New Coordinate-Proof Yardstick Measures Asteroid Encounter Uncertainty

When a near-Earth asteroid swings past our planet, a small cloud of orbital uncertainty can balloon into something enormous. A well-constrained orbit determined from telescopic observations, propagated forward through a close planetary encounter, can stretch into a wildly elongated distribution of possible trajectories, making it genuinely difficult to say how well—or how badly—we know where the object will be. A new study published in Celestial Mechanics and Dynamical Astronomy by Giacomo Tommei of the University of Pisa tackles a deceptively simple question that has long lurked beneath the surface of asteroid impact monitoring: when we say an orbit prediction is uncertain, are we measuring something real about the asteroid, or just an artifact of the mathematical coordinates we happen to use?

The problem is more than academic pedantry. Standard tools in orbit determination rely on matrices—the state-transition matrix and the target-sensitivity matrix—that describe how small changes in an asteroid’s current position and velocity translate into changes at some future target time. Analysts often inspect the eigenvalues of these matrices to gauge how fast uncertainty grows. But as Tommei’s paper demonstrates, those raw eigenvalues depend on the coordinate system and the physical units chosen. Express positions in kilometers instead of astronomical units, or mix Cartesian coordinates with orbital elements, and the same physical situation produces different numbers. Worse, such eigenvalues conflate two distinct things: the intrinsic sensitivity of the dynamics to initial conditions, and the actual uncertainty that has been propagated from the observations.

The proposed solution is a diagnostic built from a comparison that anyone can state physically: take the covariance matrix of the orbit solution—the formal description of how the observational errors spread into state uncertainty—propagate it forward to the target quantity of interest, and compare it against a tolerance that the analyst explicitly declares. The scalar summary of that comparison is the largest generalized eigenvalue of the propagated covariance paired with the tolerance matrix. In plain terms, the diagnostic answers the question: relative to the accuracy I actually need, how badly has my knowledge of the target state degraded? Because both the covariance and the tolerance are stated in the same units and coordinates, their ratio is meaningful in a way that a bare eigenvalue never is.

The mathematical heart of the paper is an invariance theorem. Tommei proves that this scalar diagnostic, and the time at which it crosses its threshold, do not change under smooth reparametrizations of the state and the target quantity. Switch from Cartesian coordinates to Keplerian elements, from geocentric to heliocentric frames, from position to range and range-rate: within the local covariance model, the diagnostic reports the same number. This is the coordinate-invariance that the title advertises, and it elevates the diagnostic from a convenient heuristic to a genuine physical quantity, one that belongs to the encounter itself rather than to the analyst’s choice of bookkeeping.

The work also revisits a classical construct of impact monitoring: the Line of Variations, the curved path through orbit space that traces the spine of the uncertainty region and along which impact solutions are traditionally searched. Existing theory treats this line under assumptions of linear behavior and non-degenerate geometry. Tommei establishes a local theorem for the Line of Variations that incorporates the residual curvature of the least-squares objective—the second-order structure left over after the best-fit orbit is found—and the transverse non-degeneracy condition required when constrained differential corrections are applied. These refinements matter in exactly the situations that are hardest to handle: weakly determined orbits and encounters where the linear approximation strains at its limits.

Connecting the diagnostic to the physics of close encounters, the paper derives a lower bound on a finite post-encounter interval that links the target-uncertainty diagnostic to encounter stretching, the rapid separation of initially nearby trajectories under the planet’s gravitational perturbation. The bound combines coordinate-covariant state metrics with an explicit target-visibility condition—a criterion stating when the target quantity can actually be resolved from the state at all. When visibility fails, no amount of propagation refinement can recover the target information, and the diagnostic flags the breakdown. A controlled nonlinear map, deliberately constructed for the purpose, verifies all three failure modes the theory warns about: violations of invariance, failure of target visibility, and the collapse of first-order covariance propagation when nonlinearity dominates.

