Effective Stability of Near-Rectilinear Halo Orbits in the Earth-Moon System

TL;DR

This paper develops a rigorous framework using normal form theory and Nekhoroshev estimates to characterize the effective stability regions of near-rectilinear halo orbits in the Earth-Moon system, relevant for long-term space missions.

Contribution

It introduces a novel approach combining high-order Taylor expansions, polynomial normal forms, and Nekhoroshev estimates to quantify the nonlinear confinement of NRHOs in the CR3BP.

Findings

Effective stability regions are governed by the normal form's analytic domain, not exponential drift.

Spatial envelopes for stability are explicitly mapped into physical coordinates.

For mission durations of 10-50 years, stability is limited by the normal form's convergence, not long-term instability.

Abstract

Near-rectilinear halo orbits (NRHOs) around Earth-Moon L2 in the Circular Restricted 3-Body Problem (CR3BP) exhibit a complex dynamical landscape, featuring a band of normally elliptic orbits embedded within regions of strong instability. This coexistence of stable and unstable dynamics, amplified by the numerical sensitivity associated with close lunar passages, makes the long-term behavior of trajectories near NRHOs a delicate and intrinsically nonlinear problem. Understanding the effective stability of these elliptic orbits is therefore a critical challenge, lying at the intersection of local normal form theory and global instability mechanisms. To quantify finite-time confinement, we formulate a rigorous framework for effective stability using discrete Poincar\'e maps. By employing jet transport to compute high-order Taylor expansions, we construct explicit polynomial normal forms. We derive discrete Nekhoroshev-type estimates by identifying the normalization order, which balances the asymptotic convergence of the map's analyticity domain against the cumulative penalty of low-order small divisors. Applying this framework to the Earth-Moon system, we map the resulting geometric limits directly into physical spatial coordinates. Crucially, we demonstrate that for practical mission lifetimes (e.g., 10-50 years), the required stability is vastly shorter than the characteristic Nekhoroshev accumulation time. Consequently, the effective stability region is not constrained by the time-dependent exponential drift, but is instead governed entirely by the maximum analytical domain of the optimized normal form. These derived spatial envelopes establish explicit geometric boundaries for the intrinsic local stability of elliptic NRHOs, providing a rigorous mathematical characterization of their nonlinear confinement within the CR3BP.