Context. We have known about the Trojan group of asteroids since 1906. With a suspected number comparable to that of the mainbelt, they move around the Lagrangian points L 4 and L 5 of the Sun-Jupiter system. Aims. During the past century the Trojan asteroids were the topic of many studies of their orbital dynamics and stability. In recent years it has been possible to perform extensive numerical simulations of the motion of these asteroids and thus also to investigate the size and structure of

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... the stability region around the Lagrangian points. The aim of this work is to find new numerical estimates of the size and shape of the stability region. Additionally, the problem of the asymmetry between the two Trojan groups (there seem to be more asteroids in L 4 than in L 5 ) is investigated. Methods. We investigate the extension of this regions in the (σ − e) and (σ − i) plane (where σ = (ω + Ω + M) − (ω JUP + Ω JUP + M JUP )) and show how the initial inclination and initial eccentricity change the size and shape of this region. The problem of Trojan asymmetry is investigated by directly analyzing the evolution of orbital parameters. Results. We find both that there is a "critical" eccentricity of e INI ≈ 0.15 that plays an important role in the dynamics of Trojan asteroids and influences the structure of the stability regions and that the region of the small amplitudes of ∆σ = σ max − σ min is larger for L 4 due to the influence of Saturn and thus the L 4 group of Trojans seems to be more stable than the L 5 group.