# Half-Light Radii for Various Profiles

For a radial profile of $I(r)$, the enclosed flux within the radius $r$ is given by

$F(r) = \int_{0}^{2\pi} d\phi \int_{0}^{r} dr r I(r,\phi) \ ,$

I’m only concerned about azimuthal symmetric cases, so $F(r) = 2\pi \int_{0}^{r} dr r I(r)$.  The idea is to solve

$\frac{F(r_{1/2})}{F(\infty)} = \frac{1}{2}$

for $r_{1/2}$.

### Gaussian Profile

Say the Gaussian intensity profile is given by

$I(r) = I_0 \exp{\left(-\frac{r^2}{2\sigma^2}\right)} \ .$

The enclosed flux within $r$ is given by

$F(r) = 2\pi I_0 \int_0^r dr r \exp{\left(-\frac{r^2}{2\sigma^2}\right)} = 2\pi I_0 \sigma^2 \left[1 - \exp{\left(-\frac{r^2}{2\sigma^2}\right)}\right]\ .$

Hence

$\boxed{r_{1/2} = \sqrt{-2 \ln{0.5}} \sigma \ .}$

### Exponential Profile

The radial surface-brightness profile $I(r)$ of a disk galaxy is often fitted by the exponential profile:

$I(r) = I_d \exp{\left(-\frac{r}{r_d}\right)} \ ,$

where $r_d$ is the disk scale length. The flux enclosed within the radius $r$ is given by

$F(r) = 2\pi I_d \int_0^r dr r \exp{\left(-\frac{r}{r_d}\right)} \ .$

Solving for the half-light radius $r_{1/2}$ leads to the following:

$\left(1 + \frac{r_{1/2}}{r_d}\right) \exp{\left(-\frac{r_{1/2}}{r_d}\right)} = \frac{1}{2} \ .$

A relevant numerical solution is

$\boxed{r_{1/2} = 1.67835 r_d \ .}$

### Sersic/de Vaucouleurs Profile

The radial surface-brightness profile $I(r)$ of a spheroidal galaxy is often fitted by the Sersic (de Vaucouleurs when $n = 4$) profile:

$I(r) = I_{1/2} \exp{\left( -k \left( \left(\frac{r}{r_{1/2}}\right)^{1/n} - 1 \right) \right)} \ .$

where $k = 1.992 n - 0.3271$ (Capaccioli 1989). This expression explicitly parametrizes the profile in terms of half-light radius.

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### 2 Responses to Half-Light Radii for Various Profiles

1. Is that formula 1.992 n − 0.3271 an approximation? It must be…

• Taro says:

David — Yes it must be. It’s a parametrization also used in GIM2D by Luc Simard. It’s probably just an empirical calibration to fit a wide variety of morphology, while sweeping all the (important) details under the rug.

By the way your (mini)blog is the best one by astrophysicists!! I enjoy reading about your ambitious research endeavors.