New ML-FAIR framework rapidly detects eccentric mean-motion resonances

Astronomers have unveiled a machine learning framework that can hunt for gravitational resonances among near-Earth asteroids without the painstaking manual analysis that such searches have always required. The new tool, called ML-FAIR, combines a clever geometric technique for spotting resonant orbits with unsupervised clustering algorithms, and in its first major test it automatically classified more than 85 percent of nearly 2,900 asteroid cases—possibly identifying the first asteroids ever caught in resonant lockstep with the planet Mercury.

The work, published in the journal Celestial Mechanics and Dynamical Astronomy by a team led by Valerio Carruba of São Paulo State University in Brazil, addresses a problem that has grown steadily more pressing as asteroid catalogs balloon in size. Mean-motion resonances occur when the orbital period of an asteroid forms a simple integer ratio with that of a planet. When two bodies are commensurable in this way, their repeated gravitational tugs accumulate coherently rather than canceling out, and the asteroid’s orbit responds with periodic oscillations. These resonances can protect asteroids in stable niches, or, more commonly, pump up their orbital eccentricity until they are flung onto planet-crossing trajectories. Understanding where and how many near-Earth objects are trapped in resonances is therefore essential for modeling how these bodies evolve—and for assessing which of them might eventually pose an impact hazard.

The traditional way to identify a resonance is to compute a resonant argument, an angle built from combinations of the asteroid’s and planet’s orbital longitudes, and then check whether that angle librates, oscillating around an equilibrium value such as 0 or 180 degrees, rather than circulating freely through the full 360-degree range. The difficulty is sheer numbers. In the near-Earth region alone, restricted to semi-major axes below 4 astronomical units and resonances up to order 20, there are 225 possible mean-motion resonances with Mercury, 177 with Venus, 127 with Earth, and 66 with Mars. Even focusing on the main argument of each resonance leaves roughly 595 cases to check in the region probed by the new study. Manually inspecting hundreds of time series for thousands of asteroids is simply not a scalable strategy, particularly as the Vera C. Rubin Observatory’s Legacy Survey of Space and Time is expected to multiply near-Earth object detections fivefold.

The foundation for the new approach was laid in 2018, when researchers introduced the FAst Identification of mean-motion Resonances (FAIR) method, which sidesteps resonant arguments entirely by exploiting geometry. When the angles describing a resonant orbit are plotted against one another—in practice, the difference in mean longitudes between asteroid and planet is plotted against the asteroid’s mean anomaly—a resonant object does not fill the plot uniformly the way a non-resonant one does. Instead, its points align along oblique stripes. The elegance of the method lies in what those stripes encode: the number of times they intersect the horizontal axis equals the resonance order q, while the number of intersections with the vertical axis yields p or p+q, the integers that define the resonance ratio. For a given planet, only two such plots are needed to screen for every possible resonance. An asteroid inside the 3E:4 resonance with Earth, for example, produces one horizontal crossing and four vertical ones, immediately revealing the 3:4 commensurability without any computation of resonant angles.

What remained laborious in the original FAIR procedure was the counting itself, which had been done largely by eye. The Brazilian-led team automated it by reframing the problem as one of detecting peaks in periodic, one-dimensional distributions. They defined two variables that collect the angles at which the stripes cross the axes—values of the resonant angle at mean anomaly zero, and values of mean anomaly where the resonant angle is zero—and then applied unsupervised machine learning to find statistically significant concentrations in those circular datasets. A first stage uses angular coverage statistics to weed out circulating, non-resonant orbits: for objects on circulating paths, the sampled angles blanket nearly the entire circle, with the 25th percentile of coverage exceeding 94.6 percent in the team’s calibration sample, so any asteroid above that cutoff is classified as non-resonant outright.

For the remaining candidates, the team tested several density-based clustering algorithms on orbits whose stripe counts had been established by visual inspection. The clear winners were OPTICS—an algorithm that orders points by reachability to expose clusters at varying densities—DBSCAN, and circular kernel density estimation with peak detection. After optimizing hyperparameters through a rigorous nested stratified cross-validation with five outer folds repeated ten times, the best performance came for low-order resonances with fewer than five stripe crossings, where mean accuracy and F1 scores all exceeded 0.885. Accuracy dropped for higher-order resonances with five to ten crossings, and both methods stumbled on switching orbits that drift in and out of resonance. Crucially, the two algorithms tended to fail on different cases, so the researchers built an ensemble: when KDE and OPTICS agree, the stripe counts are accepted automatically; when they disagree, the case is flagged for a human in the loop. A final consistency check uses Kepler’s third law to verify that the inferred resonance actually sits near the asteroid’s semi-major axis, rejecting mismatches larger than 0.4 astronomical units.

