= Inverse-angular-momentum velocity integration
{title2=$F(E,L)=L^{-1}f_E(E)$}
In a spherical <galactic distribution function> with $F=L^{-1}f_E(E)$, write $L=rv_T$ and $d^3v=dv_r\,v_Tdv_T\,d\psi$. 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 $1/2$.
Back to article page
