Abstract
We study the long-term dynamics of a planetary system composed of a star and a planet. Both bodies are considered as extended, non-spherical, rotating objects. There are no assumptions made on the relative angles between the orbital angular momentum and the spin vectors of the bodies. Thus, we analyze full, spatial model of the planetary system. Both objects are assumed to be deformed due to their own rotations, as well as due to the mutual tidal interactions. The general relativity corrections are considered in terms of the post-Newtonian approximation. Besides the conservative contributions to the perturbing forces, there are also taken into account non-conservative effects, i.e., the dissipation of the mechanical energy. This dissipation is a result of the tidal perturbation on the velocity field in the internal zones with non-zero turbulent viscosity (convective zones). Our main goal is to derive the equations of the orbital motion as well as the equations governing time-evolution of the spin vectors (angular velocities). We derive the Lagrangian equations of the second kind for systems which do not conserve the mechanical energy. Next, the equations of motion are averaged out over all fast angles with respect to time-scales characteristic for conservative perturbations. The final equations of motion are then used to study the dynamics of the non-conservative model over time scales of the order of the age of the star. We analyze the final state of the system as a function of the initial conditions. Equilibria states of the averaged system are finally discussed.
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Acknowledgments
I would like to thank Benoît Noyelles, Michael Efroimsky and the anonymous reviewer for the informative reviews that improved the manuscript and Sylvio Ferraz-Mello for comments on the averaging theory. Many thanks to Krzysztof Goździewski for a discussion and corrections of the manuscript. This work was supported by the Polish Ministry of Science and Higher Education grant N/N203/402739. The author is a recipient of the stipend of the Foundation for Polish Science (programme START, editions 2010 and 2011). This research was carried out with the support of the ”HPC Infrastructure for Grand Challenges of Science and Engineering” Project, co-financed by the European Regional Development Fund under the Innovative Economy Operational Programme.
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Migaszewski, C. The generalized non-conservative model of a 1-planet system revisited. Celest Mech Dyn Astr 113, 169–203 (2012). https://doi.org/10.1007/s10569-012-9413-3
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DOI: https://doi.org/10.1007/s10569-012-9413-3