Orbital evolution of asteroids during depletion of the solar nebula

The pumping-up of inclinations and eccentricities caused by sweeping secular resonances in the asteroid belt is studied through numerical orbital integration and linear analysis. The present asteroids have large mean eccentricities and inclinations that cannot be explained by planetary perturbations alone. Sweeping of secular resonances associated with the depletion of the primitive solar nebula has been proposed for the origin of asteroids' orbits (e.g., by Ward, Colombo, and Franklin in 1976). We performed three-dimensional orbital integrations of asteroids under gravitational forces by Jupiter, Saturn, and the solar nebula. The asteroids' motions are also affected by hydrodynamic gas drag. We consider three types of nebula depletion models: (1) the uniform depletion model, in which the nebula density decreases exponentially with time and uniformly throughout the nebula; (3) the inside-out depletion model, in which nebula gas is depleted from the inside region; (3) the gap-opening model, in which a gap centered at Jupiter gradually expands. Previous studies have concentrated on the two-dimensional uniform depletion model. Our simulation shows that inclinations of asteroids are not pumped up enough in the first model to account for the observed magnitude. Moreover, most asteroids spiral into the Sun by gas drag if the depletion timescale is longer than 10(5) yr because pumped-up eccentricity induces strong gas drag. On the other hand, in the second and third models, inclinations are pumped up. Our linear analysis shows that the nonuniform depletion model is essential for the secular resonances pumping up inclination to sweep in the asteroid belt. In the case of the inside-out depletion model, both eccentricity and inclination are pumped up enough to be consistent with the observed magnitude in the entire asteroid belt, if the nebula depletion timescale (which is the time required for the nebula edge to migrate by the distance of 5 AU) is longer than 3 x 10(5) yr. Furthermore, since secular resonances pass after nebula gas has already been depleted in the passing region, gas drag does not damp the pumped-up eccentricities and inclinations and the semimajor axes of the asteroids. Therefore, the resultant eccentricity and inclination in the inside-out depletion model are consistent with observed ones for the nebula depletion time inferred from observed T Tauri stars (10(6)-10(7) yr). Our gap-opening model shows similar results, although eccentricity and inclination are slightly smaller in the outer asteroid belt region. The essentially required condition to pump up both eccentricity and inclination highly enough in the asteroid region is that the nebula edge moves outward from 5 to 10 AU. The gap formation caused by Jupiter's tidal perturbations may be similar to our inside-out or gap-opening models, inside 10 AU. Therefore, it would be responsible for the high eccentricities and inclinations in the asteroid belt.