Jaffe profile

The Jaffe profile (or Jaffe model) is a mathematical function that is used to describe the distribution of mass or light in elliptical galaxies and the bulges of spiral galaxies. It was proposed by Walter Jaffe in 1983.[1] Its usefulness derives from the fact that it accurately reproduces a de Vaucouleurs profile when projected onto the sky.

The density in the Jaffe model is given by

${\displaystyle \rho (r)={\rho _{0} \over 4\pi }\left({r \over r_{0}}\right)^{-2}\left(1+{r \over r_{0}}\right)^{-2}.}$

In this equation, ${\displaystyle \rho _{0}}$ and ${\displaystyle r_{0}}$ are parameters that can be varied to fit the observed density.

Jaffe described how he arrived at his model:[1]

[The formula] was derived heuristically from the observation that the brightness profiles of spherical galaxies seem to run as ${\displaystyle r^{-3}}$ and ${\displaystyle r^{-1}}$ in at least some parts of their envelopes and cores, respectively. This would imply that the spatial density runs as ${\displaystyle r^{-4}}$ and ${\displaystyle r^{-2}}$.

Variations on Jaffe's law include the Hernquist profile, the Dehnen profile and the NFW profile, which have a similar functional form as Jaffe's law but which use different values for the two exponents.