D.1 Sersic Law

The properties of galaxy surface brightness radial profiles can be derived in a general form using Sersic law, first introduced by [Sersic 1968] and also known as $r^{1/n}$ law or generalized de Vaucouleurs law. This can be written as

$$\Sigma(r)=\Sigma_e \,\exp \left( -b_n \left[ \left( \frac{r}{r_e} \right)^{1/n} -1 \right] \right)~.$$ (D.1)

where $r_e$ is the effective radius, or the radius within which the galaxy emits half its brightness, $\Sigma_e$ is the surface brightness at $r_e$ and $b_n$ is a positive parameter that, for a given $n$, can be determined from the definition of $r_e$ and $\Sigma_e$. The value of $n$ determines the degree of concentration of the profile, quantified e.g. by the fraction of energy emitted within a given number of effective radii, the profile being steeper or less concentrated for higher $n$ and conversely flatter or less concentrated for lower $n$. Particularly interesting special cases are the bulge-like $r^{1/4}$ profile for $n=4$ and the disk-like exponential profile for $n=1$, which will be discussed in greater detail in Sections D.2 and D.3.

According to Equation D.1, the brightness integrated within a given radius $r$ is given by

$$\begin{split}F(r)&= \int_0^r 2\pi \,r' \,\Sigma(r') \,\mathrm... ...\, b_n\,\left(\frac{r}{r_e}\right)^{1/n} \right)\\ \end{split}$$ (D.2)

where $\gamma$ is the incomplete gamma function. The total brightness predicted by the profile is

$$\begin{split}F_{tot}&= \lim_{r \rightarrow \infty} F(r) = 2\p... ..._e^2 \,\Gamma(2n) \equiv k_n \,\Sigma_e \,r_e^2 \\ \end{split}$$ (D.3)

where $\Gamma$ is the gamma function. This relation, remembering that, by definition of effective radius, it is $F(r_e)=F_{tot}/2$, can be used to obtain an equation linking $b_n$ and $n$. After cancellation of common terms, one obtains

$$\Gamma(2n) - 2 \gamma(2n \, , \, b_n)= 0~,$$ (D.4)

a non-linear equation which can only be solved numerically, e.g. via the Newton Method (see Section 9.7 in [Press et al. 1996]). Values of $b_n$ and $k_n$ corresponding to integer values of $n$ from 1 to 10 are given in Table D.1.

Table D.1: Values of $b_n$ and $k_n$ for different values of $n$.

$n$	$b_n$		$k_n$
1	1	.	6783470	11.948495
2	3	.	6720608	16.310881
3	5	.	6701554	19.743758
4	7	.	6692495	22.665234
5	9	.	6687149	25.251949
6	11	.	668363	27.597728
7	13	.	667757	29.759676
8	15	.	667704	31.774676
9	17	.	667636	33.669429
10	19	.	667567	35.463170