Surface photometry of galaxies (see e.g. [Jedrzejewski 1987] for E galaxies and [Kent 1985] for D galaxies) is usually analysed by fitting ellipses to the isophotes and by plotting their surface brightness versus their radius, which is defined as the geometric mean of the ellipse's semi-axes and , i.e. . The resulting plot is then called the surface brightness radial profile of the galaxy. In this context, the effective radius of the galaxy is defined as the radius of the isophote encircling half of the light emitted by the galaxy, also called the effective isophote. The effective radius and the effective surface brightness, the latter being the surface brightness of the effective isophote, are usually indicated with and , respectively.
Until the launch of HST, accurate measurements of the small angular sizes of faint galaxies were made virtually impossible by the phenomenon of seeing. The Medium Deep Survey ([Ratnatunga et al. 1999]), the first survey project to be carried out with HST superb instrumentation, has recently brought to an end this long-standing lack of meaningful data, while [Im et al. 1995] have demonstrated the potential of angular size measurements to discriminate between currently competing cosmological models.
[Casertano et al. 1995] have obtained effective radii for about 10,000 galaxies from Wide Field and Planetary Camera (WF/PC) parallel observations of random fields in the band. As shown in their Figure 6, the observed angular size distribution as function of magnitude shows a large scatter about the median value, mainly due to the intrinsic scatter in linear size and redshift distribution. The same figure also shows that the observed relation between the median effective radius and magnitude is well-fit by the theoretically predicted relation for galaxies of constant central surface brightness mag/arcsec and absolute magnitude in the context of a mild luminosity evolution scenario. This latter relation asymptotically approaches the linear relation in vs. that is measured in local samples of bright spiral galaxies following Freeman's law, and was therefore taken as a description of the relation between the galaxy effective radius and magnitude in our model. Least-square polynomial fit showed that its accurate description required a fourth-degree polynomial, which is represented in Figure 4.3, together with the Euclidean extrapolation to faint magnitudes of the local linear relation.