Point mass model

If we assume there is no dark halo, then it is easy to find a relation between the circular velocity $V_c$ and the escape speed $v_e$. If we assume all mass $M$ of the MW is interior to the solar radius $R_\odot$ ^7.3, then $V_c^2=GM/R_\odot$, and the potential $\Phi=-GM/R_\odot=-V_c^2$. The velocity of a star just able to escape from the gravitational well is the escape speed, $v_e$. So since that star has zero total energy,

$$0=E={1\over 2}v_e^2+\Phi={1\over 2} v_e^2-V_c^2 .$$ (7.1)

So in this case, $v_e=2^{1/2} V_c\approx 311{\hbox{\rm km s$^{-1}$}}$ (Using $V_c=220{\hbox{\rm km s$^{-1}$}}$.) So for a point mass model of the MW, these high velocity stars cannot be bound to the MW - suggestingthe presence of more mass in the outskirts of the MW than expected, given the rapid decrease in the distribution of light.