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

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 $M$.