Phase portrait of the planetary 2:1 resonance
Uranus-Neptune and 6 pairs amongst the more than 400 exoplanets currently known show periods close to a 2:1 commensurability. In the most conspicuous case, Gliese 876, the two planets are very close to a stationary solution in which both orbits have aligned semi-major axis and the motion on them is such that the conjunction of the two planets occurs when the two planets are at pericenter (symmetric conjunction). This solution is referred in the literature as an “apsidal corotation resonance” (the two bodies corotate with the features of the changing mutual gravitational field). The analysis of these motions shows other possibilities which may be anti-symmetric (the pericenters lie on opposite directions) or asymmetric (the semi-axes are frozen but at angles different of 0 or ). However, ACRs are not the only possible resonant motions. In some actual cases, the planets are in resonant motion but not in a stationary configuration. The best known example is the pair of planets of HD 82943. In this case, the motion of the planets is a composition of two different proper oscillations around two families of periodic solutions which intersect at an ACR. The region where the motion occurs is surrounded by a very chaotic domain (in fact the published best fit solution lies in this chaotic domain). We intend to present the results of an exploration of the phase space around stable and unstable ACRs in two main cases: The case of HD 82943 and that of two planets with the same masses as Jupiter and Saturn. We show how the phase portrait is different following the more massive planet is the outer or the inner one in the pair, and we discuss the details of these portraits.