A..3 Disk Profile

For $n=1$, Equation 17 can be rewritten as the exponential law

$$\begin{split}\Sigma_d(r) & = \Sigma_e \,\exp\left[-1.6783 \le... ...ght) = \Sigma_0 \,\exp\left(-\frac{r}{r_s}\right)~, \end{split}$$ (26)

which characterizes the profile of disk components of disk galaxies, where $\Sigma_0$ is the central surface brightness and $r_s$ is referred to as the disk scale length. The relations between these two quantities and $\Sigma_e$ and $r_e$ are respectively

$$\Sigma_0 = 5.3567 \,\Sigma_e~,~~~~~~r_s=\frac{r_e}{1.6783}~.$$ (27)

According to this profile, the total brightness of the galaxy can be written as

$$F_{d,tot} = 11.948 \,\Sigma_e \,r_e^2~,$$ (28)

while the average surface brightness inside the effective radius $<\Sigma>_e$ is related to $\Sigma_e$ by

$$<\Sigma>_e=\frac{F_{d,tot}/2}{\pi\,r_e^2}=1.9016\,\Sigma_e~.$$ (29)

When put on a magnitude scale, the disk profile given by Equation 26 becomes

$$\begin{split}\mu_d(r)&=-2.5\,\log\left(\frac{\Sigma_d(r)}{\Si... ...frac{r}{r_e}-1\right) ~\mathrm{[mag~arcsec^{-2}]}~, \end{split}$$ (30)

while Equation 28 can be trivially manipulated to obtain for $\Sigma_e$ the expression

$$\begin{split}\mu_e&=-2.5\,\log\left(\frac{\Sigma_e}{\Sigma_{z... ... +I_{\mathrm{[mag]}}~~~\mathrm{[mag~arcsec^{-2}]}~. \end{split}$$ (31)