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A..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

$\displaystyle \Sigma(r)=\Sigma_e \,\exp \left( -b_n \left[ \left( \frac{r}{r_e} \right)^{1/n} -1 \right] \right)~,$ (17)

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 A.2 and A.3.

According to Equation 17, the brightness integrated within a given radius $ r$ is given by

\begin{displaymath}\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}\end{displaymath} (18)

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

\begin{displaymath}\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}\end{displaymath} (19)

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

$\displaystyle \Gamma(2n) - 2 \gamma(2n \, , \, b_n)= 0~,$ (20)

a non-linear equation which can only be solved numerically, e.g. via the Newton method (see Section 9.7 in [Press et al.(1992)]). Values of $ b_n$ and $ k_n$ corresponding to integer values of $ n$ from 1 to 10 are given in Table 6.

Table 6: 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


next up previous
Next: A..2 Bulge Profile Up: A. Galaxy Surface Brightness Previous: A. Galaxy Surface Brightness
Mattia Vaccari 2002-01-31