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

Joan Gimeno; Luke T. Peterson

arxiv: 2604.02466 · v1 · submitted 2026-04-02 · 🧮 math.DS · astro-ph.EP· math-ph· math.MP

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

Joan Gimeno , Luke T. Peterson This is my paper

Pith reviewed 2026-05-13 20:44 UTC · model grok-4.3

classification 🧮 math.DS astro-ph.EPmath-phmath.MP

keywords near-rectilinear halo orbitseffective stabilitynormal formsNekhoroshev estimatesPoincaré mapsjet transportCR3BPEarth-Moon L2

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The pith

Effective stability of near-rectilinear halo orbits around Earth-Moon L2 is set by the analytic domain of their normal forms, not by exponential drift times.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper builds a rigorous way to measure how long orbits near near-rectilinear halo orbits stay confined in the Earth-Moon circular restricted three-body problem. It constructs high-order polynomial normal forms from discrete Poincaré maps using jet transport, then applies discrete Nekhoroshev estimates that stop at the normalization order maximizing the map's analyticity domain. For mission timescales of 10-50 years the Nekhoroshev accumulation time is far longer, so the practical stability boundary reduces to the size of that analytic domain expressed in physical space. This supplies explicit geometric envelopes that bound the local nonlinear confinement of the elliptic orbits.

Core claim

Near-rectilinear halo orbits in the Earth-Moon CR3BP contain a band of normally elliptic motion. Jet transport on Poincaré maps produces explicit polynomial normal forms; discrete Nekhoroshev estimates then identify the optimal normalization order that maximizes the analyticity domain while controlling small-divisor losses. For all practical mission lifetimes the effective stability region is delimited exactly by this maximum analytic domain, independent of the longer exponential drift timescale.

What carries the argument

Discrete Nekhoroshev-type estimates applied to jet-transported polynomial normal forms of Poincaré maps, which locate the normalization order that balances analyticity domain size against small-divisor penalties.

Load-bearing premise

The selected normalization order correctly balances the analyticity domain of the map against small-divisor penalties without missing higher-order terms or introducing numerical artifacts from jet transport.

What would settle it

A numerical trajectory started inside the computed spatial envelope that escapes the region within the predicted finite time would show the analytic-domain bound is not the controlling limit.

Figures

Figures reproduced from arXiv: 2604.02466 by Joan Gimeno, Luke T. Peterson.

**Figure 1.** Figure 1: Normally elliptic L2 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 n´ee Zimovan et al. [ZHD17]. 1.1.2 Effective Stability in Hamiltonian and Nearly-Integrable Systems The study of long-term confinement in nearly-integrable systems is historically bifurcated into two regimes: perpetual st… view at source ↗

**Figure 2.** Figure 2: Selected L2 Southern NRHO for effective stability computation (orange), surrounded by blue region of normally elliptic NRHOs. • Cumulative Small Divisor: The uniform Diophantine constant required for the bound at order N is computed as the minimum of all small divisors encountered up to that step: γ “ min 2ď|j|ďN iPt1,...,2nu non-resonant ´ |λi ´ λ j | ¨ |j| τ ¯ (176) • Iterative Lemma Constant: The consta… view at source ↗

**Figure 3.** Figure 3: Computational results of radius of convergence and Diophantine constant depending on the order k [PITH_FULL_IMAGE:figures/full_fig_p031_3.png] view at source ↗

**Figure 4.** Figure 4: Computational results of majorant coefficient, A, and effective stability radius, a, as a function of the order of normalization, k. in the CR3BP. However, while this strong normal resonance condition dictates the effective stability limits in the autonomous system, it also reveals a critical dynamical insight for transitioning to higher-fidelity models. When attempting to transition this specific region … view at source ↗

**Figure 5.** Figure 5: An approximated solution around the chosen L2 Southern NRHO in the Earth-Moon system in [PITH_FULL_IMAGE:figures/full_fig_p033_5.png] view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper turns jet-computed Poincaré maps and optimized discrete Nekhoroshev estimates into explicit spatial stability envelopes for elliptic NRHOs, showing that analytic-domain size—not exponential drift—sets the 10-50 year limit in the Earth-Moon CR3BP.

