🔭 astrophysics

Resonance effects in orbits around the Moon

This paper analyzes resonant orbits around the Moon by combining numerical and analytical models to identify specific semi-major axis and inclination values that either amplify orbital eccentricity for space debris removal or stabilize orbits to extend spacecraft lifetimes, while mapping regions where resonance effects should be avoided.

Original authors: Jean Paulo Carvalho, Daniel Casanova

Published 2026-08-14

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CC BY 4.0

Original authors: Jean Paulo Carvalho, Daniel Casanova

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the Moon not as a silent, cratered rock, but as a bumpy, lumpy dance floor where gravity doesn't pull straight down everywhere. In the world of space travel, this uneven gravity is just one of the forces acting on a satellite. There's also the "third-body" tug of the Earth, which is constantly pulling on our lunar neighbors, and the gentle but persistent push of sunlight, known as Solar Radiation Pressure (SRP). Think of SRP like a constant, invisible breeze; for a heavy, dense object, it's a whisper, but for a lightweight object with a large surface area—like a satellite with a solar sail—it becomes a strong wind.

When these forces mix in just the right way, something magical and dangerous happens: resonance. In everyday life, resonance is what happens when you push a child on a swing at exactly the right moment, making them go higher and higher with very little effort. In space, resonance occurs when the timing of a satellite's orbit lines up perfectly with the rhythm of the Sun's position or the Moon's wobbly gravity. When this happens, the satellite doesn't just drift; it gets a massive boost in its orbital "wiggle," causing its path to stretch out until it might crash into the Moon or fly away. Scientists care about this because it's a double-edged sword: for a mission-critical satellite, resonance is a disaster waiting to happen, but for space junk, it's a free ticket to a clean-up crew that doesn't need fuel.

This paper by Jean Paulo Carvalho and Daniel Casanova dives deep into the Moon's orbital dance floor to map out exactly where these dangerous (or helpful) resonances hide. The researchers built a sophisticated computer model to simulate how satellites move around the Moon, accounting for the Moon's lumpy gravity (up to 18 different "layers" of gravitational bumps), the Earth's pull, and the push of sunlight. They didn't just look at one type of resonance; they investigated six specific ways the satellite's orbit could sync up with the Sun's motion.

The team ran thousands of simulations to see what happens to a satellite's orbit over five years, specifically looking at how its shape (eccentricity) changes. They found that if you place a satellite at certain heights and angles, the resonance acts like a cosmic slingshot. In particular, they discovered that orbits tilted at roughly 60 degrees (going forward) and 120 degrees (going backward) are hotspots for chaos. In these zones, the satellite's path stretches out dramatically, causing it to crash into the lunar surface much faster than expected.

However, the story isn't just about crashes. The simulations also revealed "safe zones." For instance, a specific polar orbit at a semi-major axis of 1938 km remained stable and didn't crash within the five-year timeframe, even with all the gravitational bumps and solar pushes. The authors suggest that by understanding these maps, mission planners can do two things: first, they can steer their precious satellites away from the 60 and 120-degree resonance zones to keep them safe; second, they could intentionally park space debris in these "chaotic corridors" to let the Moon's own gravity and the Sun's breeze do the work of cleaning up the orbit.

It is important to note that these findings come from computer simulations using a specific model (Model 18×3) and a specific ratio of solar sail area to mass (0.012 m²/kg). The paper suggests that while the Earth's gravity adds a little extra complexity, the main drivers of these resonances are the Moon's own gravity and the Sun's pressure. The authors conclude that these dynamic maps provide a clear guide for where to fly and where to avoid, turning the Moon's chaotic gravitational landscape from a hidden hazard into a predictable tool for keeping space clean.

Technical Summary: Resonance Effects in Orbits Around the Moon

Problem Statement
The long-term evolution of orbits around the Moon is governed by a complex interplay of gravitational and non-gravitational perturbations. Specifically, low-altitude lunar trajectories are susceptible to severe resonance conditions arising from the coupling between the Moon's asymmetric gravitational field (selenopotential) and Solar Radiation Pressure (SRP). Understanding these resonant interactions is critical for two distinct operational goals: ensuring the stability of operational spacecraft by avoiding regions of catastrophic eccentricity growth, and facilitating the passive removal of space debris by exploiting resonant mechanisms to accelerate reentry into the lunar surface. While similar resonance corridors ("deorbit corridors") have been mapped for Earth orbits (e.g., via the ReDSHIFT project), a comprehensive analysis of these specific SRP-induced resonances in the lunar environment, particularly considering higher-order gravitational harmonics, is required for robust mission design.

Methodology
The authors employ a hybrid analytical and numerical approach to investigate six distinct resonance cases in low-altitude lunar orbits.

