Surface density contrast in an eccentric ring

A higher orbital line density at apoapsis does not always give a higher surface density of a disk: the local ring width also changes. In a narrow aligned eccentric ring, neglecting disc twist and keeping first-order orbital eccentricity, ${\Sigma }_{\mathrm{peri}}\propto 1+ade/da$ and ${\Sigma }_{\mathrm{apo}}\propto 1-ade/da$. Constant eccentricity therefore gives constant surface density to this order.