Past exam of the astrophysics course of the University of Oxford / 2015 / 7 / b / 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 givesThe 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.