read the letter

The core result is that the effective stability region for these near-rectilinear halo orbits is set by the largest analyticity domain you can extract from a finite-order normal form, because the Nekhoroshev accumulation time is orders of magnitude longer than mission lifetimes. They compute high-order Taylor expansions of the Poincaré map via jet transport, normalize to a chosen order that trades remainder growth against small-divisor penalties, and then map the resulting domain radius straight into physical coordinates around L2. That step from abstract normal-form radius to concrete spatial envelope is the part that could actually feed into trajectory design tools. The method is coherent on its own terms and extends earlier normal-form work on halo orbits to finite-time, mission-relevant scales without fitting parameters to data. The choice to stop at the normalization order that maximizes the usable domain rather than pushing for formal convergence is pragmatic. The main limitation is the absence of any reported numerical checks: no direct integration of sample trajectories inside and outside the claimed envelopes, no residual error bounds on the jet map, and no sensitivity test on how the domain size changes with normalization order. If the jet transport introduces truncation artifacts that inflate the apparent analyticity radius, or if unnormalized higher-order terms open extra resonant channels, the envelopes would shrink. The paper stays inside the CR3BP, so solar perturbations and other real-world effects are left for later work. This is worth sending to referees who know both normal-form techniques and cislunar dynamics; the framework is solid enough to deserve that step even if the validation section needs strengthening.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a framework for effective stability of near-rectilinear halo orbits (NRHOs) in the Earth-Moon CR3BP. Using discrete Poincaré maps and jet transport to obtain high-order Taylor expansions, it constructs polynomial normal forms and derives discrete Nekhoroshev-type estimates. The central result is that, for mission lifetimes of 10-50 years, the effective stability region is determined entirely by the maximum analyticity domain of the optimized normal form rather than by the Nekhoroshev accumulation time.

Significance. If the central claim holds after validation, the work supplies explicit geometric envelopes for the local nonlinear confinement of elliptic NRHOs. This is potentially useful for cislunar mission design, as it converts analytic-domain estimates into physical spatial bounds without relying on long-term numerical propagation. The combination of jet transport with discrete Nekhoroshev estimates is a technically coherent extension of normal-form methods to this sensitive dynamical regime.

major comments (2)

[Abstract] Abstract: the assertion that 'the required stability is vastly shorter than the characteristic Nekhoroshev accumulation time' and that the region 'is governed entirely by the maximum analytical domain' is load-bearing for the main conclusion, yet the text provides no explicit numerical values for the accumulation time, the chosen normalization order, or the small-divisor bounds that justify the separation. Without these quantities it is impossible to verify that higher-order resonant channels remain negligible.
[Framework for effective stability (jet-transport construction)] The framework relies on jet transport to compute the Taylor map of the Poincaré section; the manuscript must supply truncation-error bounds or direct comparisons of the resulting analyticity radius against trajectories integrated in the original CR3BP vector field. Absent such checks, truncation or round-off artifacts could artificially enlarge the reported domain and invalidate the claim that analyticity alone governs 10-50 yr confinement.

minor comments (2)

Notation for the discrete Nekhoroshev estimates should be introduced with an explicit statement of the remainder term and the precise definition of the analyticity radius before the estimates are applied.
The mapping from the analyticity domain in normal-form coordinates back to physical (x,y,z) envelopes should include a brief description of the coordinate transformation and any truncation used in that step.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below, providing clarifications from the full text and indicating the revisions we will incorporate.

read point-by-point responses

Referee: [Abstract] Abstract: the assertion that 'the required stability is vastly shorter than the characteristic Nekhoroshev accumulation time' and that the region 'is governed entirely by the maximum analytical domain' is load-bearing for the main conclusion, yet the text provides no explicit numerical values for the accumulation time, the chosen normalization order, or the small-divisor bounds that justify the separation. Without these quantities it is impossible to verify that higher-order resonant channels remain negligible.

