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Two-layer model via non-quasi-periodic normal form theory.

Gabriella Pinzari1, Benedetto Scoppola2, Matteo Veglianti2

  • 1Dipartimento di Matematica "Tullio Levi-Civita", Università degli Studi di Padova, Via Trieste, 63, 35131 Padova, Italy.

Celestial Mechanics and Dynamical Astronomy
|April 20, 2026
PubMed
Summary

This study analyzes the two-layer model, revealing that resonant motion is maintained by exponentially-long trapping times. These findings offer rigorous mathematical conditions for resonant dynamics in celestial mechanics.

Keywords:
Capture into resonanceFrictionNon–quasi–periodic normal form theory

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Area of Science:

  • Celestial Mechanics
  • Dynamical Systems Theory
  • Astrophysics

Background:

  • The two-layer model describes a celestial body with a core and viscous fluid layer, relevant to understanding planetary dynamics.
  • Resonance phenomena are crucial in celestial mechanics, influencing orbital stability and evolution.
  • Previous work explored resonance capture qualitatively and numerically.

Purpose of the Study:

  • To rigorously investigate the mathematical conditions maintaining resonant motion in the two-layer model.
  • To provide an explicit estimation of the resonance trapping time.
  • To reframe the model using a non-quasi-periodic normal form theory.

Main Methods:

  • Application of a specialized non-quasi-periodic normal form theory.
  • Mathematical analysis of a 2+1/2 degrees-of-freedom non-autonomous dynamical system.
  • Derivation of resonance trapping time estimates.

Main Results:

  • The resonance trapping time is estimated to be exponentially long with respect to system parameters.
  • Explicit mathematical conditions for maintaining resonant motion are provided.
  • The study offers a rigorous framework for analyzing resonant dynamics.

Conclusions:

  • The two-layer model exhibits stable resonant motion due to exponentially long trapping times.
  • The developed mathematical framework provides precise conditions for resonance maintenance.
  • This research advances the understanding of resonant dynamics in celestial bodies.