Journal Article

A secular theory of coplanar, non-resonant planetary system

Cezary Migaszewski and Krzysztof Goździewski

in Monthly Notices of the Royal Astronomical Society

Published on behalf of The Royal Astronomical Society

Volume 388, issue 2, pages 789-802
Published in print August 2008 | ISSN: 0035-8711
Published online July 2008 | e-ISSN: 1365-2966 | DOI:
A secular theory of coplanar, non-resonant planetary system

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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 semimajor 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 semimajor 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.

Keywords: methods: analytical; methods: data analysis; methods: N-body simulations; techniques: radial velocities; celestial mechanics; stars: individual: HD 37124

Journal Article.  10463 words.  Illustrated.

Subjects: Astronomy and Astrophysics

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