For a radial profile of , the enclosed flux within the radius is given by

I’m only concerned about azimuthal symmetric cases, so . The idea is to solve

for .

### Gaussian Profile

Say the Gaussian intensity profile is given by

The enclosed flux within is given by

Hence

### Exponential Profile

The radial surface-brightness profile of a disk galaxy is often fitted by the exponential profile:

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

Solving for the half-light radius leads to the following:

A relevant numerical solution is

### Sersic/de Vaucouleurs Profile

The radial surface-brightness profile of a spheroidal galaxy is often fitted by the Sersic (de Vaucouleurs when ) profile:

where (Capaccioli 1989). This expression explicitly parametrizes the profile in terms of half-light radius.

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

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.

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..

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.

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

http://adsabs.harvard.edu/abs/2005PASA…22..118G

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?

advsci — You’d have to use the standard uncertainty propagation formula: https://en.wikipedia.org/wiki/Propagation_of_uncertainty. I don’t remember off the top of my head how the computation comes out…