Interior gravitational force of an exterior thin ring
= Interior gravitational force of an exterior thin ring
{title2=$\partial_R\psi_{\mathrm{ring}}=GmR/(2a^3)+\cdots$}
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>.