Constant-anisotropy distribution function

A spherical constant-anisotropy distribution function hasIts velocity moments have the radius-independent velocity-anisotropy parameter $\beta$.

In a spherical galactic distribution function with $F=L^{-1}{f}_E(E)$, write $L=r{v}_T$ and $d^{3}v=d{v}_r{v}_{T}d{v}_{T}d\psi$. The velocity Jacobian cancels the apparent singularity at $v_T=0$. Integration of Gaussian energy components then reduces to one full-line and one half-line Gaussian integral. The angular average makes each tangential velocity dispersion half the radial one, so the velocity-anisotropy parameter is $1/2$.

For $f(E,L)=f_E(E)/L$, with $L=rv\sin \eta$ and $E=\psi -v^2/2$, velocity integration gives $r\rho =2{\pi }^2{\int }_0^{\psi }{f}_E(E)dE$. Thus $f_E(E)=(2{\pi }^2{)}^{-1}d(r\rho )/d\psi$ at $\psi =E$. The factor $1/L$ is locally integrable under the velocity-volume measure. Its angular measure is uniform in $\eta$; averages of ${\cos }^2\eta$ and ${\sin }^2\eta$ are both $1/2$, so the total tangential second moment equals the radial second moment and the velocity-anisotropy parameter is $\beta =1/2$. At the largest allowed radius $\psi (r_E)=E$, the derivative becomes $[\rho +r(d\rho /dr)]/(d\psi /dr)$, not $[\rho +d\rho /d\psi ]/(d\psi /dr)$. Positivity requires $d(r\rho )/d\psi \ge 0$.

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.