Spherical mass profile for a flat rotation curve ($M(<r)=v_0^{2}r/G,\rho (r)=v_0^2/(4\pi G{r}^2)$)

In a spherically dominated Newtonian gravitational field, the circular orbit speed satisfies $v_c^2=GM(<r)/r$. A flat galaxy rotation curve therefore implies $M(<r)=v_0^{2}r/G$. Differentiating the enclosed mass gives $\rho =(4\pi r^2{)}^{-1}dM/dr$, yielding the inverse-square density profile. This is a halo-region approximation, not a mass inversion for a flattened stellar disk. A flat curve extending to infinite radius would imply infinite mass, so a real halo profile must eventually change or truncate.

A spherical halo with flat circular speed $v_c$ has $M(r)=v_c^{2}r/G$ and $\rho =v_c^2/(4\pi G{r}^2)$. If it is sharply truncated at $r_{\ast }$, the work per unit mass from $r<r_{\ast }$ to the boundary is $v_c^2\log (r_{\ast }/r)$ 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 $r_{\ast }$ rather than an exact halo edge.

Subtracting a finite stellar mass from the distant spherical mass required for a flat galaxy rotation curve gives the displayed asymptotic offset. Extending it to $r=0$ would give negative enclosed mass, so it cannot define a physical inner halo. Keeping only its positive leading term instead gives a singular isothermal halo with density proportional to $r^{-2}$ and a nonzero limiting central circular speed. These are different extrapolations, and neither is forced by an outer rotation curve.