Conditions of dynamical stability for the HD 160691 planetary system

In our previous paper we showed that the currently determined orbital parameters placed four recently announced planetary systems (HD 12661, HD 38529, HD 37124, and HD 160691) in very different situations from the point of view of dynamical stability. In the present paper we deal with the last of these systems, whose orbital parameters of the outer planet are yet uncertain. We discover a stabilizing mechanism that could be the key to its existence. The paper is devoted to the study of this mechanism by a global dynamics analysis in the orbital parameter space related to the HD 160691 system. We obtained our results using a new technique called the mean exponential growth factor of nearby orbits (MEGNO), and verified them with the fast Lyapunov indicator technique (FLI). In order to be dynamically stable, the HD 160691 planetary system has to satisfy the following conditions: (1) it should have a 2 : 1 mean motion resonance, (2) combined with an apsidal secular resonance, (3) in a configuration P_c(ap)-S-P_b(ap) (which means that the planets c and b may be considered as initially located at their apoastron around the central star S), and (4) it must satisfy specific conditions for the respective sizes of the eccentricities. High eccentricity for the outer orbit (e_c > 0.52) is the most probable necessary condition, while the eccentricity of the inner orbit e_b becomes relatively unimportant when e_c > 0.7. We also show that there is an upper limit for planetary masses (in the interval permitted by the undetermined line-of-sight inclination factor sin i_l) due to the dynamical stability mechanism. More generally, in this original orbital topology, where the resonance variables θ₁ and θ₃ librate about 180° while θ₂ librates about 0°, the HD 160691 system and its mechanism have revealed aspects of the 2 : 1 orbital resonances that have not been observed nor analyzed before. The present topology with antialigned apsidal lines, combined with the 2 : 1 resonance, is indeed more wide-ranging than the particular case of the HD 160691 planetary system. It is a new theoretical possibility that is suitable for a stable regime despite relatively small semimajor axes with respect to the important masses in interactions.