Abstract
Extrasolar systems with planets on eccentric orbits close to or in mean-motion resonances are common. The classical low-order resonant Hamiltonian expansion is unfit to describe the long-term evolution of these systems. We extend the Lagrange-Laplace secular approximation for coplanar systems with two planets by including (near-)resonant harmonics and realize an expansion at high order in the eccentricities of the resonant Hamiltonian both at orders one and two in the masses. We show that the expansion at first order in the masses gives a qualitative good approximation of the dynamics of resonant extrasolar systems with moderate eccentricities, while the second order is needed to reproduce more accurately their orbital evolutions. The resonant approach is also required to correct the secular frequencies of the motion given by the Laplace-Lagrange secular theory in the vicinity of a mean-motion resonance. The dynamical evolutions of four (near-)resonant extrasolar systems are discussed, namely GJ 876 (2:1 resonance), HD 60532 (3:1), HD 108874 and GJ 3293 (close to 4:1).
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Notes
Here, we expand around the initial values, but the average values over a long-term numerical integration could also be considered (see, e.g., Sansottera et al. (2013)).
Let us note that Rivera et al. (2010) have revealed the presence of an additional planet in a three-body Laplace resonance with the previously two known giant planets.
Let us stress that, to better visualize the evolution of \(\sigma _1\), we plot the evolution on a much smaller timescale.
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Acknowledgements
The authors seize the opportunity of the Topical Collection for the 50th birthday of CM&DA to dedicate this paper to the memory of Jacques Henrard. This work follows the path traced in his two contributions published in the first volume of Celestial Mechanics. The work of M. S. has been partially supported by the National Group of Mathematical Physics (GNFM-INdAM). Computational resources have been provided by the PTCI (Consortium des Équipements de Calcul Intensif CECI), funded by the FNRS-FRFC, the Walloon Region, and the University of Namur (Conventions No. 2.5020.11, GEQ U.G006.15, 1610468 et RW/GEQ2016).
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Appendix Low-order expansions of \({\mathcal {H}}_{\mathrm{res}}^{({\mathcal {O}}1)}\) and \({\mathcal {H}}_{\mathrm{res}}^{({\mathcal {O}}2)}\) for GJ 876
Appendix Low-order expansions of \({\mathcal {H}}_{\mathrm{res}}^{({\mathcal {O}}1)}\) and \({\mathcal {H}}_{\mathrm{res}}^{({\mathcal {O}}2)}\) for GJ 876
We report here the low-order expansion of \({\mathcal {H}}_{\mathrm{res}}^{({\mathcal {O}}1)} = {\overline{{\mathcal {H}}}}^{({\mathcal {T}})} + {\widetilde{{\mathcal {H}}}}^{({\mathcal {T}})}\) and \({\mathcal {H}}_{\mathrm{res}}^{({\mathcal {O}}2)} = {\overline{{\mathcal {H}}}}^{({\mathcal {O}}2)} + {\widetilde{{\mathcal {H}}}}^{({\mathcal {O}}2)}\) (see (5) and (10), respectively) for GJ 876. We refer to Table 1 for the physical and orbital parameters of the system and Subsect. 4.2 for a complete description of the system (Tables 2, 3).
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Sansottera, M., Libert, AS. Resonant Laplace-Lagrange theory for extrasolar systems in mean-motion resonance. Celest Mech Dyn Astr 131, 38 (2019). https://doi.org/10.1007/s10569-019-9913-5
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DOI: https://doi.org/10.1007/s10569-019-9913-5