High-order expansions of multi-revolution elliptic Halo orbits in the elliptic restricted three-body problem

Abstract: Multi-revolution elliptic Halo (ME-Halo) orbits are a special class of symmetric and periodic solutions within the framework of the elliptic restricted three-body problem (ERTBP). During a single period, an M:N ME-Halo orbit completes $M$ revolutions around a libration point and the primaries revolve N times around each other. Owing to the repeated configurations, ME-Halo orbits hold great promise as nominal trajectories for space mission design. However, a major challenge associated with ME-Halo orbits lies in their mathematical description. To this end, we propose a novel method to derive high-order analytical expansions of ME-Halo orbits in the ERTBP by introducing two correction terms into the equations of motion in the y- and z-directions. Specifically, both the coordinate variables and correction terms are expanded as power series in terms of the primary eccentricity, the in-plane amplitude, and the out-of-plane amplitude. High-order approximations are constructed using a perturbation method, and their accuracy is validated through numerical analysis. Due to the inherent symmetry, ME-Halo orbits can be classified into four distinct families: southern/northern and periapsis/apoapsis groups. The analytical approximations developed in this study not only provide high-accuracy initial guesses for the numerical computation of ME-Halo orbits, but also offer new insights into the dynamical environment near collinear libration points in the ERTBP, thereby advancing practical applications in mission design.