The new work, published in the journal Celestial Mechanics and Dynamical Astronomy, comes from Rita Mastroianni of the European Space Agency’s Advanced Concepts Team, Edoardo Legnaro of the University of Genoa, and Christos Efthymiopoulos of the University of Padova and Aristotle University of Thessaloniki. Their paper, “Fully analytical propagator for lunar satellite orbits in closed form,” describes a theory capable of tracking the long-term motion of artificial satellites around the Moon with an accuracy sufficient for many practical applications, and it does so through the elegant machinery of Hamiltonian perturbation theory.

At the heart of the method is a carefully constructed physical model of the forces acting on a lunar satellite. The team incorporated the twelve most important harmonics of the lunar gravity field, the terms in the mathematical expansion of the Moon’s gravitational potential that describe its deviations from a perfect sphere. These harmonics capture the mass concentrations, or “mascons,” and the pronounced flattening and irregularity of the lunar body, all of which dominate the way a low lunar orbit drifts and deforms. On top of this they added the tidal perturbation caused by Earth’s quadrupole, computed using a precise representation of the Earth’s lunicentric ephemeris, meaning the exact position of our planet as seen from the Moon at every moment of the satellite’s life.

Numerical tests show that this moderately sized model is enough. When compared against propagation using high-order gravity models derived from NASA’s GRAIL mission, which mapped the Moon’s gravity field in exquisite detail, the analytical theory achieves satisfactory precision for all trajectories at altitudes ranging from roughly 300 kilometres to 3000 kilometres above the lunar surface, as well as for frozen orbits at lower altitudes. Frozen orbits are those special trajectories whose average orbital elements remain essentially constant over long periods, making them prized targets for science and navigation missions that need stable, predictable paths. The framework is also deliberately open-ended: additional gravitational terms can be incorporated in a straightforward manner whenever a mission’s requirements demand higher fidelity.

The mathematics underpinning the new propagator is a “closed form” solution of the secular equations of motion obtained through a Hamiltonian normal form approach. In plain terms, instead of tracking the rapid, orbit-by-orbit oscillations of a satellite’s position, the theory first distils the dynamics down to its slow, long-term trends, and then solves those trends exactly, using symbolic transformations rather than numerical stepping. The procedure employs two successive canonical transformations, each together with its inverse. The first converts osculating orbital elements, the instantaneous Keplerian elements that oscillate rapidly due to perturbations, into mean orbital elements, which smooth away the short-period wiggles. The second converts mean elements into proper elements, a concept borrowed from asteroid dynamics, which are quasi-constants of the motion, quantities that remain nearly fixed over decades even as the actual orbit evolves.

Proper elements deserve particular attention because they are the true secret of the method’s longevity. Once the proper elements of a lunar satellite are computed from its initial conditions, the entire future and past evolution of its mean orbit can be recovered analytically, because the slow secular frequencies of the motion follow directly from the transformed Hamiltonian. The theory then works backwards through the two transformations, from proper to mean elements and from mean to osculating elements, finally reconstructing the satellite’s Cartesian position and velocity in the lunicentric reference frame, known as PALRF, for any time t. No intermediate numerical propagation of the initial conditions is required at any stage. The answer comes out of the equations themselves.

The practical payoff is enormous. A closed-form propagator can be evaluated in a fraction of the time needed for high-fidelity Cartesian integration, and it can be embedded in onboard autonomy software, preliminary mission design tools and Monte Carlo studies that need millions of trajectory evaluations. It also gives mission designers something numerical integrators cannot easily provide: understanding. Because the secular solution is analytical, the connection between a satellite’s initial orbit and its long-term fate is visible in the structure of the formulas, revealing which orbits remain stable, which drift slowly, and which are doomed to impact the lunar surface.

That last category matters. The authors show that the propagator is valid over timescales of several decades for all non-impacting orbits, with one important exception: narrow regions associated with specific secular resonances, where slow commensurabilities between the satellite’s orbital motion and the perturbing frequencies of the Earth and the lunar gravity field can drive dramatic changes, most notably growth in orbital eccentricity that can send an otherwise stable satellite crashing into the Moon. Earlier work by Legnaro and Efthymiopoulos had mapped these inclination-dependent lunisolar resonances and studied the natural lifetimes of lunar satellites, and the new closed-form theory inherits that insight, flagging precisely where its own validity must yield to more careful treatment.

To quantify the accuracy of the method, the team performed extensive numerical comparisons against full Cartesian propagation using the complete force model. These tests assess how the analytical solution degrades over time and across orbital regimes, and the results indicate that for the vast majority of the targeted altitude range, the agreement remains well within the tolerances needed for practical orbit prediction and mission analysis. The precision achieved against GRAIL-based gravity models is particularly significant, because it demonstrates that a compact, analytically tractable model of just twelve harmonics plus the Earth’s tidal quadrupole is not a crude approximation but a genuinely competitive tool.

The lineage of this achievement stretches back decades. Analytical satellite theories date to Dirk Brouwer’s classic 1959 solution for artificial Earth satellites, and semi-analytic theories for lunar satellites were explored as early as 1970 by Giacaglia and collaborators. More recent efforts by De Saedeleer, Lara, Ferrer and others pushed lunar analytic theory further, but always with limitations in the gravity field model or the treatment of third-body perturbations. The new propagator represents a step change: a modern normal-form construction, validated against the best gravity data available, valid for decades, and implemented in publicly available software. The authors have released their code, AnalyticCisProp, on GitHub, making the technology immediately accessible to the astrodynamic community.

The timing could hardly be better. A renewed wave of lunar exploration is under way, with crewed missions, cargo landers, communication and navigation constellations, and science orbiters all planned for the coming decade. Every one of these endeavours needs orbit prediction, station-keeping analysis and collision-avoidance assessment around the Moon, where there is no atmosphere to provide natural decay and no global tracking network comparable to Earth’s. A tool that computes decades of orbital evolution from closed-form expressions, and that can quantify, analytically, the lifetime of a satellite before it impacts the surface, speaks directly to the needs of that new lunar economy.

There is also a deeper scientific appeal. The method revives a grand tradition in celestial mechanics, the dream of solving orbital motion by mathematics rather than by machine, and shows that with the right modern techniques, Hamiltonian normal forms, proper elements and careful model reduction, that dream remains alive and useful even in the messy gravitational environment of the Moon. For a problem once thought to demand brute-force computation, the elegant answer, it turns out, can be written in closed form.