Celestial Mechanics and Dynamical Astronomy

, Volume 126, Issue 1–3, pp 31–60 | Cite as

Complete spin and orbital evolution of close-in bodies using a Maxwell viscoelastic rheology

  • Gwenaël BouéEmail author
  • Alexandre C. M. Correia
  • Jacques Laskar
Original Article


In this paper, we present a formalism designed to model tidal interaction with a viscoelastic body made of Maxwell material. Our approach remains regular for any spin rate and orientation, and for any orbital configuration including high eccentricities and close encounters. The method is to integrate simultaneously the rotation and the position of the planet as well as its deformation. We provide the equations of motion both in the body frame and in the inertial frame. With this study, we generalize preexisting models to the spatial case and to arbitrary multipole orders using a formalism taken from quantum theory. We also provide the vectorial expression of the secular tidal torque expanded in Fourier series. Applying this model to close-in exoplanets, we observe that if the relaxation time is longer than the revolution period, the phase space of the system is characterized by the presence of several spin-orbit resonances, even in the circular case. As the system evolves, the planet spin can visit different spin-orbit configurations. The obliquity is decreasing along most of these resonances, but we observe a case where the planet tilt is instead growing. These conclusions derived from the secular torque are successfully tested with numerical integrations of the instantaneous equations of motion on HD 80606 b. Our formalism is also well adapted to close-in super-Earths in multiplanet systems which are known to have non-zero mutual inclinations.


Restricted problems Extended body Dissipative forces Planetary systems Rotation HD 80606 b 



GB is grateful to Dan Fabrycky for the fruitful discussions which lead to this work. We acknowledge support from CIDMA strategic project UID/MAT/04106/2013.


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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.IMCCE, Observatoire de ParisUPMC Univ. Paris 6ParisFrance
  2. 2.Departamento de Física, CIDMAUniversidade de AveiroAveiroPortugal

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