Symmetric and asymmetric librations in extrasolar planetary systems: a global view

Abstract

We present a global view of the resonant structure of the phase space of a planetary system with two planets, moving in the same plane, as obtained from the set of the families of periodic orbits. An important tool to understand the topology of the phase space is to determine the position and the stability character of the families of periodic orbits. The region of the phase space close to a stable periodic orbit corresponds to stable, quasi periodic librations. In these regions it is possible for an extrasolar planetary system to exist, or to be trapped following a migration process due to dissipative forces. The mean motion resonances are associated with periodic orbits in a rotating frame, which means that the relative configuration is repeated in space. We start the study with the family of symmetric periodic orbits with nearly circular orbits of the two planets. Along this family the ratio of the periods of the two planets varies, and passes through rational values, which correspond to resonances. At these resonant points we have bifurcations of families of resonant elliptic periodic orbits. There are three topologically different resonances: (1) the resonances (n + 1):n, (2:1, 3:2, …), (2) the resonances (2n + 1):(2n − 1), (3:1, 5:3, …) and (3) all other resonances. The topology at each one of the above three types of resonances is studied, for different values of the sum and of the ratio of the planetary masses. Both symmetric and asymmetric resonant elliptic periodic orbits exist. In general, the symmetric elliptic families bifurcate from the circular family, and the asymmetric elliptic families bifurcate from the symmetric elliptic families. The results are compared with the position of some observed extrasolar planetary systems. In some cases (e.g., Gliese 876) the observed system lies, with a very good accuracy, on the stable part of a family of resonant periodic orbits.

Resonant periodic orbits in the exoplanetary systems

“…had studied the dynamics of many MMR in the planar case by extracting an appropriate averaged Hamiltonian and computing the families of its stationary points. Modeling a two-planet system with the general three body problem (GTBP), we can study the dynamics of the non-averaged system by computing families of periodic orbits in a suitable rotating frame (Voyatzis and Hadjidemetriou, 2005;Hadjidemetriou, 2006;Voyatzis, 2008;Voyatzis et al, 2009). These families of periodic orbits should coincide with the families of stationary points, provided that the averaged Hamiltonian is sufficiently correct.…”

Trapping in three-planet resonances during gas-driven migration

“…However, an unstable crowded system -the starting point of typical planet-planet scattering simulations -is not the unique result of formation in a gas-disc, since resonant interactions during the gas phase should lead to stable resonant systems, such as the ones observed (e.g. Michtchenko et al 2006a, Hadjidemetriou 2006 for studies of mean-motion resonant systems).…”

Orbital stability of coplanar two-planet exosystems with high eccentricities

“…The planet P2 moves on the plane Oxy and its position is given by (x2, y2). The angle, ϑ, which defines the orientation of the rotating frame, is an ignorable variable and therefore, we obtain -further to the energy integral-the integral of angular momentum and the system is reduced to three degrees of freedom (Hadjidemetriou 2006b).…”