Symmetric and asymmetric librations in extrasolar planetary systems: a global view
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 − 1t), (3:1, 5:3, ...) and (3) all other resonan 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.
KeywordsPeriodic orbits Resonances Extrasolar planetary systems
Unable to display preview. Download preview PDF.
- Beauge, C., Callegari, N., Ferraz-Mello, S., Michtchenko, T.: Resonances and stability of extra-solar planetary systems. In: Knezevic Z., Milani A. (eds.) Dynamics of Populations of Planetary Systems. Cambridge University Press, Cambridge, p. 3Google Scholar
- Hadjidemetriou, J.D.: Periodic orbits in gravitational systems. In: Proceedings of the Cortina meeting 2004, 2006Google Scholar
- Hadjidemetriou, J.D., Psychoyos, D.: Dynamics of extrasolar planetary systems: 2/1 resonant motion. In: G. Contopoulos and N. Voglis (eds.), Lecture Notes in Physics: Galaxies and Chaos, Vol. 626, pp. 412–432, Springer-Verlag, Berlin (2003)Google Scholar
- Lee, M.H., Peale, S.J.: Extrasolar planets and mean motion resonances. In Deming, D., Seager, S. (eds) Scientific Frontiers in Research of Extrasolar planets. ASP, 197 (2003)Google Scholar
- Peale, S., Lee, M.: Extrasolar planets and the 2:1 orbital resonances, In DDA 33rd Meeting, BAAS 34, 933 (2002)Google Scholar
- Schneider, J.: http://www.obspm.fr/encycl/catalog.html, (2006)Google Scholar
- Voyatzis, G., Hadjidemetriou, H.D.: Symmetric and asymmetric 3:1 resonant periodic orbits: an application to the 55Cnc extra-solar system. Cel. Mech. Dyn. Astr. this issue (2006)Google Scholar