NFW mass and circular-speed profile ($M(r)=4\pi {\rho }_s{r}_s^3[\log (1+x)-x/(1+x)]$)

For an NFW profile, $x=r/r_s$ and $v_c^2=GM(r)/r$. The circular speed rises as $r^{1/2}$ at small radius and eventually decreases as $[\log (r/r_s)/r{]}^{1/2}$. Its maximum occurs at $x\simeq 2.16258$, with $v_{c,\max }^2\simeq 2.71706G{\rho }_s{r}_s^2$.