Secular evolution of resonant planets in the coplanar case

Application to the systems HD 73526 and HD 31527

Departamento de Astronomía, Instituto de Física, Facultad de Ciencias, UdelaR, Iguá 4225, Montevideo 11400, Uruguay
e-mail: juan.pons.93@gmail.com; gallardo@fisica.edu.uy

Received: 24 October 2023
Accepted: 18 January 2024

Abstract

Aims. We study the secular evolution of two planets in mutual deep mean-motion resonance (MMR) in the planar elliptic three-body problem framework for different mass ratios. We do not consider any restriction in the eccentricity of the inner planet e₁ or in the eccentricity of the outer planet e₂.

Methods. The method we used is based on a semi-analytical model that consists of calculating the averaged resonant disturbing function numerically. It is assumed for this that all the orbital elements (except for the mean longitudes) of both planets are constant on the resonant timescale. In order to obtain the secular evolution inside the MMR, we make use of the adiabatic invariance principle, assuming a zero-amplitude resonant libration. We constructed two phase portraits, called the ℋ₁ and ℋ₂ surfaces, in the three-dimensional spaces (e₁, Δϖ, σ) and (e₂, Δϖ, σ), where Δϖ is the difference between the planetary longitude of perihelia and σ is the critical angle. These surfaces, which are related through the angular moment conservation, allow us to find the apsidal corotation resonances (ACRs) and to predict the secular evolution of e₁, e₂, Δϖ, and σ (libration center).

Results. While studying the 1:1, 2:1, 3:1, and 3:2 MMR, we found that large excursions in eccentricity can exist in some particular cases. We compared the secular variations of e₁, e₂, Δϖ, and σ predicted by the model with a numerical integration of the exact equations of motion for different mass ratios. We obtained good matches. Finally, the model was applied to study the secular evolution of the resonant exoplanet systems HD 73526 and HD 31527. They both have a pair of planets and are very close to the deep MMR condition. In the first system, we found that the pair of planets that constitutes the system evolves in a symmetrical ACR, whereas in the second system, we found that planets c and d, which are in an unusual 16:3 MMR, are close to an ACR, but outside its dynamical region, where Δϖ circulates.