Analytical and Semi-Analytical Approaches to the Third-Body Perturbation in Nearly Co-Orbital Regimes

This paper studies a range of formulations for third-body motion, based on the disturbing function derived from the Hamiltonian of the Circular Restricted Three-Body Problem (CR3BP). The main one is the well known Keplerian Map (KM), derived from a first-order Picard iteration on the Lagrange Planetary Equations. Three additional strategies to model the third-body effect are generated. The first is the Periapsis-Apoapsis-Periapsis Keplerian Map (PAPKM), a semi-analytical formulation using the eccentric anomaly as the independent variable. The second is the Euler Keplerian Map (EK Map), a model for long time propagation that uses an Euler method to obtain an accurate evolution of the orbital elements throughout the motion. Finally, the third method is an approximate analytical solution for the evolution of the semi-major axis of the motion, obtained via a Taylor expansion of the eccentricity. These formulations are contrasted with the original KM model and show similar dynamical behaviour, with a decrease in computational time. Furthermore, all of them prove to be more accurate within their application limits.