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

For a spherical mass distribution, the shell theorem says the circular gravitational acceleration is $GM(<R)/R^2$. Balancing it against the centripetal acceleration $V_c^2/R$ givesAt $R=16\mathrm{kpc}=4r_0$, the galaxy rotation curve gives $V_c=200(1-e^{-4})=196.337\mathrm{km}{\mathrm{s}}^{-1}$. With $R=4.93708\times 10^{20}\mathrm{m}$,This is the total dynamically inferred enclosed mass, including any dark matter; the galaxy rotation curve alone does not separate its components.