Celestial Mechanics and Dynamical Astronomy

, Volume 123, Issue 4, pp 453–479 | Cite as

The resonance overlap and Hill stability criteria revisited

Original Article


We review the orbital stability of the planar circular restricted three-body problem in the case of massless particles initially located between both massive bodies. We present new estimates of the resonance overlap criterion and the Hill stability limit and compare their predictions with detailed dynamical maps constructed with N-body simulations. We show that the boundary between (Hill) stable and unstable orbits is not smooth but characterized by a rich structure generated by the superposition of different mean-motion resonances, which does not allow for a simple global expression for stability. We propose that, for a given perturbing mass \(m_1\) and initial eccentricity e, there are actually two critical values of the semimajor axis. All values \(a < a_\mathrm{Hill}\) are Hill-stable, while all values \(a > a_\mathrm{unstable}\) are unstable in the Hill sense. The first limit is given by the Hill-stability criterion and is a function of the eccentricity. The second limit is virtually insensitive to the initial eccentricity and closely resembles a new resonance overlap condition (for circular orbits) developed in terms of the intersection between first- and second-order mean-motion resonances.


Eccentric orbits Mean-motion resonances Resonance overlap criterion Stability Three-body problem 


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Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Instituto de Astronomía Teórica y Experimental (IATE), Observatorio AstronómicoUniversidad Nacional de CórdobaCórdobaArgentina
  2. 2.Complejo Astronómico El Leoncito (CASLEO-CONICET)San JuanArgentina

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