Osipkov-Merritt distribution function ($Q=E-L^2/(2r_a^2)$)

An Osipkov-Merritt spherical distribution has $F=f(Q)$ for $Q=E-L^2/(2r_a^2)>0$ and vanishes otherwise. Rescaling transverse velocities by $\sqrt{1+r^2/r_a^2}$ makes its velocity integral isotropic in the rescaled variables. Its velocity-anisotropy parameter is $\beta =r^2/(r^2+r_a^2)$.

In unit Plummer model scales, $F=3Q^{3/2}/(\sqrt{2}{\pi }^3)$ for $Q=E-L^2/2>0$ gives $\rho =3{\psi }^3/[4\pi (1+r^2)]=3{\psi }^5/(4\pi )$. It reproduces the isotropic Plummer density but has anisotropy $r^2/(1+r^2)$ and different second velocity moments.