Explain stable commensurabilities without resonant-angle libration

Determine why some circumbinary planetary systems exhibit stable orbital-period commensurabilities without libration of the associated resonant angles.

Background

The study identifies mean-motion resonances by verifying libration of resonant angles rather than relying solely on near-integer period ratios. In the sequential-migration simulations, some systems retain stable planetary architectures with orbital period commensurabilities even though no corresponding resonant-angle libration is detected. This challenges a straightforward interpretation of such commensurabilities as genuine mean-motion resonances and leaves their dynamical origin unresolved.

References

This highlights a phenomenon that remains difficult to explain: some systems exhibit stable architectures with orbital period commensurabilities, yet without libration of the resonant angles (15 simulations in the sequential scenario and 7 in the simultaneous scenario).

— Stability of circumbinary planets: the role of binary properties and migration scenarios  (2609.05134 - Grane et al., 4 Sep 2026) in Section 3.3, “MMRs observed between planets, and between a planet and the binary star”

Admittedly, it remains observationally unclear whether the commensurabilities observed in exoplanet systems are genuinely in resonant locking states, as measuring resonant angles requires high-precision transit-timing or radial-velocity data.

— N-Body Dynamics as a Mechanism for Mean Motion Resonance Formation in the Post-Gas Era  (2609.31244 - Sun et al., 25 Sep 2026) in Section 4, subsection “Comparison with Gas-Driven Migration”