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A Global Study of the 3:1 Resonance Neighborhood: A Search for Unstable Asteroids

The recent discovery of two asteroids with unstable orbits in the neighborhood of the 5 : 2 resonance raises the question whether there exist unstable asteroids in the neighborhood of the more important 3 : 1 asteroidal resonance. The two asteroids have short dynamical lifetimes of about 10 and 50 Myr. Semimajor-axis drift due to the Yarkovsky effect can explain their existence in such a short-lived orbit. Since this effect is not specific to the 5 : 2 resonance, a natural question is whether there exist short-lived asteroids near other resonance borders. This question is interesting because by studying such objects we can better understand the production of near-Earth asteroids (NEAs) via the collision and Yarkovsky processes. We search for short-lived asteroids in the neighborhood of the 3 : 1 resonance. The borders of this resonance and the positions of the asteroids with respect to them are determined by using the mean elliptic planar restricted three-body problem and a method that invokes the concept of a "representative plane." A semianalytical global study of the orbits for this nonlinear dynamical system is made through this convenient plane. The limit between the circulation and libration or chaotic orbits in this simple problem is taken as the edge of the resonance. We then choose 41 numbered asteroids that are closest to the resonance borders and have inclinations smaller than 10°. The orbits of these asteroids are numerically integrated using the full equations of motion in the gravitational field of the Sun and the planets Jupiter, Saturn, Mars, Earth, and Venus. We find that about 40% of the studied asteroids enter into the 3 : 1 resonance within 100 Myr and are subsequently dynamical eliminated. The existence of a nonnegligible number of asteroids in the 3 : 1 resonance neighborhood implies that a process must be continually acting to replenish this region. We suggest that these asteroids may have wandered to the resonance borders through small semimajor-axis displacements due to the Yarkovsky effect. This result also confirms that asteroids are continually feeding the chaotic 3 : 1 resonance region and consequently the Mars crossers and NEA population.