Secondary resonances

From the catalogue we know that the size of the stable region mainly depends on the initial eccentricity and the mass ratio of the primary bodies. But also the secondary resonances play an important role when we investigate the stability region around the equilibrium points.
In the circular restricted three-body problem, motions around L4 has two frequencies, ns and nl, corresponding to the short and long period components of libration. In the ERTBP, however, the point L4 has an elliptic motion itself, similar to the elliptic motion of the primaries, and a corresponding mean motion n. This frequency combines with the frequencies of the short and long period librations, and thus motions around L4 have four frequencies (and linear combinations of them) which for e → 0 approach the limit values ns, n - nl, nl, and n - ns. Taking n as unit frequency, the normalized frequencies ns and nl are given by (Rabe 1970, 1973):

where


References:
Rabe E., 1973, in Tapley E.D., Szebehely V., eds, Recent Advances in Dynamical Astronomy. D. Reidel Publ. Co., Dordrecht, p. 156.
Rabe E., 1970, in Giacaglia G.E.O., ed., Periodic Orbits, Stability and Resonances. D. Reidel Publ. Co., Dordrecht, p. 33.