To show the diagnostic working on real data, Tommei turns to the most closely watched close approach in modern astronomy: the April 2029 encounter of asteroid 99942 Apophis with Earth. Apophis, roughly 340 meters across, will pass within about 31,000 kilometers of Earth’s surface—closer than many geostationary satellites—an event that has been tracked with extraordinary precision since its discovery in 2004. The benchmark uses published numerical target-uncertainty values from earlier work on Apophis’s non-zero Yarkovsky acceleration, the tiny thermal recoil force that subtly alters an asteroid’s orbit over years, together with data from NASA/JPL’s Small-Body Database Close-Approach API. No raw astrometry is reprocessed; the point is to demonstrate how the diagnostic responds as observations accumulate and as the analyst tightens or relaxes the stated tolerance.

The Apophis exercise yields a nuanced picture. As observational refinement shrinks the covariance, the diagnostic falls, quantifying exactly how much better the 2029 encounter is known than earlier predictions were. As the tolerance selection changes, the diagnostic rescales accordingly, making explicit a choice that is usually left implicit in uncertainty reporting. Crucially, Tommei is careful about what the diagnostic is not: it is a normalized summary of the target covariance, a single number that condenses a complicated situation, not a replacement for the b-plane analysis—the standard target-plane formalism used to compute impact probabilities—or for full nonlinear sampling of the uncertainty region. The diagnostic tells you when to worry and how much; it does not by itself tell you whether an impact will occur.

The paper closes with practical guidance that should make the method usable in operational settings. Threshold calibration is discussed, addressing how analysts should choose the tolerance that gives the diagnostic its meaning. The growth-bound constants appearing in the finite-interval lower bound are examined for practical evaluation, and invariant alternatives based on the trace and determinant of the covariance–tolerance pair are offered for situations where a single largest eigenvalue is too coarse a summary. Reproducibility is built in: the scripts that generate the paper’s figures and tables, along with all derived numerical values, are supplied as supplementary online material, and the synthetic data are produced by the supplied script itself.

The broader significance reaches beyond a single asteroid. Impact monitoring systems such as ESA’s Aegis services at the NEO Coordination Centre and NASA’s Sentry computations continuously propagate thousands of near-Earth objects and must decide, automatically, which encounters deserve human scrutiny. A coordinate-invariant, physically interpretable measure of target uncertainty offers such systems a principled ranking criterion—one that does not shift when software, coordinates, or units change, and that flags when linear covariance propagation can no longer be trusted. The work draws on deep connections to information geometry, optimal experimental design, and the theory of Lagrangian coherent structures in fluid dynamics, fields that have independently grappled with how to measure stretching and information loss in ways that survive changes of description. For a discipline whose highest-stakes judgments—impact probabilities, keyhole passages, resonant returns—rest on knowing exactly how well an orbit is known, a yardstick that measures the asteroid rather than the mathematics is a quietly powerful addition to the toolkit. The research was supported by the Italian Space Agency through the Monitoring Asteroids project, and the paper’s single-author provenance, from conception through final verification, underscores the individual mathematical labor behind what appears, on the surface, to be a modest scalar. Sometimes the most useful number is the one that stays the same no matter how you look at it.

Subject of Research: A coordinate-invariant measure of target uncertainty propagation for near-Earth asteroid close encounters

Article Title: Coordinate-invariant target-uncertainty diagnostic for near-earth object encounters

Article References: Tommei, G. (2026). Coordinate-invariant target-uncertainty diagnostic for near-earth object encounters. Celestial Mechanics and Dynamical Astronomy, 138(5), Article 62. https://doi.org/10.1007/s10569-026-10335-w

Image Credits: AI Generated

DOI: 10.1007/s10569-026-10335-w

Keywords: near-Earth objects, orbit determination, close encounters, uncertainty propagation, covariance, Line of Variations, Apophis, impact monitoring, generalized eigenvalues, celestial mechanics, Yarkovsky effect, invariance