Enclosed mass does not determine a disc rotation curve
= Enclosed mass does not determine a disc rotation curve
The <spherical shell theorem> yields $v_c^2=GM(<r)/r$ only in spherical geometry. A razor-thin axisymmetric disc instead requires $v_c^2=R\partial_R\Phi(R,0)$, including forces from exterior annuli. The <interior gravitational force of an exterior thin ring> gives a concrete counterexample with identical enclosed mass but different circular speeds.