New Orbital Coordinates Could Make Spacecraft Tracking Near the Moon Far More Reliable

As space agencies and private companies prepare to establish a lasting human presence around the Moon, a quiet mathematical problem has become one of the most pressing challenges in modern astrodynamics: how to keep track of spacecraft in the vast, chaotic region of space between Earth and the Moon. A new study published in the journal Celestial Mechanics and Dynamical Astronomy by Maaninee Gupta and Kyle J. DeMars of Purdue University offers a promising answer. Their research demonstrates that a specialized set of orbital coordinates, known as the Modified Generalized Equinoctial Orbital Elements, or M-GEqOEs, can dramatically improve how uncertainty in a spacecraft’s position and velocity is predicted in cislunar space, the spherical volume enclosed by the Moon’s orbit around Earth.

The stakes are considerable. Space Domain Awareness, the practice of observing, tracking, and maintaining custody of objects in orbit, has historically focused on the region near Earth, where ground-based radars and telescopes can readily monitor satellites. Cislunar space is a different beast entirely. The region is enormous compared to the near-Earth regime, and objects within it are governed by the competing gravitational pulls of both Earth and the Moon, with the Sun adding further complications. Small errors in a spacecraft’s estimated state can balloon into vastly different predicted trajectories, making it extraordinarily difficult to know where a satellite was, is, or will be.

At the heart of the problem lies an assumption that underpins much of modern space tracking: that uncertainty in a spacecraft’s state can be described by a Gaussian, or bell-curve, probability distribution. Near Earth, this assumption works reasonably well. In cislunar space, however, the dynamics are so nonlinear that Gaussian distributions quickly distort into curved, elongated, and multi-lobed shapes that no single bell curve can capture. When navigation and risk-assessment systems rely on Gaussian assumptions that no longer hold, errors creep into catalog maintenance, collision avoidance, and maneuver detection, potentially compromising the safety of missions such as NASA’s Gateway lunar outpost.

Gupta and DeMars approached this challenge by changing the language in which spacecraft motion is described. Conventional tracking uses Cartesian coordinates, essentially position and velocity in three-dimensional space. The alternative is to use orbital elements, quantities that describe the size, shape, and orientation of an orbit. The researchers built upon the Generalized Equinoctial Orbital Elements introduced by Baù and colleagues, which are characterized by the total energy of the orbiting system. This energy-based formulation allows perturbing forces, whether conservative forces derived from gravitational potentials, such as lunar and solar gravity, or non-conservative forces, to be embedded directly into the equations that evolve the orbital elements, rather than being bolted on afterward through complicated variational equations.

The modification at the core of the new work overcomes a fundamental limitation of the original generalized elements, which are only valid for trajectories with negative total energy. By replacing the semi-major axis with the generalized semi-latus rectum and the mean longitude with the true longitude, the M-GEqOE set remains robust across the wider variety of trajectories found in cislunar space. The six elements include the generalized semi-latus rectum, two parameters describing a generalized eccentricity vector, two parameters orienting the orbital plane, and the true longitude, which varies rapidly along the orbit. Together, they describe a non-osculating ellipse that carries the imprint of the perturbing environment within its very definition.

To test the approach under realistic conditions, the researchers propagated trajectories under a high-fidelity model incorporating the gravitational influences of Earth, Moon, and Sun, using planetary ephemerides from NASA’s Jet Propulsion Laboratory. One subtle numerical obstacle had to be overcome: the generalized angular momentum, which depends on an effective potential energy, must remain nonnegative, yet the third-body gravitational potential can push this quantity below zero even when the underlying physics remains perfectly valid. The team’s solution was to apply a constant offset to the potential, computed once before propagation, which shifts the reference level of potential energy without altering the equations of motion, since those depend only on the gradient of the potential.

