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

Joan Gimeno, Luke T. Peterson

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 Poincaré 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.

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

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 Poincaré 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.

Paper Structure

This paper contains 29 sections, 9 theorems, 154 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Motivation and Background
  3. Near-Rectilinear Halo Orbits in the Earth-Moon System
  4. Effective Stability in Hamiltonian and Nearly-Integrable Systems
  5. Normal Forms and Long-Time Stability in Celestial Mechanics
  6. Main Questions and Summary of Results
  7. Preliminaries and Normal Form Construction
  8. The Poincaré Map and Jet Transport
  9. Normal Forms for the Poincaré Map Near an Elliptic Fixed Point
  10. Linear Normalization
  11. Finite-Order Normal Form Construction
  12. Canonicity of Transformations
  13. Definition of the Remainder
  14. Invariant Structures of the Truncated Normal Form
  15. Remainder Estimates and Optimal Normalization
  16. ...and 14 more sections

Key Result

Lemma 1

Assume that $G(s)$ is analytic on the complex polydisc $\mathcal{D}_\rho \coloneqq \{ |s|_\infty < \rho \}$. Let $C_k \coloneqq \sum_{|j|=k} |G_j^k|$. Then there exists computable constants $\tilde{C} > 0$ and $\tilde{\rho} > 0$ such that $\blacktriangleleft$$\blacktriangleleft$

Figures (5)

  • Figure 1: Normally elliptic $L_2$ halo orbits (blue) surrounded by linearly unstable orbits. Note that the gray bar in the right figure indicates the radius of the Moon. Right figure inspired by Spree née Zimovan et al. zimovan2017near.
  • Figure 2: Selected $L_2$ Southern NRHO for effective stability computation (orange), surrounded by blue region of normally elliptic NRHOs.
  • Figure 3: Computational results of radius of convergence and Diophantine constant depending on the order $k$.
  • Figure 4: Computational results of majorant coefficient, $A$, and effective stability radius, $a$, as a function of the order of normalization, $k$.
  • Figure 5: An approximated solution around the chosen $L_2$ Southern NRHO in the Earth-Moon system in Table \ref{['tab:OrbitData']} on the Poincaré section $\Sigma$. First row in Normal Form coordinates, and second row in E-M coordinates.

Theorems & Definitions (18)

  • Lemma 1: Cauchy Estimates
  • proof
  • Lemma 2: Coordinate Transformation
  • proof
  • Lemma 3: Composition and Remainder Bound
  • proof
  • Lemma 4: Iterative Lemma
  • proof
  • Lemma 5: Truncated Remainder Bound
  • proof
  • ...and 8 more