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Cislunar Mean-Motion Resonances: Definitions, Widths, and Comparisons with Resonant Satellites

Anjali Rawat, Bhanu Kumar, Aaron J. Rosengren, Shane D. Ross

TL;DR

This work develops a global, geometry-driven portrait of cislunar mean-motion resonances beyond GEO using the planar PCR3BP. By constructing Poincaré maps on a perigee section and computing stable resonant islands and their unstable manifolds, it quantifies stable resonance widths and expansive chaotic zones for the 2:1 and 3:1 MMRs, across a range of Jacobi constants. The study shows that PCR3BP widths exceed semi-analytical predictions and align with high-fidelity ephemeris propagations for spacecraft such as TESS, IBEX, and Spektr-R, reinforcing the relevance of MMRs in xGEO dynamics and mission design. It further demonstrates how perigee mappings to geocentric elements and the $(\varpi,a)$ plane can validate the resonance character of real trajectories, suggesting broad applicability for cislunar SDA and trajectory planning. Future directions include extending to 3D dynamics, heteroclinic transfers among resonances, and four-dimensional Poincaré maps to capture full spatial effects.

Abstract

Lunar mean-motion resonances (MMRs) significantly shape cislunar dynamics beyond GEO, forming stable-unstable orbit pairs with corresponding intermingled chaotic and regular regions. The resonance zone is rigorously defined using the separatrix of unstable resonant periodic orbits surrounding stable quasi-periodic regions. Our study leverages the planar, circular, restricted three-body problem (PCR3BP) to estimate the (stable) resonance widths and (unstable) chaotic resonance zones of influence of the 2:1 and 3:1 MMRs across various Jacobi constants, employing a Poincaré map at perigee and presenting findings in easily interpretable geocentric orbital elements. An analysis of the semi-major axis versus eccentricity plane reveals broader regions of resonance influence than those predicted by semi-analytical models based on the perturbed Kepler problem. A comparison with high-fidelity 3-dimensional ephemeris propagation of several spacecraft - TESS, IBEX, and Spektr-R - in these regions is made, which shows good agreement with the simplified CR3BP model.

Cislunar Mean-Motion Resonances: Definitions, Widths, and Comparisons with Resonant Satellites

TL;DR

This work develops a global, geometry-driven portrait of cislunar mean-motion resonances beyond GEO using the planar PCR3BP. By constructing Poincaré maps on a perigee section and computing stable resonant islands and their unstable manifolds, it quantifies stable resonance widths and expansive chaotic zones for the 2:1 and 3:1 MMRs, across a range of Jacobi constants. The study shows that PCR3BP widths exceed semi-analytical predictions and align with high-fidelity ephemeris propagations for spacecraft such as TESS, IBEX, and Spektr-R, reinforcing the relevance of MMRs in xGEO dynamics and mission design. It further demonstrates how perigee mappings to geocentric elements and the plane can validate the resonance character of real trajectories, suggesting broad applicability for cislunar SDA and trajectory planning. Future directions include extending to 3D dynamics, heteroclinic transfers among resonances, and four-dimensional Poincaré maps to capture full spatial effects.

Abstract

Lunar mean-motion resonances (MMRs) significantly shape cislunar dynamics beyond GEO, forming stable-unstable orbit pairs with corresponding intermingled chaotic and regular regions. The resonance zone is rigorously defined using the separatrix of unstable resonant periodic orbits surrounding stable quasi-periodic regions. Our study leverages the planar, circular, restricted three-body problem (PCR3BP) to estimate the (stable) resonance widths and (unstable) chaotic resonance zones of influence of the 2:1 and 3:1 MMRs across various Jacobi constants, employing a Poincaré map at perigee and presenting findings in easily interpretable geocentric orbital elements. An analysis of the semi-major axis versus eccentricity plane reveals broader regions of resonance influence than those predicted by semi-analytical models based on the perturbed Kepler problem. A comparison with high-fidelity 3-dimensional ephemeris propagation of several spacecraft - TESS, IBEX, and Spektr-R - in these regions is made, which shows good agreement with the simplified CR3BP model.
Paper Structure (16 sections, 1 theorem, 28 equations, 15 figures)

This paper contains 16 sections, 1 theorem, 28 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Planar, Circular, Restricted Three-Body Problem
  3. Jacobi Constant and the Energy Manifold.
  4. Tisserand Approximation.
  5. The Poincaré Map and Key Dynamical Features
  6. Surface of Section at Perigee
  7. Poincaré Map on the Poincaré Section
  8. Resonance in xGEO
  9. Unstable Periodic Orbits and Manifolds
  10. Semi-Analytical Approach to Identify Resonance Widths
  11. Direct Identification of Resonance Widths and Chaotic Zones
  12. Comparison with the Ephemeris Model
  13. Conclusion
  14. Continuous Function in l for Perigee Detection
  15. Transformation from the Inertial Ecliptic frame (ECLIPJ2000) to the Non-Dimensional Synodic Frame
  16. ...and 1 more sections

Key Result

Lemma 1

Suppose $q \in W^u(p_i) \bigcap W^s(p_j)$ is a PIP; then $P^k(q)$ is a PIP for all $k \in \mathbb{Z}$.

Figures (15)

  • Figure 1: (a) Non-dimensional barycentered co-rotating $(x,y)$ frame. (b) The geocentric osculating orbital elements showing inertial longitude of perigee ($\varpi_{iner}$) and synodic longitude of perigee ($\varpi$).
  • Figure 2: Poincaré map $P$ on a Poincaré section $\Sigma_C$ in the CR3BP. The unit vector $\hat{\Sigma}$ gives the sense in which trajectories are crossing $\Sigma_C$.
  • Figure 3: (a) PIP ($q$) and secondary intersection point ($\bar{q}$, not a PIP). (b) A BIP $q$ defining a local boundary $B$ between two regions $R_1$ and $R_2$.
  • Figure 4: Construction of a top and bottom boundary to a resonance region $R_1$.
  • Figure 5: Resonance regions for a period-3 orbit $\mathcal{O}(p_3)=\{p_1,p_2,p_3\}$ and a neighbouring period-1 orbit $\mathcal{O}(\bar{p}_1)=\{\bar{p}_1 \}$. The "identify" indicates that the synodic longitude of perigee, is an angle.
  • ...and 10 more figures

Theorems & Definitions (5)

  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 1
  • Definition 4