Analysis of Near-Polar and Near-Circular Periodic Orbits Around the Moon with J2, C22 and Third-Body Perturbations

In the Moon–Earth elliptic restricted three-body problem, near-polar and near-circular lunar-type periodic orbits are numerically continued from Keplerian circular orbits using Broyden’s method with line search. The Hamiltonian system, expressed in Cartesian coordinates, is treated via the symplectic scaling method. The radii of the initial Keplerian circular orbits are then scaled and normalized. For cases in which the integer ratios {j/k} of the mean motions between the inner and outer orbits are within the range [9,150], some periodic orbits of the elliptic restricted three-body problem are investigated. For the middle-altitude cases with j/k∈[38,70], the perturbations due to J2 and C22 are incorporated, and some new near-polar periodic orbits are computed. The orbital dynamics of these near-polar, near-circular periodic orbits are well characterized by the first-order double-averaged system in the Poincaré–Delaunay elements. Linear stability is assessed through characteristic multipliers derived from the fundamental solution matrix of the linear varational system. Stability indices are computed for both the near-polar and planar near-circular periodic orbits across the range j/k∈[9,50].