Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 349 / 1

With the convention in the question, the constant velocity-anisotropy parameter is $\beta =1-{\sigma }_t^2/{\sigma }_r^2$. Since $V_c^2=rd\Phi /dr$, the Spherical Jeans equation becomesFor $\nu ={\nu }_0{r}^{-\alpha }$ and $V_c^2=V_0^2{r}^{2\gamma }$, its general integrating factor solution isThe homogeneous term represents a boundary pressure. For an extended scale-free system the physical boundary condition normally removes it, leavingPositivity requires $\alpha -2\beta -2\gamma >0$. A complete physical model must also have a nonnegative galactic distribution function and sensible inner and outer boundary behaviour; for example, strong radial anisotropy is restricted by density-slope--anisotropy inequalities.

Observationally, the tracer density can be estimated from star counts only after correcting distances, extinction, survey selection, and incompleteness. Spectroscopy supplies mainly line-of-sight velocities; proper motions add transverse information but become less precise for distant halo stars. The equation shows the mass--anisotropy--density degeneracy directly: the same measured ${\sigma }_r$ can result from a larger $V_c$, a steeper tracer slope $\alpha$, or a different $\beta$. Even globally constant power laws therefore do not determine the galactic mass profile unless some of these quantities are independently constrained.

If $\alpha$, $\beta$, or $\gamma$ changes near a break radius, the solution at one radius also depends on the outer boundary integral. A break in observed dispersion may be attributed to a mass-profile feature, a tracer-density break, or a change in orbital anisotropy. Separate tracer populations, full three-dimensional velocities, higher velocity moments, and measurements over a wide radial range help break this degeneracy.

More flexible alternatives model a nonnegative solution of the Collisionless Boltzmann equation itself. An action-based galactic distribution function gives an analytic or parametrized $f(\mathbf{J})$; a Schwarzschild orbit-superposition model assigns nonnegative weights to an orbit library; and a made-to-measure stellar-dynamical model adjusts particle weights to reproduce observations. These methods retain more phase-space information than Jeans moments, although their flexibility introduces model choices and regularization.