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A study of the 1/2 retrograde resonance: periodic orbits and resonant capture

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Abstract

We describe the families of periodic orbits in the 2-dimensional 1/2 retrograde resonance at mass ratio \(10^{-3}\), analyzing their stability and bifurcations into 3-dimensional periodic orbits. We explain the role played by periodic orbits in adiabatic resonance capture, in particular how the proximity between a stable family and an unstable family with a nearly critical segment, associated with Kozai separatrices, determines the transition between distinct resonant modes observed in numerical simulations. Combining the identification of stable, critical and unstable periodic orbits with analytical modeling, resonance capture simulations and computation of stability maps helps to unveil the complex 3-dimensional structure of resonances.

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Notes

  1. Morais and Namouni (2013b) showed that the retrograde resonant angles may be obtained from the standard prograde disturbing function by applying a canonical transformation \(\uplambda _p^*=-\uplambda _p\), \(\omega ^*=\omega -\pi \), \(\varOmega ^*=-\varOmega -\pi \) which is equivalent to inverting the planet’s motion, hence swap** ascending and descending nodes.

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Acknowledgements

Bibliography access was provided by CAPES-Brazil. M.H.M. Morais research had financial support from São Paulo Research Foundation (FAPESP/2018/08620-1) and CNPQ-Brazil (PQ2/304037/2018-4).

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Authors and Affiliations

  1. Instituto de Geociências e Ciências Exatas, Universidade Estadual Paulista (UNESP), Av. 24-A, 1515 13506-900, Rio Claro, SP, Brazil

    M.H.M. Morais

  2. Université Côte d’Azur, CNRS, Observatoire de la Côte d’Azur, CS 34229, 06304, Nice, France

    F. Namouni

  3. Department of Physics, Aristotle University of Thessaloniki, 54124, Thessaloniki, Greece

    G. Voyatzis & T. Kotoulas

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  1. M.H.M. Morais
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  2. F. Namouni
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  3. G. Voyatzis
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  4. T. Kotoulas
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Correspondence to M.H.M. Morais.

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Morais, M., Namouni, F., Voyatzis, G. et al. A study of the 1/2 retrograde resonance: periodic orbits and resonant capture. Celest Mech Dyn Astr 133, 21 (2021). https://doi.org/10.1007/s10569-021-10020-0

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  • DOI: https://doi.org/10.1007/s10569-021-10020-0

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