Dynamical Model: The study utilizes a potential function ( $U$ $U$ ) comprising three components: the non-spherical gravitational potential of the Moon ( $U_{\text{Moon}}$ $U_{Moon}$ ), the third-body gravitational perturbation from Earth ( $U_{\oplus}$ $U_{\oplus}$ ), and the disturbing potential from Solar Radiation Pressure ( $U_{\text{SRP}}$ $U_{SRP}$ ).
- Two lunar gravity models are compared: a simplified zonal model truncated at degree 18 (Model 18×0) and a more complex hybrid model including zonal terms up to degree 18, plus sectoral and tesseral terms up to order 3 (Model 18×3).
- The SRP is modeled assuming a spacecraft with a fixed orientation perpendicular to the Sun, potentially coupled with a solar sail to increase the area-to-mass ratio ( $A/m$ ). The study adopts a baseline $A/m$ of $0.012 \, \text{m}^2/\text{kg}$ .
Analytical Framework: The authors derive an averaged disturbing potential for SRP expanded in Legendre polynomials. This expansion identifies six specific resonant angles ( $\psi_1$ $ψ_{1}$ through $\psi_6$ $ψ_{6}$ ) resulting from the geometric coupling between the spacecraft's orbital elements (argument of periselene $g$ $g$ , longitude of the ascending node $h$ $h$ ) and the Sun's ecliptic longitude ( $\lambda_{\odot}$ $λ_{⊙}$ ).
- Resonance conditions are defined where the time derivative of these angles approaches zero (e.g., $\dot{\psi}_1 = \dot{h} + \dot{g} - \dot{\lambda}_{\odot} \approx 0$ ).
- Lagrange's planetary equations are applied to the simplified potential ( $J_2 + \text{SRP}$ ) to isolate analytical expressions for the resonant semi-major axis as a function of inclination and eccentricity.
Numerical Simulation: To validate and expand upon the analytical findings, the authors perform numerical integrations of the equations of motion over a 5-year period. They generate dynamic maps plotting the amplitude of eccentricity variation ( $Amp(e) = e_{\text{max}} - e_{\text{min}}$ ) and satellite lifetime against the semi-major axis ( $a$ ) and inclination ( $i$ ). Simulations test the impact of adding higher-degree zonal harmonics, tesseral/sectoral terms, and Earth's third-body perturbation.

Key Contributions and Results
The study systematically maps the phase-space geometry of low-altitude lunar orbits under the influence of SRP and lunar gravity, yielding the following results:

Identification of Resonance Zones: The analysis confirms that specific combinations of semi-major axis and inclination create resonant conditions where the rate of change of orbital eccentricity is amplified.
Impact of Gravitational Harmonics:
- Using the simplified Model 18×0 + SRP, resonance effects produce only modest eccentricity variations, primarily affecting very low inclinations ( $<20^\circ$ ).
- Incorporating higher-degree zonal harmonics (Model 18×0 + SRP extended to $J_{18}$ ) significantly alters the dynamical structure. This reveals strong eccentricity growth regions near critical inclinations of approximately 60° (prograde) and 120° (retrograde), corresponding to the critical inclination region.
- The inclusion of sectoral, tesseral terms, and Earth's perturbation (Model 18×3 + Earth + SRP) further modifies the dynamics, confirming that the irregular lunar gravitational field strongly influences orbital stability.
Critical Inclination Instability: The dynamic maps (Figures 4 and 5) highlight that semi-major axes between 2000 km and 2300 km exhibit significant eccentricity growth when the inclination is near the critical values (63.43° and 116.57°). In these zones, the resonance mechanism destabilizes a broad range of semi-major axes, accelerating the potential for lunar surface impact.
Stable vs. Unstable Regions:
- Debris Removal: The identified regions of high eccentricity variation (particularly near critical inclinations) are proposed as "corridors" for passive debris removal, where natural resonances can be exploited to shorten orbital lifetimes.
- Operational Stability: Conversely, the maps identify regions where eccentricity remains stable. Notably, a specific polar orbit with a semi-major axis of 1938 km is identified as an exception that remains stable and does not collide with the lunar surface within the 5-year analysis window, even under the influence of higher-order perturbations.
Third-Body Effects: The inclusion of Earth's gravitational perturbation acts as a second-order effect, inducing minor localized increases in eccentricity and slightly broadening resonance boundaries, but it does not fundamentally alter the global structure of the dynamical regime established by the lunar selenopotential and SRP.

Significance
The paper claims that its primary significance lies in providing a systematic mapping of six key resonance cases in low-altitude lunar orbits. By delineating "forbidden operational regions" and "highly active chaotic escape corridors," the generated dynamic maps offer practical guidelines for mission planners. Specifically, the work enables the selection of long-term stable orbits for lunar constellations by avoiding resonance zones, while simultaneously offering a strategy for orbital safety and debris mitigation by identifying regions where resonant structures can be utilized to facilitate the passive reentry of space debris. The results align with and provide more detailed confirmation of previous findings regarding resonant inclinations around 60° and 120°, extending the understanding of how higher-order lunar gravity terms interact with SRP in the lunar environment.

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