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

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


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

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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7 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.

      • Knusper says:

        Well, I think nothing is swept under the rug… You arrive at those constants by solving the integral for the Profile. At one step I guess the integral has to be evaluated numerically. So if you count numerical integration “approximation” … well, than it is an approximation..

        • Taro says:

          Thanks for the reference. I think what I meant was the limitation of fitting a fairly generic parametric model to galaxy morphology, not the estimation of the parameter itself.

  2. Knusper says:

    Here is a reference where they derive it — it can almost be done analytically until the last step:


  3. advsci says:

    Hey Taro, if I fit an exponential function to a profile and get a value and standard deviation for parameter r_d, then r_h = 1.67835*r_d, but does the same formula apply to the error as well?

    So does the standard deviation become sd_h = 1.67835*sd_d?

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