# What is a “scale length”, and how do I calculate it for galaxies?

I'm trying to work through, understand, and apply concepts regarding mass models of galaxies. Looking at the Hernquist model, I'm finding the equation $$Φ(r)=−\frac{GM}{r+a},$$ where a is the scale length. I've tried looking up what "scale length" is, and I get confusing answers. More details about relevant mass models linked above.

A "scale length" (or "scale" anything) is a "characteristic" length for the particular physical scenario at hand. Of course, that's another unhelpful term for the same thing, but what it means is: the typical length scale of interest for whatever is being considered. In the case of the Hernquist model, usually applied to stellar density profiles, $a$ is the typical radius of the stellar core. For $r \ll a$ the potential is roughly constant, which translates to a density $\rho \propto 1/r$. For $r \gg a$, the potential falls off inversely with distance, or the density $\rho \propto 1/r^4$. The radius $a$ is roughly the transition between the two, and I say roughly because there isn't an 'exact' transition point---it's a smooth transition, as are most situations with "characteristic" quantities.
For stellar distributions in galaxies, $a$ varies by a few orders of magnitude from system to system, but is usually on the order of a kiloparsec or so.