A secular theory of coplanar, non-resonant planetary system

Abstract

We present the secular theory of coplanar $N$-planet system, in the absence of mean motion resonances between the planets. This theory relies on the averaging of a perturbation to the two-body problem over the mean longitudes. We expand the perturbing Hamiltonian in Taylor series with respect to the ratios of semi-major axes which are considered as small parameters, without direct restrictions on the eccentricities. Next, we average out the resulting series term by term. This is possible thanks to a particular but in fact quite elementary choice of the integration variables. It makes it possible to avoid Fourier expansions of the perturbing Hamiltonian. We derive high order expansions of the averaged secular Hamiltonian (here, up to the order of 24) with respect to the semi-major axes ratio. The resulting secular theory is a generalization of the octupole theory. The analytical results are compared with the results of numerical (i.e., practically exact) averaging. We estimate the convergence radius of the derived expansions, and we propose a further improvement of the algorithm. As a particular application of the method, we consider the secular dynamics of three-planet coplanar system. We focus on stationary solutions in the HD 37124 planetary system.

14 pages, 4 figures, MATHEMATICA file expansion.m, accepted to Monthly Notices of the Royal Astronomical Society. (minor corrections of symbols)

References in corpus (16)

• Catalog of Nearby Exoplanets
• An extrasolar planetary system with three Neptune-mass planets
• The Kozai Mechanism and the Evolution of Binary Supermassive Black Holes
• The HARPS search for southern extra-solar planets XI. Super-Earths (5 & 8 M_Earth) in a 3-planet system
• Five Planets Orbiting 55 Cancri
• Secular Evolution of Hierarchical Planetary Systems
• Stability Limits in Extra-solar Planetary Systems
• Secular dynamics of the three body problem: application to the upsilon Andromedae planetary system
• The HARPS search for southern extra-solar planets. IX. mu Ara, a system with four planets
• Extra-solar planets around HD196050, HD216437 and HD160691
• Modeling the 3--D Secular Planetary Three-Body Problem. Discussion on the outer $\upsilon$ Andromedae Planetary System
• On the extrasolar multi-planet system around HD160691
• Detection of a Third Planet in the HD 74156 System Using the Hobby-Eberly Telescope
• The Secular Evolution and Dynamical Architecture of the Neptunian Triplet Planetary System HD 69830
• The long-term stability of extrasolar system HD 37124. Numerical study of resonance effects
• Extrasolar planetary dynamics with a generalized planar Laplace-Lagrange secular theory

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