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

\begin{equation*} F( r) = \int_{0}^{2 \pi} d\phi \int_{0}^{r} dr r I(r, \phi) \ . \end{equation*}

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

\begin{equation*} \frac{ F( r_{ 1 / 2 } ) }{ F( \infty ) } = \frac{1}{2} \end{equation*}

for \(r_{1/2}\).

Gaussian Profile

Say the Gaussian intensity profile is given by

\begin{equation*} I( r) = I_0 \exp{\left(-\frac{r^2}{2\sigma^2}\right)} \ . \end{equation*}

The enclosed flux within \(r\) is given by

\begin{equation*} 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]\ . \end{equation*}

Hence

\begin{equation*} \boxed{ r_{1/2} = \sqrt{-2 \ln{0.5}} \sigma \ . } \end{equation*}

Exponential Profile

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

\begin{equation*} I( r) = I_d \exp{\left(-\frac{r}{r_d}\right)} \ , \end{equation*}

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

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

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

\begin{equation*} \left(1 + \frac{r_{1/2}}{r_d}\right) \exp{\left(-\frac{r_{1/2}}{r_d}\right)} = \frac{1}{2} \ . \end{equation*}

A relevant numerical solution is

\begin{equation*} \boxed{r_{1/2} = 1.67835 r_d \ .} \end{equation*}

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):

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

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