Sérsic profile

Remarks

The total luminosity is $$L = \int_0^\infty 2\pi R \, I(R) \, dR = 2 \pi n \, I_e R_e^2 \, \frac{e^{b_n}}{b_n^{2n}} \, \Gamma(2n),$$ by the substitution $u = b_n (R/R_e)^{1/n}$. The same substitution shows that the light inside $R_e$ is $L \, \gamma(2n, b_n)/\Gamma(2n)$, so the choice of $b_n$ makes $R_e$ the radius that contains half of the light, and $I_e = I(R_e)$. A good approximation is $b_n \approx 2n - 1/3$ for $n$ not too small. In magnitudes per unit area, the profile is a straight line in $R^{1/n}$.

Example: for $n = 1$ the profile is the exponential disk $I(R) = I_0 e^{-R/h}$ with $I_0 = I_e e^{b_1}$ and $h = R_e / b_1$. The condition is $\gamma(2, b_1) = 1 - (1 + b_1) e^{-b_1} = \frac{1}{2}$, which gives $b_1 \approx 1.678$.

Non-example: a power law $I(R) = I_e (R/R_e)^{-2}$ is not a Sérsic profile for any $n$, since its integral $\int_0^\infty 2\pi R \, I(R) \, dR$ diverges, while every Sérsic profile has finite total light.