Sharp-edge force singularity of a thin disk ($g_R\sim G{\Sigma }_{\mathrm{edge}}\log (\ell /ϵ)$)

At an abruptly truncated razor-thin disk with nonzero edge surface density of a disk, the midplane radial force diverges logarithmically. Locally, mass occupies a half-plane and the inward force contains a nonzero angular factor multiplying $G{\Sigma }_{\mathrm{edge}}{\int }_{ϵ}^{\ell }ds/s$. The opposite exterior half-plane is absent, so its cancellation does not occur. Finite thickness or a taper taking the edge density to zero regularizes the force. Therefore a finite-edge galaxy rotation curve requires more than the ideal scale-free Mestel disk normalization.