Past exam of the astrophysics course of the University of Oxford / 2021 / 6 / c / Solution

At large radius $V_c\to V_0$, so the spherical enclosed mass is $M(<r)\simeq V_0^{2}r/G$. A spherical mass density satisfies $dM/dr=4\pi r^2\rho (r)$, and henceEquivalently, a local power law $\rho \propto r^{-p}$ with enclosed mass dominated by that extended profile has $M\propto r^{3-p}$ and $V_c^2\propto r^{2-p}$. A flat galaxy rotation curve requires $p=2$.

Differentiating the exact mass profile verifies the limit and the finite central density:As $x\to 0$, the bracket is $3x^2+O(x^3)$, so $\rho (0)=3V_0^2/(4\pi G{r}_0^2)$. As $x\to \infty$, it tends to one, recovering the $r^{-2}$ tail. Extending the flat galaxy rotation curve to infinity makes total mass diverge linearly; a real finite halo must eventually depart from this idealization.