Secular Resonances in Mean Motion Commensurabilities: The 2/1 and 3/2 Cases

Abstract

The dynamic behavior of asteroids orbiting near the mean motion commensurabilities 2/1 and 3/2 with Jupiter is strongly affected by secular perturbations due to the outer planets, which are not present in the usual models based on the restricted three-body problem. Moreover, many interesting and puzzling phenomena involve inclined asteroid orbits and cannot be studied by planar models. Therefore, we have investigated the location and the dynamic effects of the three main secular resonances v 5, v 6 , and v 16, inside the mean motion commensurabilities. This is done by means of a seminumerical method, which allows us to obtain a 3-D picture of the dynamics, including the case of asteroids at high inclinations. Our findings throw new light on the existence of the Hecuba gap and the Hilda group in the asteroid belt. In particular, we have found that: (i) The interaction between the v 5 and the v 6 resonances gives rise to large chaotic zones for eccentricities >0.25 and >0.45 in the 3/2 and 2/1 resonances, respectively, which may cause close encounters with Jupiter and ejection from the solar system—however, the Hilda asteroids avoid this chaotic zone due to a phase-related protection mechanism. (ii) At large eccentricities the resonance causing a libration of the argument of perihelion is also present and can force strong secular changes of the eccentricity and the inclination. (iii) Although the Hilda group is bounded below the v 16 resonance curve in the inclination vs. eccentricity plane, such a boundary is not present in the 2/1 resonance, where v 16 affects zero-inclination orbits at eccentricities of about 0.2. However, this resonance can substantially increase the inclination but not the eccentricity and does not affect nearly circular orbits; therefore its location is not sufficient to explain the formation of the Hecuba gap.