Biboroku

Half-Light Radii for Various Profiles

Written by Taro Sato, on . Tagged: astro math

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

F(r)=02πdϕ0rdrrI(r,ϕ) .

I’m only concerned about azimuthal symmetric cases, so F(r)=2π0rdrrI(r) . The idea is to solve

F(r1/2)F()=12

for r1/2.

Gaussian Profile

Say the Gaussian intensity profile is given by

I(r)=I0exp(r22σ2) .

The enclosed flux within r is given by

F(r)=2πI00rdrrexp(r22σ2)=2πI0σ2[1exp(r22σ2)] .

Hence

r1/2=2ln0.5σ .

Exponential Profile

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

I(r)=Idexp(rrd) ,

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

F(r)=2πId0rdrrexp(rrd) .

Solving for the half-light radius r1/2 leads to the following:

(1+r1/2rd)exp(r1/2rd)=12 .

A relevant numerical solution is

r1/2=1.67835rd .

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)=I1/2exp(k((rr1/2)1/n1)) .

where k=1.992n0.3271 (Capaccioli 1989). This expression explicitly parametrizes the profile in terms of half-light radius.