4.1 Elliptical Galaxies

The surface brightness radial profiles of elliptical galaxies are in general reasonably well described by de Vaucouleurs, or $r^{1/4}$, law, first introduced by [de Vaucouleurs(1948)]

$$\Sigma_E(r) = \Sigma_{E,e} \,\exp\left(-7.6692 \left[\left(\frac{r}{r_e}\right) ^{1/4}-1\right] \right)~,$$ (6)

where the effective surface brightness is labelled with an additional ``$E$'' because in our model this quantity, unlike the effective radius, will in general be different for E and D galaxies. This law has succeeded in reproducing, with a remarkable accuracy, the profiles of quite a few E galaxies. For instance, [Capaccioli et al.(1990)] found that the $r^{1/4}$ fit of the surface brightness radial profile of the nearby standard elliptical NGC 3379 give residuals smaller than 0.08 mag over a 10 magnitude range. [Makino et al.(1990)], however, found from dynamical arguments that the $r^{1/4}$ law bore little physical significance, though it is the best-fitting function, and that $r^{1/n}$ laws with $n$ in the range 3-10 gave almost as good fits for a range of $r$ of about 100. More recently, [Caon et al.(1993)] showed that the best-fitting $n$ correlates with the galaxy linear effective radius and luminosity, while [Andredakis et al.(1995)] found that the light profiles of the bulges of disk galaxies, which are also usually modelled with an $r^{1/4}$ law, are in fact best-fitted by $r^{1/n}$ profiles with an $n$ correlating with the galaxy morphological type. Nevertheless, the empirical fitting function given by Equation 6 is useful for characterizing the global properties of galaxies, and by that token in this study elliptical galaxies and bulges of disk galaxies will both be modelled with $r^{1/4}$ laws.

On a magnitude scale, Equation 6 becomes

$$\mu_E(r)=\mu_{E,e}+8.3268\,\left[\left(\frac{r}{r_e}\right)^{1/4}-1\right]~,$$ (7)

where $\mu_{E,e}$ is the effective surface brightness of Es expressed in mag/arcsec$^2$. This latter quantity can be expressed as function of $r_e$ and $I$, and thus, via Equation 3, of $I$ only, obtaining

$$\mu_{E,e}=2.5\,\log\left(22.665\right)+5\,\log\left(r_{e,\mathrm{[as]}}(I) \right)+I_{\mathrm{[mag]}}~~~\mathrm{[number~deg^{-2}~mag^{-1}]}~.$$ (8)

Values of $\mu_{E,e}$ are given in Table 5.