= Escape speed of a truncated flat-rotation halo
{title2=$v_{\rm esc}^2=2v_c^2[1+\log(r_*/r)]$}
A spherical halo with flat circular <speed> $v_c$ has $M(r)=v_c^2r/G$ and $\rho=v_c^2/(4\pi Gr^2)$. If it is sharply truncated at $r_*$, the work per unit mass from $r<r_*$ to the boundary is $v_c^2\log(r_*/r)$ and from the boundary to infinity is $v_c^2$. Adding them gives the displayed <escape velocity>. A bound star's observed radial speed is only a lower bound on its total speed and on the escape speed, so one maximum radial measurement generally gives a lower bound on $r_*$ rather than an exact halo edge.
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