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 datasets. We fit this function to a recent measurement of the BBD based on 45,000 galaxies from the two-degree field Galaxy Redshift Survey (Cross et al. 2001). Using a BBF we explore the impact of the limiting detection isophote on classical measures of the galaxy luminosity distribution. If Gaussian corrected magnitudes are used these change to $\Delta M^*_{b_j} \sim 0.38 $ mags, $\Delta \phi^* \sim 11%$ and $\Delta \alpha < 0.01$ for $\mu_{lim,b_j}=24.0$ mag arcsec$^{-2}$. Hence while the faint-end slope, $\alpha$, appears fairly robust to surface brightness issues, both the $M^*$ and $\phi^*$ 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 ($1.1<j_{b_j}<2.2\times10^8h L_{\odot}$Mpc$^{-3}$ see Cross et al. 2001). We conclude that surface brightness selection effects are primarily responsible for this variation. After due consideration of these effects, we derive a value of $j_{b_j}=2.16\times10^8h L_{\odot}$Mpc$^{-3}$.