= Solution
For spherical gravity, $d\Phi/dr=GM(\le r)/r^2$. Inserting $\beta=1-\sigma_t^2/\sigma_r^2$ into the <Spherical Jeans equation> and solving for the total enclosed <mass> gives
$$
\boxed{M(\le r)=-\frac{r^2}{G\nu}\left[\frac{d(\nu\sigma_r^2)}{dr}+\frac{2\beta\nu\sigma_r^2}{r}\right]
=-\frac{r\sigma_r^2}{G}\left(\frac{d\ln\nu}{d\ln r}+\frac{d\ln\sigma_r^2}{d\ln r}+2\beta\right)}.
$$
The <mass> includes all sources of the potential, including <dark matter> or a central compact object; the stellar tracer density need not follow total <mass>. The bracket must yield nonnegative enclosed <mass> in a physically admissible model. Density, radial stress and anisotropy are independent observational/model inputs before this equation becomes a <mass> estimate.
Back to article page
