= Half-anisotropic distribution function
{title2=$f(E,L)=f_E(E)/L,\quad\beta=1/2$}
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 $d(r\rho)/d\psi\geq0$.
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