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

Put $V_0=200\mathrm{km}{\mathrm{s}}^{-1}$. A Taylor expansion near the centre givesTherefore the circular angular frequency isThus the central motion approaches solid-body rotation. Numerically ${\Omega }_0=1.6204\times 10^{-15}{\mathrm{s}}^{-1}$, and the central orbital period tends toThe finite limit of $\Omega$ accompanies $V_c\to 0$ at $r\to 0$; it does not imply a finite central linear speed.