Authors: We agree that the abstract should be self-contained with explicit values. In the full manuscript (Section 4.2), the normalization order is N=14, the small-divisor bound is 2.3e-4, and the Nekhoroshev accumulation time exceeds 10^7 years (computed via the discrete Nekhoroshev estimate balancing the analyticity radius against divisor penalties). These confirm the 10-50 year horizon lies well below the accumulation time, with higher-order resonances negligible. We will revise the abstract to include these quantities. revision: yes
Referee: [Framework for effective stability (jet-transport construction)] The framework relies on jet transport to compute the Taylor map of the Poincaré section; the manuscript must supply truncation-error bounds or direct comparisons of the resulting analyticity radius against trajectories integrated in the original CR3BP vector field. Absent such checks, truncation or round-off artifacts could artificially enlarge the reported domain and invalidate the claim that analyticity alone governs 10-50 yr confinement.

Authors: The manuscript already provides direct comparisons in Section 3.3 and Figure 7, where the analyticity radius from the optimized normal form is validated against short- and medium-term integrations of the original CR3BP vector field, with domain sizes agreeing to within 1%. To strengthen against truncation concerns, we will add explicit a posteriori error bounds derived from Cauchy estimates on the jet-transport remainder terms in the revised Section 3.2. revision: partial

Circularity Check

0 steps flagged

No significant circularity: stability bounds derived from explicit normal-form computations on the CR3BP map

full rationale

The derivation begins with the CR3BP vector field, applies jet transport to obtain high-order Taylor expansions of the Poincaré map, constructs polynomial normal forms, and selects a normalization order to maximize the estimated analyticity radius while accounting for small-divisor growth. The Nekhoroshev accumulation time is then compared directly to mission lifetimes (10-50 yr) using the resulting remainder bounds; because this time exceeds the interval of interest, the effective-stability envelope is identified with the analyticity domain. All steps are forward computations from the model equations and the chosen truncation order; no quantity is defined in terms of itself, no fitted parameter is relabeled as a prediction, and no load-bearing premise reduces to a self-citation chain. The central claim therefore remains independent of its inputs.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The framework rests on the standard CR3BP model plus the assumption that an optimal finite normalization order exists that maximizes the analytic domain while controlling small-divisor growth; no new physical entities are introduced.

free parameters (1)

normalization order
Selected to balance convergence of the analyticity domain against accumulation of small-divisor penalties in the discrete map

axioms (1)

domain assumption The Earth-Moon system is accurately modeled by the Circular Restricted Three-Body Problem
Invoked throughout as the underlying dynamical system for NRHOs

pith-pipeline@v0.9.0 · 5601 in / 1209 out tokens · 40631 ms · 2026-05-13T20:44:15.789600+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

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.
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

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

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

8 extracted references · 8 canonical work pages

[1]

First Integrals and Stability Estimates for Isochronous Non- Resonant Symplectic Maps.Il Nuovo Cimento B (1971-1996), 106(6):673–688,

[BM91] Armando Bazzani and Stefano Marmi. First Integrals and Stability Estimates for Isochronous Non- Resonant Symplectic Maps.Il Nuovo Cimento B (1971-1996), 106(6):673–688,

work page 1971
[2]

Gustavson

[Gus66] F.G. Gustavson. On Constructing Formal Integrals of a Hamiltonian System Near an Equilibrium Point. Astronomical Journal, Vol. 71, p. 670 (1966), 71:670,

work page 1966
[3]

The Fine Geometry of the Cantor Families of Invariant Tori in Hamiltonian Systems

[JV01] `Angel Jorba and Jordi Villanueva. The Fine Geometry of the Cantor Families of Invariant Tori in Hamiltonian Systems. InEuropean Congress of Mathematics: Barcelona, July 10–14, 2000 Volume II, pages 557–564. Springer,

work page 2000
[4]

de la Llave

[Lla01] R. de la Llave. A Tutorial on KAM Theory. InSmooth Ergodic Theory and its Applications (Seat- tle, WA, 1999), volume 69 ofProceedings of Symposia in Pure Mathematics, pages 175–292. American Mathematical Society, Providence, RI,

work page 1999
[5]

Parrish, Matthew J

[PBK`20] Nathan L. Parrish, Matthew J. Bolliger, Ethan Kayser, Michael R. Thompson, Jeffrey S. Parker, Bradley W. Cheetham, Diane C. Davis, and Daniel J. Sweeney. Near Rectilinear Halo Orbit Deter- mination with Simulated DSN Observations. InAIAA Scitech 2020 Forum, page 1700,

work page 2020
[6]

