= Solution
A useful empirical profile is the Sérsic law,
$$
I(R)=I_e\exp\left\{-b_n\left[\left(\frac R{R_e}\right)^{1/n}-1\right]\right\},
$$
where $b_n$ is chosen so that $R_e$ encloses half the light. Classical bright <elliptical galaxies> are often approximated by the <de Vaucouleurs's law> case $n=4$, with $b_4\simeq7.67$. Real ellipticals span a range of Sérsic indices, so $R^{1/4}$ is not universal.
Their classical luminosity-dispersion scaling is the <Faber-Jackson relation>,
$$
\boxed{L\propto\sigma^4}
$$
as an approximate empirical trend with intrinsic scatter and sample-dependent slope. Combining size, <surface brightness> and dispersion gives the more informative <fundamental plane of elliptical galaxies>.
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