= 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 function>
$$
f(\mathcal E,L)=f_0L^{-2\beta}\mathcal E^n,\qquad \mathcal E>0,
$$
with $f=0$ for $\mathcal E\le0$ and $f_0>0$. <Velocity> integration gives the <mass density>
$$
\rho=A r^{-2\beta}\Psi^{n+3/2-\beta},\qquad
A=2^{3/2-\beta}\pi^{3/2}f_0\frac{\Gamma(1-\beta)\Gamma(n+1)}{\Gamma(n+5/2-\beta)}.
$$
For 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.<Hypervirial Models of Stellar Systems by N. W. Evans and J. An>[Section 2.2, equations (18)–(19); their energy exponent n−3/2 is renamed n here]{ref}
To derive the coefficient, write the velocity-volume element as $2\pi v^2\sin\vartheta\,dv\,d\vartheta$. The angular integral is $\sqrt\pi\Gamma(1-\beta)/\Gamma(3/2-\beta)$. In the radial integral set $x=v^2/(2\Psi)$, giving
$$
\int_0^{\sqrt{2\Psi}}v^{2-2\beta}(\Psi-v^2/2)^n\,dv
=2^{1/2-\beta}\Psi^{n+3/2-\beta}\frac{\Gamma(3/2-\beta)\Gamma(n+1)}{\Gamma(n+5/2-\beta)}.
$$
Multiplication 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>.
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