 Solution

For a steady spherical, nonrotating, bounded collisionless galaxy, the mean velocity vanishes, the mixed velocity moments vanish, and ${\sigma }_{\theta \theta }^2={\sigma }_{\varphi \varphi }^2\equiv {\sigma }_t^2$. A steady radial stellar flux would satisfy $r^2\nu {\overline{v}}_r=\mathrm{const}$; regularity and no central source/sink set that constant to zero in the equilibrium model. The Spherical Jeans equation is thereforeHere $\nu$ is the tracer number density, not necessarily the gravitating mass density. Random motions supply the velocity stress balancing the gravitational field.

Two conventions in the printed general equation need correction before interpreting it outside this special limit. Its first standalone derivative of mean radial velocity should be a time derivative, $\nu {\partial }_t{\overline{v}}_r$, not $\nu {\partial }_r{\overline{v}}_r$: the latter has incorrect acceleration dimensions. Also a velocity dispersion is the centered moment $\overline{v_i{v}_j}-{\overline{v}}_i{\overline{v}}_j$. The printed raw-moment definition equals it only when the mean velocities vanish, as they do in the equilibrium used here. These issues do not change the simplified result above.