From Babylonian lunar observations to Floquet multipliers and Conley-Zehnder Indices

Abstract: The lunar periods of our moon -- the companion of the Earth -- which date back to the Babylonians until around 600 BCE, are 29.53 days for the synodic, 27.55 days for the anomalistic and 27.21 days for the draconitic month. In this paper we define and compute these periods in terms of Floquet multipliers and Conley--Zehnder indices for planar periodic orbits in the spatial Hill lunar problem, which is a limit case of the spatial circular restricted three body problem. For very low energies, we are able to prove analytically the existence of the families of planar direct (family $g$) and retrograde periodic orbits (family $f$) and to determine their Conley-Zehnder index. For higher energies, by numerical approximations to the linearized flow, we also study other families of planar and spatial periodic orbits bifurcating from the families $g$ and $f$. Moreover, our framework provide an organized structure for the families, especially to see how they are connected to each other. Since the solutions we analyze are of practical interest, our work connects three topics: Babylonian lunar periods, symplectic geometry, and space mission design.