# Jeans equation for a fully isotropic velocity dispersion tensor

Given a known spherically symmetric gravitational potential, $$\Phi(r)$$, I need to calculate the value of the velocity dispersion tensor $$\sigma_{rr}$$. To calculate $$\sigma_{rr}$$ I am using Jean's equation for a spherically symmetric system.

$$\frac{1}{v}\frac{\partial}{\partial r}(v\sigma_{rr}^{2})+2\frac{(\sigma_{rr}^{2}-\sigma_{\theta\theta}^{{2}})}{r}=-\frac{\partial\Phi}{\partial r}$$

I am also told that I can assume a fully isotropic velocity dispersion tensor, which I interpret to mean that $$\sigma_{rr}^{2}=\sigma_{\theta\theta}^{2}$$. Given this fact, I arrive at the following equation.

$$\frac{\partial v}{\partial r}\sigma_{rr}^{2} + \frac{\partial \sigma_{rr}^{2} }{\partial r}v=-v\frac{\partial\Phi}{\partial r}$$

From here I am not sure how to solve for $$\sigma_{rr}^{2}$$. Is $$\sigma_{rr}^{2}$$ constant in the r direction? If not, how do I solve for $$\sigma_{rr}^{2}$$?