 Solution

A useful empirical profile is the Sérsic law,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 $b_4\simeq 7.67$. 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,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.