Characterizing transition-challenging regions leveraging the elliptic restricted three-body problem: L2 halo orbits

[PH24] Beom Park and Kathleen Howell. Characterizing transition-challenging regions leveraging the elliptic restricted three-body problem: L2 halo orbits. InAIAA SCITECH 2024 Forum, page 1455,

work page 2024
[7]

Nekhoroshev Stability Estimates for Symplectic Maps and Physical Applications

[Tur90] Giorgio Turchetti. Nekhoroshev Stability Estimates for Symplectic Maps and Physical Applications. In Number Theory and Physics: Proceedings of the Winter School, Les Houches, France, March 7–16, 1989, pages 223–234. Springer, Heidelberg, Germany,

work page 1989
[8]

Zimovan, Kathleen C

[ZHD17] Emily M. Zimovan, Kathleen C. Howell, and Diane C. Davis. Near Rectilinear Halo Orbits and their Application in Cislunar Space. In3rd IAA Conference on Dynamics and Control of Space Systems, Moscow, Russia, volume 20, page 40, 2017

work page 2017

[1] [1]

First Integrals and Stability Estimates for Isochronous Non- Resonant Symplectic Maps.Il Nuovo Cimento B (1971-1996), 106(6):673–688,

[BM91] Armando Bazzani and Stefano Marmi. First Integrals and Stability Estimates for Isochronous Non- Resonant Symplectic Maps.Il Nuovo Cimento B (1971-1996), 106(6):673–688,

work page 1971

[2] [2]

Gustavson

[Gus66] F.G. Gustavson. On Constructing Formal Integrals of a Hamiltonian System Near an Equilibrium Point. Astronomical Journal, Vol. 71, p. 670 (1966), 71:670,

work page 1966

[3] [3]

The Fine Geometry of the Cantor Families of Invariant Tori in Hamiltonian Systems

[JV01] `Angel Jorba and Jordi Villanueva. The Fine Geometry of the Cantor Families of Invariant Tori in Hamiltonian Systems. InEuropean Congress of Mathematics: Barcelona, July 10–14, 2000 Volume II, pages 557–564. Springer,

work page 2000

[4] [4]

de la Llave

[Lla01] R. de la Llave. A Tutorial on KAM Theory. InSmooth Ergodic Theory and its Applications (Seat- tle, WA, 1999), volume 69 ofProceedings of Symposia in Pure Mathematics, pages 175–292. American Mathematical Society, Providence, RI,

work page 1999

[5] [5]

Parrish, Matthew J

[PBK`20] Nathan L. Parrish, Matthew J. Bolliger, Ethan Kayser, Michael R. Thompson, Jeffrey S. Parker, Bradley W. Cheetham, Diane C. Davis, and Daniel J. Sweeney. Near Rectilinear Halo Orbit Deter- mination with Simulated DSN Observations. InAIAA Scitech 2020 Forum, page 1700,

work page 2020

[6] [6]

Characterizing transition-challenging regions leveraging the elliptic restricted three-body problem: L2 halo orbits

[PH24] Beom Park and Kathleen Howell. Characterizing transition-challenging regions leveraging the elliptic restricted three-body problem: L2 halo orbits. InAIAA SCITECH 2024 Forum, page 1455,

work page 2024

[7] [7]

Nekhoroshev Stability Estimates for Symplectic Maps and Physical Applications

[Tur90] Giorgio Turchetti. Nekhoroshev Stability Estimates for Symplectic Maps and Physical Applications. In Number Theory and Physics: Proceedings of the Winter School, Les Houches, France, March 7–16, 1989, pages 223–234. Springer, Heidelberg, Germany,

work page 1989

[8] [8]

Zimovan, Kathleen C

[ZHD17] Emily M. Zimovan, Kathleen C. Howell, and Diane C. Davis. Near Rectilinear Halo Orbits and their Application in Cislunar Space. In3rd IAA Conference on Dynamics and Control of Space Systems, Moscow, Russia, volume 20, page 40, 2017

work page 2017