Dynamical Behaviour of Asteroids in the 4:1 Resonance

Abstract

We study the dynamics of mean motion resonance with Jupiter in the 4:1 gap using only gravitational methods. This mechanism is capable of explaining this Kirkwood gap in an uniform way (see Ferraz-Mello, 1994; Ferraz-Mello et al., 1994; Moons, 1997; Yoshikawa, 1989). We considered the asteroidal motion in two and three dimensions and we carried out our investigations integrating numerically the full equations of motion and taking into account Mars, Jupiter and Saturn as disturbing planets. The orbital evolution of asteroids was obtained considering the elements variation. The numerical investigations were carried out using symplectic integrators. These integrations were stopped when the asteroid had close encouters with Mars or Jupiter, this occurs when the distance between the planet and the asteroid is of the order of 0.01 AU or less, or when the eccentricity in-creases up to 0.9. We studied real and fictitious asteroids on a time scale of 5 × 10⁷ yr. The initial osculating elements of perturbing planets and their inverse masses were taken from the Ephemerides of Minor Planets (EMP) at the epoch of JD 2450000.5. The initial data corresponding to the real asteroids were also taken from the EMP. The starting elements of fictitious asteroids were, in all analyzed cases, a = a _crit = 2.064 AU, i = 2°.5 and e = 0.01 (in the majority of cases). The other initial elements are shown in Table II. We have also studied fictitious asteroids with i = 0°, a = a _crit and e = 0.01 (Table I). The present analysis leads to the following results: (1) The motions are unstable. The eccentricity, in the majority of cases, has very large increase. It may grow up to 0.9 in 10⁶ yr.