Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 320 / 2

SetThe spherical Poisson equation givesThe numerator identitysplits this into two scale-free density-potential components:Thus

At small radius, $D\sim 2b$ and the first term dominates:The cusp has finite enclosed mass because $r^2\rho \sim r^{1/2}$. Moreover,so the circular speed tends to zero as $r^{1/4}$. At large radius, $D\sim r$ andSince $r^2\rho \sim r^{-3/2}$ is integrable at infinity and $\Phi \sim -GM/r$, the total mass is finite and equals $M$.

Use the relative energy $\mathcal{E}=-E=\Psi -v^2/2$. At fixed $r$, write velocity-space spherical coordinates with polar angle $\alpha$ from the radial direction, so $L=rv\sin \alpha$. ThenThe beta function integrals givewhereThus $\gamma =2\beta$ and $p=n+3/2-\beta$, with convergence for $\beta <1$ and $n>-1$.

Comparison with part a givesHence the model has the constant-anisotropy distribution functionwhere matching the two density coefficients gives

For one component with density ${\rho }_k=A_k{r}^{-2{\beta }_k}{\Psi }^{p_k}$, direct velocity integration, or the Spherical Jeans equation, givesEach component has velocity-anisotropy parameter ${\beta }_k$. Therefore the combined coefficient is the radial-pressure-weighted meanHeresoSince $q\sim \sqrt{r/b}/2$ at the origin and $q\sim \sqrt{b/r}$ at infinity,The second, less radial component reduces the anisotropy only at intermediate radii.