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

TL;DR

This work develops high-order analytical expansions for multi-revolution elliptic Halo (ME-Halo) orbits in the elliptic restricted three-body problem (ERTBP) by introducing two correction terms in the $y$- and $z$-directions and expanding all variables in $e$, $\alpha$, and $\beta$. By enforcing frequency commensurability and a 1:1 resonance through these corrections, the authors frame ME-Halo orbits as double-resonance structures and apply perturbation theory to derive third- and higher-order solutions, including explicit correction-terms expressions. The high-order series are validated against numerical integrations, showing that greater order substantially improves accuracy and provides reliable initial guesses for numerically corrected ME-Halo orbits in Sun–Jupiter and Earth–Moon systems. The approach yields analytic insight into the dynamics near collinear libration points and offers a practical, high-accuracy tool for trajectory design and mission planning, with potential extensions to more complex resonances and real-world perturbations.

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.

Figures (9)

• Figure 1: Analytical ME-Halo orbits together with the associated numerically propagated orbits (top panels), and position deviation between analytical and numerical orbits evaluated at a quarter period (bottom panels). Analytical expansions up to order $n = 5$ and $n = 15$ are considered under the dynamical model of $\mu = 0.0001$. In the top panels, the out-of-plane amplitude is fixed at $\beta = 0.1$, and it holds $(e,\alpha)=(0.109232,0.149158)$ for $n = 5$ and $(e,\alpha)=(0.112684,0.149471)$ for $n=15$.
• Figure 2: Characteristic curves of the M2N1 ME-Halo orbits produced by means of 15th-order series expansions in dynamical models with different $\mu$. The ME-Halo orbits marked by red crosses are to be presented in Figs. \ref{['fig2']} and \ref{['fig3']}.
• Figure 3: Analytical M2N1 ME-Halo orbits in the northern periapsis group, shown in different coordinate planes as $\beta$ changes from 0.04 to 0.2 in the dynamical model of $\mu = 0.0001$. The parameters of ME-Halo orbits, including $e$, $\alpha$ and $\beta$, are provided in the top-row panels (please see Fig. \ref{['fig1']} for their location on the characteristic curve).
• Figure 4: Same as Fig. \ref{['fig2']} but for the dynamical model of $\mu = 0.0005$.
• Figure 5: The numerically corrected orbits together with the analytical M2N1 ME-Halo orbits (panels in the top three rows), and the normalized error between the analytical and numerical orbits as a function of the true anomaly (bottom-row panels) in the Sun--Jupiter systems with different eccentricities.