 Jeans moments from an augmented density

With radial pressure zero at $\psi =0$, an augmented density supplies $\rho ⟨v_r^2⟩={\int }_0^{\psi }\rho (r,q)dq$ and $\rho ⟨v_{\theta }^2⟩=(2r{)}^{-1}{\int }_0^{\psi }{\partial }_r[r^2\rho (r,q)]dq$. The derivative holds $q$ fixed. Their chain-rule combination satisfies the Spherical Jeans equation, but the physical density alone does not choose an augmented extension.