Secular dynamics of a coplanar, non-resonant planetary system under the general relativity and quadrupole moment perturbations

Abstract

We construct a secular theory of a coplanar system of N planets not involved in strong mean motion resonances, and which are far from collision zones. Besides the point-to-point Newtonian mutual interactions, we consider the general relativistic corrections to the gravitational potential of the star and the innermost planet, and also a modification of this potential by the quadrupole moment and tidal distortion of the star. We focus on hierarchical planetary systems. After averaging the model Hamiltonian with a simple algorithm making use of very basic properties of the Keplerian motion, we obtain analytical secular theory of high order in the semimajor axes ratio. A great precision of the analytic approximation is demonstrated with the numerical integrations of the equations of motion. A survey regarding model parameters (the masses, semimajor axes, spin rate of the star) reveals a rich and non-trivial dynamics of the secular system. Our study is focused on its equilibria. Such solutions predicted by the classic secular theory, which correspond to aligned (mode I) or anti-aligned (mode II) apsides, may be strongly affected by the gravitational corrections. The so-called true secular resonance, which is a new feature of the classic two-planet problem discovered by Michtchenko & Malhotra, may appear in other, different regions of the phase space of the generalized model. We found bifurcations of mode II from which emerge new, yet unknown in the literature, secularly unstable equilibria and a complex structure of the phase space. These equilibria may imply secularly unstable orbital configurations even for initially moderate eccentricities. The point mass gravity corrections can affect the long-term stability in the secular time-scale, which may directly depend on the age of the host star through its spin rate. We also analyse the secular dynamics of the υ Andromedae system in the realm of the generalized model. Also in this case of the three-planet system, new secular equilibria may appear.