The luminosity function in the Petrosian r band bivariate with various galaxy parameters was computed for Main Sample galaxies in the SDSS Data Release 4, with the addition of data from the New York Value Added Galaxy Catalog (VAGC), the Pittsburgh-CMU Value-Added Galaxy Catalog (VAC), the MPA-Garching catalog 'Physical properties for galaxies and active galactic nuclei', and the SDSS Japan Participation Group.
The parameters are eyeball Hubble type, Hubble type assigned by an artificial neural network, inverse concentration index, the Sersic index of the light profile, absolute effective surface brightness, reference frame colours u-g, g-r, r-i and i-z, absolute radius, eClass spectral type, stellar mass and galaxy environment. We used the bivariate formulation of the stepwise maximum likelihood method of Efstathiou, Ellis and Peterson (1988), taking into account completeness in each bin. The magnitudes are extinction- and K-corrected, the latter using version 4.1.14 of Mike Blanton's K-correct code. The final results are described in 'Bivariate Galaxy Luminosity Functions in the Sloan Digital Sky Survey', MNRAS 373 845, or astro-ph/0507547.
Here, Figure 1 shows a typical example of a bivariate LF, this one being for the surface brightness. Numerous features are visible, including the Schechter-function-like form of the function in absolute magnitude Mr, but the Gaussian-like form in surface brightness. In Mr, an upturn from a Schechter function is evident at the faint end for moderately luminous galaxies. In surface brightness, the Gaussian is clearly wider at fainter magnitudes, although it never obviously resolves into two components as does, e.g., colour. This shows that whatever processes determine surface brightness as not as obviously bimodal as those which determine colour. Similar insights may be gained from the numerous other bivariate LFs.
Figure 1: LF bivariate with absolute effective Petrosian 50% light radius surface brightness. In the left-hand panel, the thick contours show the regions inside which volumes of 102 to 107 cubic Mpc were probed. From Ball et al. 2006 (MNRAS 373 845).
We fitted the LF bivariate with surface brightness with a Schechter-Gaussian function (Choloniewski function). The fit was found to be poor, as might be expected if there are in fact two underlying processes. In agreement with previous authors we hypothesized, from the full set of LF results, that the two processes are those associated with bulges and disks, i.e., merger and accretion respectively.
Figure 2: SWML estimate in greyscale and fitted Choloniewski function as contours for absolute effective Petrosian surface brightness. From Ball et al. 2006.