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

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 χ 2 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 χ 2 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.

Satellite orbits design using frequency analysis

“…Note that the filtered orbit e u ðtÞ; e v ðtÞ f gdoes not correspond to a real solution of the equations of motion. However, given that the main frequencies m s and m 1 are well-separated and no low-order linear combinations with any significant amplitude appear in the spectral decompositions of uðtÞ and vðtÞ, we expect that the 'corrected' initial condition e u ð0Þ; e v ð0Þ f gwill lead an orbit that is closer to the center of motion (the PO), as also shown by Couetdic et al (2010).…”

“…As shown below, this iterative scheme, which we call 'successive filtering algorithm' converges quickly to the nearest stable PO at given i 0 , starting from an initial guess (non-periodic solution) that need not be close to the PO. In the course of this work we discovered that a similar technique had actually been used by different authors, among which (Couetdic et al, 2010) for finding stable resonant periodic orbits in two-planet systems, Noyelles (2009) and Robutel et al (2011) for studying the 3:2 spin-orbit resonance or the 1:1 resonant coorbital rotation, Dufey et al (2009) in analyzing the latitudinal libration of Mercury, or Delsate (2011) to study ground-track resonances around Vesta. These authors however used the NAFF frequency analysis algorithm of Laskar (1990) that, while being very accurate, needs integrations covering at least a few periods of the slowest component of the orbit.…”

Orbital dynamics of exoplanetary systems Kepler-62, HD 200964 and Kepler-11

“…In recent years, several exoplanetary systems have been discovered and it is very interesting to know their dynamical stability. There are several authors (Fabrycky & Murray-Clay 2010;Couetdic et al 2010;Davies et al 2014;Adams & Bloch 2015;Petrovich 2015) who have studied about the stability of exoplanetary systems. Davies et al (2014) have reviewed the long-term dynamical evolution of the planetary systems.…”

A Double Planetary System Around the Evolved Intermediate-Mass Star Hd 4732

“…The stability map in the (a, e) plane of the outer planet for different values of i is shown in Figure 7. Here we plot the stability index D = | n 2 − n 1 | (in • yr −1 ) following Couetdic et al (2010), which studied stability of HD202206 system using Laskar's frequency map analysis (Laskar 1990;Laskar et al 2001). Following the analysis, we calculate an average of mean motion of the outer planet in 1000 Kepler periods ( n 1 ) and subtract it from an average of mean motion of the same planet obtained in the next consecutive 1000 Kepler periods ( n 2 ).…”