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A minimal dynamical model for tidal synchronization and orbit circularization

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Abstract

We study tidal synchronization and orbit circularization in a minimal model that takes into account only the essential ingredients of tidal deformation and dissipation in the secondary body. In previous work we introduced the model (Escribano et al. in Phys. Rev. E, 78:036216, 2008); here we investigate in depth the complex dynamics that can arise from this simplest model of tidal synchronization and orbit circularization. We model an extended secondary body of mass m by two point masses of mass m/2 connected with a damped spring. This composite body moves in the gravitational field of a primary of mass M ≫ m located at the origin. In this simplest case oscillation and rotation of the secondary are assumed to take place in the plane of the Keplerian orbit. The gravitational interactions of both point masses with the primary are taken into account, but that between the point masses is neglected. We perform a Taylor expansion on the exact equations of motion to isolate and identify the different effects of tidal interactions. We compare both sets of equations and study the applicability of the approximations, in the presence of chaos. We introduce the resonance function as a resource to identify resonant states. The approximate equations of motion can account for both synchronization into the 1:1 spin-orbit resonance and the circularization of the orbit as the only true asymptotic attractors, together with the existence of relatively long-lived metastable orbits with the secondary in p:q (p and q being co-prime integers) synchronous rotation.

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Authors and Affiliations

  1. Institute for Theoretical Physics, Eötvös University, P.O. Box 32, 1518, Budapest, Hungary

    József Vanyó & Tamás Tél

  2. Instituto Andaluz de Ciencias de la Tierra, CSIC–Universidad de Granada, 18071, Granada, Spain

    Bruno Escribano & Julyan H. E. Cartwright

  3. Departament de Física, Universitat de les Illes Balears, 07122, Palma de Mallorca, Spain

    Diego L. González

  4. Istituto IMM-CNR, Area della Ricerca CNR di Bologna, 40129, Bologna, Italy

    Oreste Piro

Authors
  1. József Vanyó
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  2. Bruno Escribano
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  3. Julyan H. E. Cartwright
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  4. Diego L. González
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  5. Oreste Piro
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  6. Tamás Tél
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Correspondence to József Vanyó.

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Vanyó, J., Escribano, B., Cartwright, J.H.E. et al. A minimal dynamical model for tidal synchronization and orbit circularization. Celest Mech Dyn Astr 109, 181–200 (2011). https://doi.org/10.1007/s10569-010-9322-2

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  • DOI: https://doi.org/10.1007/s10569-010-9322-2

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