## 1. IntroductionThe Trojan asteroids are minor planets librating in the vicinity of Jupiter's or stability point and can be regarded as realizations of the Lagrangian triangular solutions of the three-body problem. One century after the discovery of the first asteroid of the main-belt population between Mars and Jupiter - in 1801 by Piazzi - Max Wolf discovered the first asteroid in the exact 1:1 mean motion resonance with Jupiter; this asteroid, named Achilles, always librates around . Nowadays, 246 asteroids are known to be moving close to ahead of Jupiter () and 167 objects are trailing Jupiter at about (). Analytical estimates for the stability range of libration orbits in the planar circular restricted three-body problem were already carried out by Thüring (1931); only 7 Trojans were known at that time. This limit , where is the semi-major axis of Jupiter, is well above the stable libration limit found by recent methods. Numerical integrations of libration orbits around the Lagrangian point were for the first time carried out by Thüring (1959), again in the framework of this simplified model. In a further step, the stability of the long-period libration of the Trojans has been established on the basis of the linearized variational equations, valid only for infinitesimally small displacements from the exact periodic orbit, by Rabe (1961, 1962). Using this model, Rabe (1961) established a limiting curve for the stability of the Trojans, depending on the eccentricity and the libration width, which is also valid for the planar elliptic restricted problem. Since the orbits of the real Trojans show additionally short-periodic oscillations, higher-order terms had to be included in the theoretical model. These correction terms lead to the conclusion that the Trojans are stable, at least up to the third-order approximation (Rabe, 1967). More recently, Giorgilli & Skokos (1997) have proved (in the spirit of Nekhoroshev's theory) the stability of a libration region of the 1:1 resonance of the planar circular restricted problem for times equal to the Hubble time. This `effective stability' region is, of course, too narrow for the Trojan swarms to be termed `stable', managing however to include four real Trojans within its boundaries. Using more complicated dynamical models, important studies concerning the stability of the Trojans have been carried out more recently by Erdi (1984, 1988, 1996, 1997). Several numerical investigations have been also undertaken by Schubart & Bien (1984 and 1987) and by Bien & Schubart (1984 and 1987). Milani (1993 and 1994) computed the orbits of 174 Trojans in the model of the outer solar system for years and, in some cases, for years, showing that some of them are in fact lying on chaotic orbits. In a more recent numerical study by Levison et al. (1997) the orbits of 270 fictitious Trojans and of 36 real Trojans have been computed in the model of the outer solar system up to years and years, respectively. Many chaotic orbits, which escaped from the Trojan swarm, were found in these integrations and their possible connection to short-period comets was discussed. Our paper is organized as follows. Sect. 2 is devoted to describing the most important dynamical phenomena that govern the evolution of asteroids and states the problem under consideration in the present study. In Sect. 3 we describe the numerical setup of our integrations. Our results are presented in Sect. 4. Finally, in Sect. 5 we summarize our conclusions and discuss about the connection between our results and previously reported results on the Trojan problem as well as on the issue of stable chaos. © European Southern Observatory (ESO) 2000 Online publication: February 25, 2000 |