Semi-analytical computation of commensurate semimajor axes of resonant orbits including second-order gravitational perturbations

Abstract

This research work aims to understand how resonant geopotential harmonics affect the semi-major axis of GPS orbits. The study uses a second-order approximation to calculate iteratively the impact of higher zonal perturbations on the semi-major axis. In addition, Kaula's resonant perturbation theory is utilized to compute analytically the main resonant geopotential that can have significant effects on the motion. We derive and plot the drift rate as a function of the longitudinal position, aiming to identify stable and metastable positions at specific longitudes. The study also investigates motion around these points using the Poincare method, demonstrating the existence of periodic, quasi-periodic, and chaotic orbits near these positions.