Secular Dynamics of Multiplanetary Circumbinary Systems

TL;DR

This paper develops an analytical framework to study the long-term secular dynamics of multiple planets orbiting a binary star, revealing how inner planets influence precession and resonance phenomena affecting system stability.

Contribution

It introduces a canonical coordinate-based formalism and simplified analytical solutions for the secular evolution of circumbinary planetary systems, including resonance analysis.

Findings

Inner planets accelerate binary precession rates.

Secular resonances can occur at close orbital distances.

Bifurcation of fixed points influences long-term stability.

Abstract

We present an analytical formalism to study the secular dynamics of a system consisting of N-2 planets orbiting a binary star in outer orbits. We introduce a canonical coordinate system and expand the disturbing function in terms of canonical elliptic elements, combining both Legendre polynomials and Laplace coefficients, to obtain a general formalism for the secular description of this type of configuration. With a quadractic approximation of the development, we present a simplified analytical solution for the planetary orbits for both the single planet and the two-planet cases. From the two-planet model, we show that the inner planet accelerates the precession rate of the binary pericenter, which, in turn, may enter in resonance with the secular frequency of the outer planet, characterizing a secular resonance. We calculate an analytical expression for the approximate location of this resonance and apply it to known circumbinary systems, where we show that it can occur at relatively close orbits, for example at 2.4 au for the Kepler-38 system. With a more refined model, we analyse the dynamics of this secular resonance and we show that a bifurcation of the corresponding fixed points can affect the long term evolution and stability of planetary systems. By comparing our results with complete integrations of the exact equations of motion, we verified the accuracy of our analytical model.