🔭 astrophysics

Quasi-frozen polar orbits around the Moon

This paper analyzes and identifies quasi-frozen polar orbits around the Moon by developing simplified analytical models with truncated gravitational harmonics to determine initial conditions, which are then validated through numerical integration using high-fidelity selenopotential and third-body perturbation models in GMAT.

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 trying to keep a marble rolling perfectly in a circle on a bumpy, uneven trampoline. If the trampoline is perfectly smooth, the marble stays put. But if it has lumps and dips, the marble will eventually wobble, speed up, slow down, and crash into the center. This is the daily struggle for satellites orbiting the Moon. The Moon isn't a perfect, smooth ball; it's a lumpy potato with hidden pockets of heavy and light mass scattered inside. These "lumps" create invisible gravity hills and valleys that constantly tug on satellites, usually causing them to spiral out of control and smash into the lunar surface within weeks or months.

To solve this, scientists use a concept called a "frozen orbit." Think of it like finding a specific, magical speed and tilt for your marble so that the bumps in the trampoline push it one way at the exact same moment another bump pushes it back the other way. If you hit the perfect balance, the satellite's path stops changing shape over time. It doesn't crash, and it doesn't fly away; it just keeps rolling in a stable loop forever. This is crucial for anyone wanting to set up a permanent lunar base or keep a fleet of satellites circling the Moon for years without needing to constantly fire thrusters to correct their course, which would run out of fuel very quickly.

In this study, authors Jean Paulo Carvalho and Daniel Casanova act like cosmic mechanics trying to find that perfect balance for satellites flying over the Moon's poles (the North and South poles). They built two different mathematical models to predict how the Moon's lumpy gravity would affect a satellite. The first model, which they call "Model 18 × 0," is a simplified version that looks at the Moon's gravity as if it were made of 18 different layers of rings (called zonal harmonics). The second model, "Model 18 × 3," is a bit more complex; it adds in some extra, wobbly gravity terms that account for the Moon's uneven shape in more detail.

Using these models, the team searched for the exact starting conditions—specifically how fast the satellite should go, how tilted its path should be, and where its closest point to the Moon should be located—to create a "frozen" path. They discovered that for a satellite to stay stable over the poles, it can't just be a perfect circle; it needs a tiny bit of an oval shape (eccentricity). Their calculations showed that if a satellite orbits at a distance of 1,938 km from the Moon's center with an eccentricity of 0.0241 and a tilt of exactly 90 degrees, it enters a state of "quasi-frozen" stability. This means the orbit won't change wildly, even though the Moon's gravity is trying to mess with it.

However, the authors were careful to note that their simplified models are just a starting point. When they tested these findings using a super-detailed computer simulation called GMAT (which includes a much more complex 50x50 gravity map and the pull of the Earth), they found that the orbit wasn't perfectly frozen. Instead, the satellite's path would wiggle slightly, like a tightrope walker balancing with a small sway. Despite this wobble, the orbit remained stable for a very long time—over 1,000 days in their simulations—without crashing.

Interestingly, they also found that if you try to do this at a lower altitude (1,838 km), the satellite needs to be much more oval-shaped to stay stable. But even then, the extra wobbly gravity terms in their more complex model would eventually cause that lower satellite to crash into the Moon in just over 250 days. This tells us that while the "magic recipe" for a stable polar orbit exists, it is very sensitive to how high you fly and how perfectly you can calculate the Moon's lumpy gravity. The paper concludes that by using their specific starting numbers (a = 1,938 km, e = 0.0241, i = 90°), mission planners can design lunar satellites that stay in orbit for a long time with very little need for fuel, effectively turning the Moon's chaotic gravity into a stable highway for future explorers.

Technical Summary: Quasi-frozen Polar Orbits around the Moon

Problem Statement
The study addresses the challenge of maintaining long-term stable trajectories for artificial satellites in low-altitude lunar orbits. Due to the Moon's highly non-uniform mass distribution (mascons), unmanaged orbits are subject to severe perturbations that typically cause trajectories to collapse into the lunar surface within short timeframes. The primary objective is to identify specific orbital configurations that prevent premature impact events by balancing secular drifts in orbital elements.