The team unleashed the pipeline on the Atira and Aten populations—asteroids orbiting wholly or mostly inside Earth’s own orbit—and integrated their motions under the influence of all the planets and the Moon for 1,200 years using the REBOUND N-body code with a time step of 2.6 hours. That integration length, long enough to capture resonant libration but limited by the chaos inherent in these orbits, whose typical Lyapunov times are around 150 years, produced more than five million sampling points per asteroid. Across 2,882 analyzed cases, ML-FAIR automatically screened more than 85 percent, leaving only a few hundred images for human review—and those, tellingly, were dominated by morphologically complex, high-order resonances with many stripe intersections.

The most striking result concerns Mercury. No known asteroid orbits entirely within Mercury’s path, so all resonances with the innermost planet are of the external type. ML-FAIR flagged 171 candidate asteroids, concentrated mainly in resonances up to order 11, and follow-up analysis of the resonant arguments confirmed two genuine librators: asteroid 466507 (2014 FK33), oscillating around the 270-degree equilibrium point of the 1Mer:4 resonance, and 2006 SE6, similarly trapped at 270 degrees in the 1Mer:3 resonance. Libration around 270 degrees is a rare and unusual configuration, and if confirmed by longer integrations, these two objects would be the first asteroids ever identified in mean-motion resonances with Mercury.

For Venus, Earth, and Mars, the team benchmarked ML-FAIR against an independent, long-timescale method that integrates orbits for 100,000 years and applies statistical tests to resonant angle time series and periodograms. The agreement was qualitatively strong. Among Venus crossers, ML-FAIR identified 141 candidate resonances, nine of them with populations of ten or more asteroids, and every heavily populated resonance found by the benchmark method—such as the 2V:3, 3V:4, and 4V:5—was also recovered by ML-FAIR, with 21 of the 25 resonances identified by the benchmark approach also appearing in the new analysis. For Earth, the most populated resonances were the 4E:5 and 6E:7 with 17 candidates each, followed by the 7E:8, 9E:10, and 5E:6, all clustered close to Earth’s orbit. For Mars, the 1M:2 resonance dominated with 111 candidates, though most proved to be on switching or circulating orbits, with eleven confirmed librators—a figure matching the benchmark method’s count.

The two approaches did not agree perfectly, which the researchers attribute to the fundamentally chaotic nature of Aten and Atira orbits and the very different integration timescales involved. A handful of high-order Mars resonances found by the benchmark method were missed by ML-FAIR, and the 3M:7 and 2M:5 resonances stood out as the main cases of disagreement among well-populated resonances. The authors emphasize that ML-FAIR’s automation rate is a measure of efficiency rather than a guarantee of correctness: every candidate still passes through resonant-argument verification before being accepted as a genuine resonant object. The method also deliberately excludes co-orbital 1:1 configurations, which produce a distinctive single-band morphology requiring separate treatment.

The significance of the work extends well beyond the roughly 2,900 asteroids studied here. The inner solar system region occupied by Atiras and Atens is a dense “forest” of weak resonances where nearly a thousand distinct commensurabilities compete, and it is precisely the kind of environment where manual methods collapse under their own complexity. As next-generation surveys deliver high-quality orbits for millions of small bodies—including vast numbers of small near-Earth objects relevant to the longstanding congressional goal of cataloging 90 percent of potentially hazardous asteroids larger than 140 meters—fully automated dynamical classification will shift from convenience to necessity. The ML-FAIR package has been released on GitHub, and the authors note that extending the framework to inclination-type resonances and to resonances involving the planet’s own perihelion longitude is a natural next step. For now, the study demonstrates that a decades-old geometric insight, married to modern density-based clustering, can convert one of celestial mechanics’ most tedious classification chores into a fast, scalable pattern-recognition problem.