Secular dynamics of multiplanetary circumbinary systems: stationary solutions and binary-planet secular resonance

  • Eduardo Andrade-InesEmail author
  • Philippe Robutel
Original Article


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 quadratic 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.


Circumbinary planets Secular dynamics Secular resonance Analytical development 


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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.IMCCE, Observatoire de Paris - PSL Research UniversityUPMC Univ. Paris 06, CNRSParisFrance
  2. 2.Instituto de Astronomia, Geofísica e Ciências Atmosféricas Rua do Matão 1226Universidade de São PauloSão PauloBrazil

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