Methodology
The authors employ a dual-framework approach combining analytical derivation and numerical validation:

Dynamical Modeling:
- Perturbations: The model accounts for the Moon's non-uniform gravity field and third-body gravitational perturbations from the Earth. Solar Radiation Pressure (SRP) is deemed negligible for typical satellites and is excluded.
- Model 18 × 0 (Analytical): A simplified analytical model incorporating zonal harmonics up to order 18 ( $J_2$ to $J_{18}$ ). This model uses a single-averaged method to eliminate short-periodic terms.
- Model 18 × 3 (Enhanced Analytical/Numerical): An expanded model that includes the zonal harmonics up to $J_{18}$ plus specific sectoral and tesseral terms of order and degree three ( $C_{22}, S_{22}, C_{31}, S_{31}, C_{32}, S_{32}, C_{33}, S_{33}$ ).
- Third-Body Treatment: The Earth's influence is modeled using a single-averaged formulation for an elliptical, inclined orbit.
Frozen Orbit Analysis:
- The study defines "frozen orbits" as configurations where secular variations in eccentricity ( $\dot{e}$ ) and inclination ( $\dot{i}$ ) are zero.
- The analysis focuses on fixing the argument of perilune ( $g$ ) at $90^\circ$ and $270^\circ$ .
- Because the analytical model with high-degree zonal harmonics makes deriving an explicit $e = f(i, a)$ expression mathematically intractable, the authors use a semi-analytical numerical root-finding sweep to solve $\dot{g} = 0$ and map equilibrium contours in the $(e, i)$ phase plane.
Validation:
- Numerical Integration: Equations of motion derived from Model 18 × 3 are integrated using the Runge-Kutta-Fehlberg method (rkf45) in Maple.
- High-Fidelity Benchmarking: Results are cross-validated using NASA's General Mission Analysis Tool (GMAT). The GMAT configuration utilizes a high-fidelity 50 × 50 order and degree selenopotential model and DE405 planetary ephemerides for Earth's perturbation.

Key Results

Frozen Orbit Conditions: The analysis identifies that stable polar configurations exist where lunar zonal gravity coefficients balance secular drifts. The study analyzes both $g = 90^\circ$ and $g = 270^\circ$ configurations, noting that while both yield frozen solutions in the analytical model, the $g = 270^\circ$ configuration is specifically highlighted for generating stable, frozen polar orbits within the lunar environment when considering the full dynamical context.
Semi-Major Axis Dependence:
- At a semi-major axis of 1838 km (approx. 100 km altitude), the required frozen eccentricity is high ( $e \approx 0.049$ ). Numerical propagation under Model 18 × 3 reveals that even with these initial conditions, the high eccentricity makes the orbit susceptible to rapid impact (within ~250 days) due to non-zonal perturbations.
- At a semi-major axis of 1938 km (approx. 200 km altitude), the required eccentricity relaxes to $e \approx 0.0241$ . This configuration demonstrates significantly higher stability.
Quasi-Frozen Behavior: When propagating the initial conditions derived from the analytical Model 18 × 0 using the more complex Model 18 × 3, the orbits do not remain perfectly frozen but exhibit bounded, low-amplitude oscillations in eccentricity and argument of perilune. These are classified as quasi-frozen orbits.
GMAT Verification: The GMAT simulations (50 × 50 model + Earth) confirm that the quasi-frozen configuration identified at $a = 1938$ km maintains a bounded trajectory over long durations (1000 days tested). The simplified semi-analytical framework (Model 18 × 3 + Earth) closely matches the high-fidelity GMAT results, showing only minor deviations in eccentricity and perilune argument.

Significance and Claims
The paper claims to provide a robust, semi-analytical methodology for identifying low-altitude polar frozen orbits around the Moon. By utilizing high-order zonal formulations (up to $J_{18}$ ) and validating them against high-fidelity numerical models, the authors establish a reliable baseline for mission design.

The primary contribution is the identification of specific initial conditions ( $a = 1938$ km, $e = 0.0241$ , $i = 90^\circ$ , $g = 270^\circ$ , $h = 90^\circ$ ) that minimize the need for station-keeping maneuvers. The authors assert that these results offer a practical foundation for selecting operational orbits for permanent lunar constellations and for predicting "graveyard" trajectories, thereby reducing fuel consumption compared to neighboring orbital states. The study emphasizes that while the analytical model is a simplification, it captures the primary dynamics with high fidelity, making it suitable for preliminary mission design.

Technical Summary: Quasi-frozen Polar Orbits around the Moon

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