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Numerical study of the geometry of the phase space of the Augmented Hill Three-Body problem

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Abstract

The Augmented Hill Three-Body problem is an extension of the classical Hill problem that, among other applications, has been used to model the motion of a solar sail around an asteroid. This model is a 3 degrees of freedom (3DoF) Hamiltonian system that depends on four parameters. This paper describes the bounded motions (periodic orbits and invariant tori) in an extended neighbourhood of some of the equilibrium points of the model. An interesting feature is the existence of equilibrium points with a 1:1 resonance, whose neighbourhood we also describe. The main tools used are the computation of periodic orbits (including their stability and bifurcations), the reduction of the Hamiltonian to centre manifolds at equilibria, and the numerical approximation of invariant tori. It is remarkable how the combination of these techniques allows the description of the dynamics of a 3DoF Hamiltonian system.

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Notes

  1. The frequency vector of a quasi-periodic trajectory is determined up to an unimodular transformation.

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Acknowledgements

The research of A.F. and A.J. has been supported by the Spanish Grant MTM2015-67724-P (funded by MINECO/FEDER, UE) and the Catalan Grant 2014 SGR 1145. The research of J.M.M. has been supported by the Spanish Grants MTM2013-41168-P, MTM2014-52209-C2-1-P, MTM2016-80117-P.

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Authors and Affiliations

  1. Institut de Matemàtiques de la Universitat de Barcelona, Universitat de Barcelona, Gran Via de les Corts Catalanes 585, 08007, Barcelona, Spain

    Ariadna Farrés

  2. Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Gran Via de les Corts Catalanes 585, 08007, Barcelona, Spain

    Àngel Jorba

  3. IEEC-CERES & Departament de Matemàtiques, Universitat Autónoma de Barcelona, Av. de l’Eix Central, Edifici C, Bellaterra, 08193, Bellaterra, Spain

    Josep-Maria Mondelo

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  1. Ariadna Farrés
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  3. Josep-Maria Mondelo
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Correspondence to Ariadna Farrés.

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Farrés, A., Jorba, À. & Mondelo, JM. Numerical study of the geometry of the phase space of the Augmented Hill Three-Body problem. Celest Mech Dyn Astr 129, 25–55 (2017). https://doi.org/10.1007/s10569-017-9762-z

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  • DOI: https://doi.org/10.1007/s10569-017-9762-z

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