Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 66 / 4 / ii / Solution

Use $M$ for the mass parameter throughout; the PDF uses $m$ in its formulas and $M$ in the subsequent prose for the same quantity. Set $a=r_j>0$. The Jaffe model density simplifies toIts enclosed mass isIt tends to $M$ as $r\to \infty$, verifying the total mass. It tends to zero at the center, so the cusp carries no extra point mass.

Spherical symmetry gives ${\Phi }^{\prime }(r)=GM(r)/r^2=GM/[r(r+a)]$. Integrating with $\Phi (\infty )=0$ givesThis also directly satisfies the Poisson equation for Newtonian gravity: differentiating $r^2{\Phi }^{\prime }=GMr/(r+a)$ gives $4\pi G{r}^2\rho (r)$. The circular speed isFor $r\ll a$, $v_c=\sqrt{GM/a}[1-r/(2a)+O((r/a{)}^2)]$ is approximately constant. For $r\gg a$, $v_c=\sqrt{GM/r}[1-a/(2r)+O((a/r{)}^2)]\propto r^{-1/2}$. Thus the finite-mass model has an inner nearly flat galaxy rotation curve and an outer Keplerian decline.