Stability and eccentricity for two planets in a 1:1 resonance, and their possible occurrence in extrasolar planetary systems

The nonlinear stability domain of Lagrange's celebrated 1772 solution of the three-body problem is obtained numerically as a function of the masses of the bodies and the common eccentricity of their Keplerian orbits. This domain shows that this solution can be realized in extrasolar planetary systems similar to those that have been discovered recently with two Jupiter-size planets orbiting a solar-size star. For an exact 1:1 resonance, the Doppler shift variation in the emitted light would be the same as for stars that have only a single planetary companion. But it is more likely that in actual extrasolar planetary systems there are deviations from such a resonance, raising the interesting prospect that Lagrange's solution can be identified by an analysis of the observations. The existence of another stable 1:1 resonance solution that would have a more unambiguous Doppler shift signature is also discussed.