Scale-free density-potential component

Let $\Psi (r)>0$ be the relative potential of a spherical stellar system, $\mathcal{E}=\Psi -v^2/2$ the specific binding energy, and $L=rv\sin \vartheta$ the specific angular momentum. Consider the galactic distribution functionwith $f=0$ for $\mathcal{E}\le 0$ and $f_0>0$. Velocity integration gives the mass densityFor this ordinary locally integrable distribution, the conditions are $\beta <1$ and $n>-1$. They ensure convergence at purely radial directions and at the escape-energy boundary, respectively.^[ref]

To derive the coefficient, write the velocity-volume element as $2\pi v^2\sin \vartheta dvd\vartheta$. The angular integral is $\sqrt{\pi }\Gamma (1-\beta )/\Gamma (3/2-\beta )$. In the radial integral set $x=v^2/(2\Psi )$, givingMultiplication proves the result using the gamma function normalization.

The angular moments give velocity-anisotropy parameter $\beta$, explaining the name constant-anisotropy distribution function. As a check, $\beta =0$, $n=7/2$ gives an isotropic distribution with $\rho \propto {\Psi }^5$, as in the Plummer model. The formula converts a phase-space ansatz into a density-potential relation; a self-consistent stellar model must additionally solve the Poisson equation for Newtonian gravity.