Dynamical stability analysis of the HD 202206 system and constraints to the planetary orbits^*

¹ Astronomie et Systèmes Dynamiques, IMCCE-CNRS UMR 8028, Observatoire de Paris, UPMC, 77 Avenue Denfert-Rochereau, 75014 Paris, France e-mail: laskar@imcce.fr
² Departamento de Física, Universidade de Aveiro, Campus de Santiago, 3810-193 Aveiro, Portugal
³ Observatoire de Genève, 51 ch. des Maillettes, 1290 Sauverny, Switzerland

Received: 10 November 2009
Accepted: 24 April 2010

Abstract

Context. Long-term, precise Doppler measurements with the CORALIE spectrograph have revealed the presence of two massive companions to the solar-type star HD 202206. Although the three-body fit of the system is unstable, it was shown that a 5:1 mean motion resonance exists close to the best fit, where the system is stable. It was also hinted that stable solutions with a wide range of mutual inclinations and low O-C were possible.

Aims. We present here an extensive dynamical study of the HD 202206 system, aiming at constraining the inclinations of the two known companions, from which we derive possible value ranges for the companion masses.

Methods. We consider each inclination and one of the longitudes of ascending node as free parameters. For any chosen triplet of these parameters, we compute a new fit. Then we study the long-term stability in a small (in terms of O-C) neighborhood using Laskar's frequency map analysis. We also introduce a numerical method based on frequency analysis to determine the center of libration mode inside a mean motion resonance.

Results. We find that acceptable coplanar configurations (with low χ² stable orbits) are limited with respect to inclinations to the line of sight between 30° and 90°. This limits the masses of both companions to roughly twice the minimum: m_b ∈ [16.6 M_Jup; 33.5 M_Jup] and m_c ∈ [2.2 M_Jup; 4.4 M_Jup] . Non-coplanar configurations are possible for a wide range of mutual inclinations from 0° to 90°, although ΔΩ = 0 [ π]  configurations seem to be favored. We also confirm the 5:1 mean motion resonance to be most likely. In the coplanar edge-on case, we provide a very good stable solution in the resonance, whose χ² does not differ significantly from the best fit. Using our method for the determination of the center of libration, we further refine this solution to obtain an orbit with a very low amplitude of libration, as we expect that dissipative effects have dampened the libration.