ZLK resonances and their coupling with mean-motion resonances: from the Kuiper belt to the inner Solar System

Posted on:August 18, 2026 at 12:00 PM

This is my talk for AMCM 2026, the Analytical Methods of Celestial Mechanics conference at the Euler International Mathematical Institute in St Petersburg. The work is joint with Ivana Milić Žitnik (Belgrade Astronomical Observatory) and covers our two 2026 papers at once, the Icarus one on trans-Neptunian objects and the A&A one on asteroids inside Jupiter’s orbit.

The talk in brief

Here are the theses, following the deck slide by slide.

The mechanism. The von Zeipel-Lidov-Kozai (ZLK) mechanism is a dynamical phenomenon in hierarchical three-body systems, where a distant perturber drives significant coupled oscillations of the eccentricity and the inclination. Throughout the talk the resonant angle is the argument of pericentre: σ ≡ ω.

Six behaviours of the resonant angle. Libration is rarely textbook-clean. Over 500 kyr one sees clear libration, changing behaviour, long apocentric libration, drift and capture, unclear behaviour, and increasing drift. This is why classification cannot be fully automated, and why a good part of the work is visual inspection.

Part I, the trans-Neptunian region. 81 of 1,037 numbered TNOs are in the ZLK resonance, roughly 7.8 per cent. And 100% of those 81 are also in mean-motion resonances (MMRs) with Neptune.

A highly resonant region. One object, several resonant angles. The TNO 523695 shows multiple captures in MMRs, including high-order and high-integer ones, with capture, release and re-capture inside a single integration.

Dynamically interesting objects. Four of them are worth naming.

Part II, inside Jupiter’s orbit. The picture inverts inside 5 au. 201 of 863,471 numbered asteroids are in the ZLK resonance, less than 0.02 per cent, and only 11 of the 201 are also in an MMR. 144 are near-Earth (q < 1.3 au), 28 are Mars-crossers (1.3 ≤ q < 1.66 au), and another 26,309 fall into the ‘possible’ category.

Challenges with identification. Unlike in the TNO region, inside Jupiter’s orbit there are false positives. The resonant angle librates, e and i are anti-correlated, and yet physically the object is outside the resonance. The trojan 591986 is the case in point.

Extra tools. Beyond a single libration test, on asteroid 3040 and then on the borderline trojan 591986.

Librators inside 5 au. Coloured by inclination, with the ZLK cycle period against the semi-major axis, and libration centres onto 90° and 270°.

Conclusions. Four of them.

  1. All TNOs trapped in ZLK are also trapped in two-body MMRs with Neptune. For 2N-3, 4N-7, 3N-5 and 4N-5 the libration centres are near 90° and 270°, as expected; for 1N-2 they shift to approximately 120°, 150°, 300° and 330°.
  2. 1N-X resonances are indeed strong: there is a TNO, (652920) 2014 GR₅₃, trapped in 1N-18.
  3. ZLK resonances within 5 au behave differently from ZLK capture in the Kuiper belt. It is an overwhelmingly planet-crossing phenomenon, decoupled from the MMR network, and it extends below the classical critical inclination.
  4. The overall number of librators is much lower than in the TNO region. The interplay and the overlapping of different resonances in the main belt appears to suppress long-lived ZLK libration.

Future directions. Where this goes next.

  1. The ‘possible’ category and the false positives. Secular resonances? And a universal algorithm to decide.
  2. Highly inclined, low-eccentricity objects: outside MMRs and not planet-crossers.
  3. The ‘interesting’ asteroids, one by one.
  4. Objects with T_J < 3.

If you were in the room, thank you for coming. If you were not, the deck plus these theses should carry the argument, and the two papers below have the full story.

References

Evgeny Smirnov — researcher, entrepreneur, and software developer based in Barcelona.More about me