A..2 Bulge Profile

For $n=4$, Equation 17 becomes de Vaucouleurs, or $r^{1/4}$, law

$$\Sigma_b(r) = \Sigma_e \,\exp\left(-7.6692 \left[\left(\frac{r}{r_e}\right) ^{1/4}-1\right] \right)~,$$ (21)

which characterizes the profiles of elliptical galaxies and bulge components of disk galaxies. According to this profile, the total brightness can be written as

$$F_{b,tot}=22.665\,\Sigma_e\,r_e^2~,$$ (22)

while the central surface brightness $\Sigma_0$ and the average surface brightness inside the effective radius $<\Sigma>_e$ are related to $\Sigma_e$ by

$$\Sigma_0=2141.4\,\Sigma_e~,~~~~~~~ <\Sigma>_e=\frac{F_{b,tot}/2}{\pi\,r_e^2}=3.6072\,\Sigma_e~.$$ (23)

The bulge profile given by Equation 21 can be put on the more common logarithmic scale writing

$$\begin{split}\mu_b(r)&=-2.5\,\log\left(\frac{\Sigma_b(r)}{\Si... ...right)^{1/4}-1\right] ~\mathrm{[mag~arcsec^{-2}]}~, \end{split}$$ (24)

where $\Sigma_{zp}$ is the zero-point of the adopted surface brightness magnitude scale. Similarly, Equation 22 can be manipulated so as to express $\Sigma_e$ as function of $r_e$ and $I$, obtaining

$$\begin{split}\mu_e&=-2.5\,\log\left(\frac{\Sigma_e}{\Sigma_{z... ... +I_{\mathrm{[mag]}}~~~\mathrm{[mag~arcsec^{-2}]}~, \end{split}$$ (25)

where $F_{zp}$ is the zero-point of the adopted brightness magnitude scale.