Together with Ivana Milić Žitnik I’m happy to share our new paper in Icarus (Volume 455, 117101, 2026) — a systematic observational census of von Zeipel–Lidov–Kozai (ZLK) resonances among trans-Neptunian objects. It is also the first study, to our knowledge, where the long-standing theoretical link between ZLK resonances and mean-motion resonances (MMRs) is confirmed empirically across an entire population.
Why ZLK matters
The ZLK mechanism is a long-period dynamical phenomenon in hierarchical three-body systems: in essence, the gravitational influence of a distant perturber drives coupled oscillations between the inclination and eccentricity of an orbit, with the argument of perihelion librating around 90° or 270° (or some other shifted center). For trans-Neptunian objects, this matters because ZLK can re-shape the perihelion distribution of highly eccentric, inclined bodies on timescales of millions of years — and yet, prior to this work, nobody had actually counted how many TNOs are caught in this regime.
What we did
We took all 1,037 numbered TNOs from the AstDyS catalog with semimajor axis a > 30 au, and integrated each of them numerically — first for 10 Myr, then progressively up to 300 Myr for the borderline cases. The integrations were done using the SABA(10,6,4) integrator inside the rebound engine, called from my open-source resonances package, which we extended for this study to handle a new resonance type — ZLK.
Each object was then classified into one of five categories based on the time evolution of the resonant angle: pure ZLK resonant, transient, non-resonant, chaotic, or controversial. The ambiguous cases were re-integrated and then visually inspected — a step where, ironically, our own LLM benchmark for resonance classification becomes directly relevant.
Main results
- 81 TNOs (≈ 7.8% of the sample) are trapped in pure ZLK resonance. Another 23 (≈ 2.2%) show transient behaviour — alternating libration and circulation.
- All ZLK-resonant TNOs are simultaneously trapped in a two-body MMR with Neptune. This is the empirical confirmation of theoretical predictions going back to Gallardo (2006) and Gomes et al. (2005): ZLK dynamics, for TNOs, lives inside mean-motion resonances rather than independently of them.
- The four leading MMRs hosting ZLK objects are 2N-3 (58 objects), 4N-7 (15), 1N-2 (9), and 3N-5 (7). For non-1N-2 resonances, ZLK libration centers cluster near 90° and 270°, as the classical theory predicts. The 1N-2 case is different — the centers are shifted, sitting near 120°, 150°, 300°, and 330°.
- Several unusual objects show up in the ZLK regime, including the retrograde TNO (585899) 2020 HM₉₈ in the 2N+9 resonance and the very distant (652920) 2014 GR₅₃ in the high-order 1N-18 MMR. The latter is interesting because it confirms that high-order and 1:N type resonances can still maintain ZLK dynamics at large heliocentric distances — something that is far from obvious from the theory alone.
Why this is interesting beyond TNOs
Two points, I think.
First, the result shows that the trans-Neptunian region is even more “structured” than we already suspected. After our 2025 Icarus paper showed that roughly half of TNOs are in some kind of mean-motion resonance, this study adds another layer: a meaningful sub-population is doing two things at once — locked into an MMR and librating in ZLK. The dynamics there are richer than the canonical picture of “Kuiper-belt objects on near-circular orbits” suggests.
Second, this is a methodological proof of concept for population-scale resonance studies. The combination of a clean integrator, a unified open-source pipeline, and disciplined classification can produce statistically meaningful catalogues — not just case studies. The same machinery is now ready to be applied to other classes of resonances and other regions of the Solar System.
The full paper is available at DOI: 10.1016/j.icarus.2026.117101. The code and the extended resonances package are on GitHub.