters were seen to vary is fully consistent with the scatter in the published values, reproducing the range of observed luminosity densities (1.1 < j(bj) < 2.2 x 10(8)h L-⊙ Mpc(-3)). If total magnitudes are recovered then there is no change in the luminosity function within the errors for μ(lim,bj) = 24.0 mag arcsec(-2). We conclude that surface brightness selection effects are primarily responsible for this variation. After due consideration of these effects, we derive a value of j(bj) = 2.16 x 10(8)h L-⊙ Mpc(-3).