The validation covered four representative orbits spanning the cislunar domain. The first was the 9:2 Near Rectilinear Halo Orbit, the baseline orbit for NASA’s Gateway station, where the M-GEqOE propagation over roughly six and a half days matched a conventional Cartesian N-body ephemeris model closely, with errors remaining bounded apart from a modest increase near the close perilune passage. The second was an Elliptical Lunar Frozen Orbit, a highly inclined, eccentric lunar orbit suited to long-term coverage of the lunar south pole and potential data-relay constellations, propagated over thirty days. The third and fourth were Earth-centered sidereal resonant orbits from the 4:1 and 3:1 families, the latter resembling the orbit used by NASA’s Interstellar Boundary Explorer during its extended mission, each requiring roughly twenty-seven days per revolution under combined Earth-Moon-Sun dynamics.

The most striking results concerned uncertainty. The researchers ran Monte Carlo simulations with ten thousand particles for each scenario, propagating identical dynamics in both M-GEqOE and Cartesian coordinates so that any differences in the resulting uncertainty distributions arose purely from the choice of coordinates. To assess Gaussianity rigorously, they employed the Henze-Zirkler test, a statistical test for multivariate normality that evaluates consistency with a Gaussian distribution directly from sample statistics, without requiring an additional density approximation. This choice matters because alternative measures, such as Kullback-Leibler divergence, depend on assumptions about the approximating distribution that can obscure whether observed deviations are intrinsic or artifacts of the approximation itself.

Across all four orbits, the pattern was consistent: uncertainty propagated in M-GEqOE coordinates preserved Gaussian behavior significantly longer than uncertainty propagated in Cartesian coordinates. Along the Near Rectilinear Halo Orbit, the Cartesian representation showed a sharp departure from Gaussianity at perilune passage, where nonlinearities are strongest, while the M-GEqOE representation maintained Gaussian behavior throughout the entire orbit. For the 4:1 resonant orbit, Gaussianity survived until the third perigee crossing at the fourteen-day mark in generalized coordinates, compared with failure at the very next perigee pass in Cartesian coordinates. Even where both representations eventually departed from Gaussianity, as on the 3:1 resonant orbit, the deviations were markedly smaller and recovered more quickly in the generalized elements. Notably, when the researchers visualized the uncertainty clouds in the eigenspace of the covariance matrix, the curvature and tail-like structures betraying non-Gaussian structure were clearly visible in the Cartesian projections, while the M-GEqOE ensembles retained smooth, Gaussian-like shapes.

The implications extend well beyond a single study. Accurate uncertainty propagation is the foundation upon which estimation, tracking, and data association are built, and the authors emphasize that by reducing departures from Gaussianity, the M-GEqOEs may also simplify non-Gaussian uncertainty representations, for example by requiring fewer components in a Gaussian mixture filter. As traffic in cislunar space grows with planned lunar outposts, commercial missions, and scientific observatories, the ability to predict where spacecraft are and how confident we can be in those predictions will determine whether operations in this new frontier remain safe and coordinated. What Gupta and DeMars have shown is that sometimes the most powerful tool for taming a chaotic environment is not a bigger computer or a more elaborate force model, but a smarter choice of coordinates, one that lets the physics of the three-body problem live inside the mathematics from the very start.

Subject of Research: Cislunar spacecraft state and uncertainty propagation using modified generalized equinoctial orbital elements

Article Title: Cislunar state and uncertainty propagation via the modified generalized equinoctial orbital elements

Article References: Gupta, M., & DeMars, K. J. (2026). Cislunar state and uncertainty propagation via the modified generalized equinoctial orbital elements. Celestial Mechanics and Dynamical Astronomy, 138(5), Article 59. https://doi.org/10.1007/s10569-026-10333-y

Image Credits: AI Generated

DOI: 10.1007/s10569-026-10333-y

Keywords: cislunar space, orbital mechanics, uncertainty propagation, equinoctial orbital elements, Space Domain Awareness, spacecraft navigation, Henze-Zirkler test, Monte Carlo simulation, three-body problem, Gateway lunar outpost, celestial mechanics, Gaussian distributions