Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2012 / S26 / 3 / 5

For a spherical mass distribution with constant circular speed $v_c$,For $v_c=220\mathrm{km}{\mathrm{s}}^{-1}$, this is about $0.0140(r/8\mathrm{kpc}{)}^{-2}{M}_{\odot }\mathrm{p}{\mathrm{c}}^{-3}$. It is the spherical model's total density; it is singular if extrapolated to the very centre and is intended only over the flat-rotation region.

For a sharp outer truncation at $r_{\ast }$, the exterior mass is fixed at $M_{\ast }=v_c^2{r}_{\ast }/G$. Integrating the gravitational work from $r$ to the boundary and then to infinity givesThusThe exterior contribution cannot be dropped: at $r=r_{\ast }$ it correctly gives the point-mass escape speed $\sqrt{2}{v}_c$.

If the fastest measured radial speed is assumed to approximate both the total speed and the local escape speed of a bound star, then at $r=8\mathrm{kpc}$,Without those extra assumptions, the measurement gives only a bound: a bound star's total speed is at least its radial component, and the sample maximum need not reach escape, so $r_{\ast }\gtrsim 19.9\mathrm{kpc}$ within this model. An unbound interloper would invalidate even that bound. This is not a determination of the real Milky Way halo edge.