of the velocity dispersion is the same as the scale length for the surface density, in accord with observations [@bottema1993]. The azimuthal velocity dispersion profile is set by the epicycle approximation (see @kuijken1995 and Section 3.2.3 of @binney2008) while the vertical velocity dispersion is determined from the local surface density and the model assumption that the disc has a constant scale height.

The DFs for the bulge and halo are functions solely of the energy and are therefore isotropic in velocity space. The bulge DF is constructed to yield a density profile that is given, approximately, by

$$\label{eq:bulgedensity}
\rho_b\left (r\right ) = \frac{v_b^2}{4\pi G R_e^2 c(n)}
\left (\frac{r}{R_e}\right )^{-p}e^{-b\left (r/R_e\right )^{1/n}}~,$$

which yields the Sérsic law for the projected surface density profile with index $n$ so long as one sets $p=1-0.6097/n +
0.05563/n^2$. The constant $b$ is adjusted so that $R_e$ encloses half the total projected mass and $c(n) = \left (nb^{n\left
  (p-2\right)}\right )\Gamma\left (n\left (2-p\right )\right )$ [@prugniel1997; @terzic2005].

The halo DF is constructed to yield the NFW profile [@navarro1996]:

$$\label{eq:halodensity}
\rho_h(r) = \frac{a_h v_h^2}{4\pi G}\frac{1}{r\left (r + a_h\right
  )^2}.$$

While the velocity distributions of the halo and bulge are isotropic, their space densities are slightly flattened due to 