The size of the stable region around the Lagrangian point L 4 in the elliptic restricted three-body problem is determined by numerical integration as a function of the mass parameter and eccentricity of the primaries. The size distribution of the stable regions in the mass parameter-eccentricity plane shows minima at certain places that are identified with resonances between the librational frequencies of motions around L 4 . These are computed from an approximate analytical equation of Rabe relating the frequency, mass parameter and eccentricity. Solutions of this equation are determined numerically and the global behaviour of the frequencies depending on the mass parameter and eccentricity is shown and discussed. The minimum sizes of the stable regions around L 4 change along the resonances and the relative strength of the resonances is analysed. Applications to possible Trojan exoplanets are indicated. Escape from L 4 is also investigated.

Third and fourth order mean motion resonances are studied in the model of the restricted three-body problem by numerical methods for mass parameters corresponding approximately to the Sun-Jupiter and Sun-Neptune systems. In the case of inner resonances, it is shown that there are two regions of libration in the 8:5 and 7:4 resonances, one at low, the other at high eccentricities. In the 9:5 and 7:3 resonances libration can exist only in one region at high eccentricities. The 5:2 and 4:1 resonances are very regular, with one librational zone existing for all eccentricities. There is no visible region of libration at any eccentricities in the 5:1 resonance, the transition between the regions of direct and retrograde circulation is very sharp. In the case of outer resonances, the 8:5 and 7:4 resonances have also two regions of libration, but the 9:5 resonance has three, the 7:3 resonance two librational zones. The 5:2 resonance is again very regular, but it is parted for two regions of libration at high eccentricities. Libration is possible in the 4:1 resonance only at high eccentricities. The 5:1 resonance is very symmetric. In the case of outer resonances, a comparison is made with trans-Neptunian objects (TNO) in higher order mean motion resonances. Several new librating TNOs are identified.

The stability of the Lagrangian point L 4 is investigated in the elliptic restricted three-body problem by using Floquet's theory. Stable and unstable domains are determined in the parameter plane of the mass parameter and the eccentricity by computing the characteristic exponents. Frequencies of motion around L 4 have been determined both in the stable and unstable domains and fitting functions for the frequencies are derived depending on the mass parameter and the eccentricity. Resonances between the frequencies are studied in the whole parameter plane. It is shown that the 1:1 resonances are not restricted only to single curves but extend to the whole unstable domain. In the unstable domains longer escape times of the test particle from the neighbourhood of L 4 are related to certain resonances, but changing the parameters the same resonances may lead to faster escape.

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