The appeal of so-called frozen orbits is well established. These are orbits whose size, shape, and orientation remain nearly constant in a frame rotating with the Moon, requiring almost no propellant to maintain. The Lunar Reconnaissance Orbiter demonstrated the payoff: by carefully selecting a polar frozen orbit, the mission used only 33.9 kilograms of hydrazine over twelve years for all of its station-keeping, phasing, and momentum management. Frozen orbits could similarly support extended surface observation campaigns, communications relays, and even the tracking of orbital debris or derelict hardware from past missions, such as the Apollo-era lunar module ascent stages that may still circle the Moon.

Traditionally, frozen orbits have been found analytically. Researchers construct an averaged dynamical model that strips out short-period oscillations, then solve for equilibrium points in the averaged equations. These classical analyses predict that frozen orbits occur under specific conditions: when the mean argument of perilune, the point of closest approach, sits near plus or minus 90 degrees; when the mean eccentricity is zero; or when the argument of perilune equals 0 or 180 degrees. The trouble is that averaging discards real physics, and the resulting predictions must be verified by propagating trajectories in a high-fidelity model that includes a detailed lunar gravity field and the true ephemerides of the Earth and Sun. Scaling that verification across the whole phase space has always been the bottleneck.

Bosanac and Miceli attacked the bottleneck directly. They defined over 1.1 million initial conditions at a fixed epoch, sampling orbital elements coarsely across semi-major axes up to 2,038 kilometers, eccentricities, inclinations from 0 to 180 degrees, and the full range of nodal longitudes and arguments of perilune. Each initial state was propagated for up to 180 days in a demanding dynamical model: a 100 by 100 degree-and-order spherical harmonic lunar gravity field, drawn from the GRGM900C model derived from GRAIL data, combined with the point-mass gravity of the Earth and Sun positioned according to the DE421 ephemerides. Integrations used a high-order Runge-Kutta scheme with tight tolerances of ten to the minus eleventh, terminating early if the spacecraft struck a spherical Moon of radius 1,738 kilometers.

Of the initial conditions tested, 638,181 trajectories survived at least three lunar rotational periods, roughly 81.9 days, and entered the analysis. Rather than storing every point along every path, the researchers represented each trajectory by its sequence of perilunes, the moments of closest approach, which capture the long-period evolution of the orbit’s eccentricity and orientation. Each perilune was described by two compact coordinates, p and q, equal to the eccentricity times the cosine and sine of the argument of perilune. These coordinates encode both the shape of the orbit and the direction in which its low point points, and they trace out distinctive loops and curves as the orbit evolves in the Moon-fixed frame.

The clustering itself relied on DBSCAN, a density-based algorithm that groups points lying in sufficiently crowded neighborhoods while labeling isolated points as noise. Because the dataset was too large to process at once, it was split into 64 partitions of roughly 10,000 trajectories each. Within each partition, trajectories were grouped if their perilune sequences moved through the pq-plane along similar paths at similar times, judged by unit tangent vectors sampled every half day. Local clusters were then merged across partitions and across different initial phasings, using shape-based feature vectors and a graph structure whose connected components defined the final groupings. The outcome was striking: 638,181 trajectories collapsed into just 885 global clusters, a reduction of three orders of magnitude in the information an analyst must examine.

From selected clusters, the team manually extracted sets of long-lifetime orbits, defined as trajectories showing limited drift in the pq-plane over the full 180 days. These orbits appear at discrete combinations of initial semi-major axis and inclination, and in many cases, shifting the initial longitude of the ascending node rotates the entire groundtrack geometry without changing the underlying orbital evolution. At a semi-major axis of 1,808 kilometers and an inclination of 85 degrees, for example, three different nodal longitudes produced long-lifetime orbits whose perilune groundtracks circled the north pole, circled the south pole, or swept across both hemispheres, all while the eccentricity and argument of perilune evolved in nearly identical fashion. That flexibility matters for mission design, because the groundtrack determines which terrain a spacecraft observes.

The comparison with classical theory was largely reassuring, with intriguing exceptions. Most of the recovered long-lifetime orbits cluster around arguments of perilune near plus or minus 90 degrees or encircle zero eccentricity in the pq-plane, exactly as analytical frozen orbit theories predict. The overall shapes of the mean eccentricity curves as a function of inclination also match the families derived by Lara in 2011 and later refined by Yarndley and colleagues. But at initial inclinations of 65 and 115 degrees, the numerical search surfaced low-drift orbits with mean arguments of perilune as far as 30 to 60 degrees away from the predicted minus 90 degrees. When the authors extended two of these trajectories to nearly twenty years, some remained tightly bounded, hinting at frozen orbits that the averaged theories miss, while others drifted substantially under longer-period oscillations governed by multiple frequencies.

The method also recovered orbits that analytical approaches cannot touch by construction. Resonant trajectories, whose perilune sequences form looping paths in the pq-plane, are typically excluded from averaged derivations, yet the clustering search found them naturally at low inclinations. Several results echoed earlier numerical work: orbits at 95 degrees inclination showed oscillation periods of 27.43 to 27.54 days, close to the 27.35-day repeat groundtrack period reported by Russell and Lara in 2007, and near-equatorial long-lifetime orbits resembled those designed for the DARE radio astronomy concept by Plice and colleagues.

The authors are careful to frame the work as a proof of concept. The initial conditions were sampled coarsely, at a single epoch, and only over 180 days, so some families may be missing and some orbits may not persist indefinitely. Extracting representative orbits from clusters is still manual, and automating that step is ongoing work. Even so, the demonstration is compelling: a data-mining pipeline can compress a million brute-force simulations into a compact, interpretable summary that agrees with decades of theory while exposing solutions theory overlooked. As NASA and its partners plan sustained lunar operations under Artemis, tools like this could quietly become as important as the rockets, ensuring that the spacecraft placed around the Moon stay there without burning fuel they cannot spare.

Subject of Research: Numerical identification of long-lifetime low lunar orbits using trajectory clustering in a high-fidelity lunar gravity model

Article Title: Numerical exploration of low lunar, long-lifetime orbits

Article References: Bosanac, N., & Miceli, G. E. (2026). Numerical exploration of low lunar, long-lifetime orbits. Celestial Mechanics and Dynamical Astronomy, 138(5), Article 61. https://doi.org/10.1007/s10569-026-10334-x

Image Credits: AI Generated

DOI: 10.1007/s10569-026-10334-x

Keywords: lunar orbits, frozen orbits, trajectory clustering, celestial mechanics, lunar gravity field, mission design, DBSCAN, perilune, space situational awareness, astrodynamics, Lunar Reconnaissance Orbiter, ephemeris model