We recover the joint and individual space density and surface brightness distribution ( s ) of galaxies from the Millennium Galaxy Catalogue .
The MGC is a local survey spanning 30.9 deg ^ { 2 } and probing approximately one–two mag arcsec ^ { -2 } deeper than either the Two-Degree Field Galaxy Redshift Survey ( 2dFGRS ) or the Sloan Digital Sky Survey ( SDSS ) .
The MGC contains 10 095 galaxies to B _ { \mbox { \tiny \sc MGC } } < 20 mag with 96 per cent spectroscopic completeness .
For each galaxy we derive individual K -corrections and seeing-corrected sizes .
We implement a joint luminosity-surface brightness step-wise maximum likelihood method to recover the bivariate brightness distribution ( BBD ) inclusive of most selection effects .
Integrating the BBD over surface brightness we recover the following Schechter function parameters : \phi ^ { * } = ( 0.0177 \pm 0.0015 ) h ^ { 3 } Mpc ^ { -3 } , M _ { B _ { \mbox { \tiny \sc MGC } } } ^ { * } -5 \log h = ( -19.60 \pm 0.04 ) mag and \alpha = -1.13 \pm 0.02 .
Compared to the 2dFGRS ( 59 ) we find a consistent M ^ { * } value but a slightly flatter faint-end slope and a higher normalisation , resulting in a final luminosity density j _ { b _ { J } } = ( 1.99 \pm 0.17 ) \times 10 ^ { 8 } h L _ { \odot } Mpc ^ { -3 } — marginally higher than , but consistent with , the earlier 2dFGRS ( 59 ) , ESP ( 79 ) , and SDSS z = 0.1 ( 9 ) results .
The MGC is inconsistent with the SDSS z = 0 result ( +3 \sigma ) if one adopts the derived SDSS evolution ( 9 ) .

The MGC surface brightness distribution is a well bounded Gaussian at the M ^ { * } point with \phi ^ { * } = ( 3.5 \pm 0.1 ) \times 10 ^ { -2 } h ^ { 3 } Mpc ^ { -3 } , \mu ^ { e* } = ( 21.90 \pm 0.01 ) mag arcsec ^ { -2 } and \sigma _ { \ln R _ { e } } = 0.35 \pm 0.01 .
The characteristic surface brightness for luminous systems is invariant to M _ { B _ { \mbox { \tiny \sc MGC } } } -5 \log h \approx - 19 mag faintwards of which it moves to lower surface brightness .
The surface brightness distribution also broadens ( \sigma _ { \ln R _ { e } } \approx 0.5 - 0.7 ) towards lower luminosities .
The luminosity dependence of \sigma _ { \ln R _ { e } } provides a new constraint for both the theoretical development ( Dalcanton , Spergel & Summers 1997 ; Mo , Mao & White 1998 ) and numerical simulations ( e.g. , 19 ) which typically predict a mass-independent \sigma _ { \ln R _ { e } } \approx 0.56 \pm 0.04 ( see 78 and 15 ) .

Higher resolution ( FWHM \ll 1 arcsec ) and deeper ( \mu _ { \mbox { \tiny \sc lim } } \gg 26 mag arcsec ^ { -2 } in the B -band ) observations of the local universe are now essential to probe to lower luminosity and lower surface brightness levels .