Circular and isotropic speed comparison in a singular isothermal sphere
= Circular and isotropic speed comparison in a singular isothermal sphere
{title2=$\langle v^2\rangle_{\mathrm{iso}}=3v_0^2/2$}
For the $r^{-2}$ <singular isothermal sphere>, the isotropic <Spherical Jeans equation> gives $\sigma_r^2=\sigma_\theta^2=\sigma_\phi^2=v_0^2/2$, so $\langle v^2\rangle=3v_0^2/2$. Randomly oriented circular orbits instead have $\sigma_r^2=0$, each transverse second moment $v_0^2/2$, and total mean square speed $v_0^2$.