Exponential-disk Keplerian asymptotic ($v_c^2=GM/R[1+9R_d^2/(2R^2)+O(R^{-4})]$)

For an exponential galactic disk, the first nonspherical term in the Legendre expansion of thin-disk gravity is positive and proportional to $R^{-3}$ in $v_c^2$. The missing enclosed mass and exterior-ring terms are exponentially small at large radius. Hence the galaxy rotation curve approaches the point-mass Keplerian curve from above.