The Combined Effects of Cold-Nebula Drag and Mean-Motion Resonances

Abstract

A model for Epstein drag forces in the Solar System's primordial dust cloud is proposed; this model is consistent with the asteroid belt forming in a cold nebula. Small bodies are assumed to move in exact Keplerian orbits in the absence of drag; the orbital average of the collisional effects leads to circularization of orbits, for any density profile of the nebula. An increase in density near Jupiter's orbit is assumed. In the presence of such a density gradient but in the absence of a massive perturber, orbital radii will increase or decrease according to whether the relative outward gradient is larger or smaller than a critical value. Given both a density gradient and a massive Jupiter, analysis of the planar restricted three-body problem so modified leads to a condition on the relative density gradient for the (initially small) eccentricity of an asteroid near a given resonance with the perturber to increase or decrease. These analytical results on the evolution of asteroid eccentricities in the combined presence of mean-motion resonances and solar-nebula drag effects are found to be in good agreement with those obtained by a seminumerical averaging scheme. Based on these analytical and seminumerical results, we formulate a self-consistent hypothesis for the general pattern of gaps and groups in the asteroid belt.