Uniformly Rotating Homogeneous Rings in Newtonian Gravity

In this paper, we describe an analytical method for treating uniformly rotating homogeneous rings without a central body in Newtonian gravity. We employ series expansions about the thin ring limit and use the fact that in this limit the cross-section of the ring tends to a circle. The coefficients can in principle be determined up to an arbitrary order. Results are presented here to the 20th order and compared with numerical results.