Extrasolar planets in mean-motion resonance:: Apses alignment and asymmetric stationary solutions

In recent years several pairs of extrasolar planets have been discovered in the vicinity of mean-motion commensurabilities. In some cases, such as the GJ 876 system, the planets seem to be trapped in a stationary solution, the system exhibiting a simultaneous libration of the resonant angle theta(1)=2lambda(2)-lambda(1)-pi(1) and of the relative position of the pericenters. In this paper we analyze the existence and location of these stable solutions, for the 2:1 and 3:1 resonances, as functions of the masses and orbital elements of both planets. This is undertaken via an analytical model for the resonant Hamiltonian function. The results are compared with those of numerical simulations of the exact equations. In the 2:1 commensurability, we show the existence of three principal families of stationary solutions: (1) aligned orbits, in which theta(1) and pi(1)-pi(2) both librate around zero, (2) antialigned orbits, in which theta(1)=0 and the difference in pericenter is 180degrees, and (3) asymmetric stationary solutions, in which both the resonant angle and pi(1)-pi(2) are constants with values different from 0degrees or 180degrees. Each family exists in a different domain of values of the mass ratio and eccentricities of both planets. Similar results are also found in the 3:1 resonance. We discuss the application of these results to the extrasolar planetary systems and develop a chart of possible planetary orbits with apsidal corotation. We estimate, also, the maximum planetary masses in order for the stationary solutions to be dynamically stable.