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 $\sigma_{rr}^{2}$?


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