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

TL;DR

This paper extends the Laplace-Lagrange secular theory to better model the long-term dynamics of extrasolar planetary systems near mean-motion resonances, incorporating high-order eccentricity and mass effects.

Contribution

It introduces a high-order resonant Hamiltonian expansion that improves the accuracy of long-term orbital evolution predictions for resonant extrasolar systems.

Findings

First-order mass expansion qualitatively matches observed dynamics.

Second-order expansion improves quantitative accuracy.

Resonant corrections refine secular frequency calculations.

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).