Interior gravitational force of an exterior thin ring (${\partial }_R{\psi }_{\mathrm{ring}}=GmR/(2a^3)+\cdots$)

A thin ring of mass $m$ and radius $a$ has interior relative potential $Gm[1+R^2/(4a^2)+\cdots ]/a$. Its leading radial acceleration is outward, $GmR/(2a^3)$, unlike the zero interior force of a spherical shell. Exterior disc annuli therefore affect a galaxy rotation curve.