Hill's Lunar Equations, Series, Convergence, Motion of the Perigee

Abstract

We investigate Hill's lunar equations, series and the motion of the perigee, and we use computers to go farther than has previously been known, calculating the coefficients of Hill's series up to order 24 in m, and the coefficients that do not depend on a_0 up to order 30. Numerical calculations indicate that the radius of convergence of Hill's series is somewhere near the value of m of the cusped orbit (0.560958), which we formulate as a conjecture. We calculate the motion of the perigee using a linearization of the equation for the anomalistic period, as in Hill's documentation, but with some discrepancies.