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

TL;DR

This paper extends calculations of Hill's lunar equations beyond previous limits, analyzing series convergence and the motion of the perigee with computer-aided methods, and proposes a conjecture on the series' radius of convergence.

Contribution

It computes higher-order coefficients of Hill's series and investigates their convergence, providing new insights into lunar motion modeling.

Findings

Calculated series coefficients up to order 24 in m

Estimated the radius of convergence near 0.560958

Identified discrepancies in perigee motion calculations

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.