Together with Ivana Milić Žitnik I’m happy to share our new paper in Astronomy & Astrophysics: a complete census of numbered asteroids inside Jupiter’s orbit that are currently trapped in the von Zeipel-Lidov-Kozai (ZLK) resonance. It is the direct sequel to our Icarus study of trans-Neptunian objects, and the honest summary is that the picture inverts once you cross inside 5 au.
The question we started with
In the trans-Neptunian region we found something remarkably clean: 81 TNOs are in the ZLK resonance, and every single one of them is simultaneously trapped in a two-body mean-motion resonance (MMR) with Neptune. That is a strong statement, and a strong statement invites the obvious follow-up question. Is this a universal property of ZLK dynamics in the Solar System, or a local feature of the Kuiper belt?
The only way to answer it was to repeat the exercise on the other side of the Solar System, and to do it exhaustively rather than on a sample.
What we did
We integrated the orbits of all 863,471 numbered asteroids with semimajor axis a < 4.9 au from the AstDyS catalogue. The model was a full N-body one (the Sun, the eight planets and Pluto), integrated with the IAS15 adaptive-step integrator inside rebound, driven by the open-source resonances package.
The resonant angle here is simply the argument of pericentre: σ ≡ ω. The work was split into three integration phases: 100 kyr automated screening for the whole catalogue, 500 kyr manual re-examination of the candidates, and a further 100 kyr run to identify mean-motion resonances for every candidate. Each object was classified as regular, irregular, transient, slow-circulating, chaotic or nonresonant. For the candidates we then searched two-body MMRs up to order 50 and three-body MMRs up to order 10.
Main results
- 201 asteroids, which is less than 0.02% of the sample, are currently in the ZLK resonance. 163 of them librate cleanly with a stable period and amplitude (regular), 38 formally librate but with evolving parameters (irregular). A further 26,309 objects land in the broader ‘possible’ category, and that category is a research topic on its own.
- The population is planet-crossing. 144 objects have perihelion q < 1.3 au (near-Earth), 28 are Mars-crossers with 1.3 ≤ q < 1.66 au. The main belt is almost empty.
- The MMR coupling does not survive the trip inwards. Only 11 of the 201 confirmed librators are in a two-body or three-body MMR. Against 100% in the trans-Neptunian region, this is not a weaker version of the same effect, it looks like a decoupling.
- The classical critical inclination is not a wall. About one third of the confirmed objects sit below the quadrupole value of ∼39°, and several librators reach c₁ = (1 − e²) cos² i > 3/5, which the textbook quadrupole picture does not allow.
- Libration centres cluster on 90° and 270°, as the classical theory predicts.
Identification is harder than it looks
One methodological point deserves its own paragraph, because it cost us the most time. Inside Jupiter’s orbit there are false positives, and they are convincing ones. The resonant angle librates, the eccentricity and the inclination are anti-correlated, everything looks like a textbook ZLK case, and yet physically the object is outside the resonance. The Jupiter trojan 591986 is our working example.
So a single libration test is not enough. We had to add a small toolbox around it: the cross-periodogram of e against i, the eccentricity vector, the filtered free ω against the osculating one on both short and long windows, and the phase portrait of the filtered σ. Only the combination of these gives a verdict one can defend.
Why this matters
Two things, I think.
First, the trans-Neptunian result should not be generalised. ZLK dynamics beyond Neptune lives inside the mean-motion resonance network; inside 5 au it does not. Our reading is that the sheer density of overlapping resonances in the main belt, mean-motion and secular alike, suppresses long-lived ZLK libration rather than protecting it. The Kuiper belt is the sparse, orderly place here, and the inner Solar System is the noisy one.
Second, this is now a complete catalogue rather than a set of case studies. 201 objects with measured libration centres and periods is a starting point for asking why these particular asteroids, and what happens to the 26,309 ‘possible’ ones when they are looked at properly. That is the next piece of work.
The full paper is available at DOI: 10.1051/0004-6361/202660737. The resonances package, extended for this study, is on github. I also presented both papers together at AMCM 2026 in St Petersburg, and the deck is there if you prefer slides to prose.