Symmetric and Asymmetric Librations in Planetary and Satellite Systems at the 2/1 Resonance

Abstract

Families of asymmetric periodic orbits at the 2/1 resonance are computed for different mass ratios. The existence of the asymmetric families depends on the ratio of the planetary (or satellite) masses. As models we used the Io-Europa system of the satellites of Jupiter for the case m 1>m 2, the system HD82943 for the new masses, for the case m 1=m 2 and the same system HD82943 for the values of the masses m 1<m 2 given in previous work. In the case m 1≥ m 2 there is a family of asymmetric orbits that bifurcates from a family of symmetric periodic orbits, but there exist also an asymmetric family that is independent of the symmetric families. In the case m 1<m 2 all the asymmetric families are independent from the symmetric families. In many cases the asymmetry, as measured by $$\varpi_2-\varpi_1$$ and by the mean anomaly M of the outer planet when the inner planet is at perihelion, is very large. The stability of these asymmetric families has been studied and it is found that there exist large regions in phase space where we have stable asymmetric librations. It is also shown that the asymmetry is a stabilizing factor. A shift from asymmetry to symmetry, other elements being the same, may destabilize the system.