Resonant Laplace-Lagrange theory for extrasolar systems in mean-motion resonance

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 Laplace-Lagrange 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).

21 pages, 7 figures

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• The HARPS search for southern extra-solar planets XXXV. Planetary systems and stellar activity of the M dwarfs GJ 3293, GJ 3341, and GJ 3543
• Kolmogorov and Nekhoroshev theory for the problem of three bodies
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• Secular dynamics of a planar model of the Sun-Jupiter-Saturn-Uranus system; effective stability into the light of Kolmogorov and Nekhoroshev theories

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