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D.3 Disk Profile

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

\begin{displaymath}\begin{split}\Sigma_d(r) & = \Sigma_e \,\exp\left[-1.6783 \le...
...ght) = \Sigma_0 \,\exp\left(-\frac{r}{r_s}\right)~, \end{split}\end{displaymath} (D.10)

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

$\displaystyle \Sigma_0=5.3567\,\Sigma_e~,~~~~~~r_s=\frac{r_e}{1.6783}~.$ (D.11)

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

$\displaystyle F_{d,tot} = 11.948 \,\Sigma_e \,r_e^2~.$ (D.12)

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

$\displaystyle <\Sigma>_e=\frac{F_{d,tot}/2}{\pi\,r_e^2}=1.9016\,\Sigma_e~.$ (D.13)

When put on a magnitude scale, the disk profile given by Equation D.10 becomes

\begin{displaymath}\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}\end{displaymath} (D.14)

while equalling Equations A.4 and D.12, one obtain for $ \Sigma_e$ the expression

\begin{displaymath}\begin{split}\mu_e&=-2.5\,\log\left(\frac{\Sigma_e}{\Sigma_{z...
... +I_{\mathrm{[mag]}}~~~\mathrm{[mag~arcsec^{-2}]}~, \end{split}\end{displaymath} (D.15)


next up previous contents
Next: D.4 Bulge+Disk Profile Up: D. Galaxy Surface Brightness Previous: D.2 Bulge Profile   Contents
Mattia Vaccari 2000-12-05