We have numerically investigated a three-body problem consisting of the Sun, an asteroid, and an infinitesimal particle initially placed about the asteroid. We assume that the asteroid has the following properties: a circular heliocentric orbit at $R=2.55 \AU$, an asteroid/Sun mass ratio of $\mu=5*10^{-12}$, and a spherical shape with radius $R_A$=100 km; these values are close to those of the minor planet 29 Amphitrite. In order to describe the zone in which circum-asteroidal debris could be stably trapped, we pay particular attention to the orbits of particles that are on the verge of escape. We consider particles to be stable or trapped if they remain in the asteroid's vicinity for at least 5 asteroid orbits about the Sun, or about 20 years. Applying this criterion to particles started on circular orbits around the asteroid, we find that, as we vary the starting distance from the asteroid, a fairly abrupt transition between trapped and untrapped objects occurs. We define the distance where the transition occurs to be the critical distance. Our orbital plots for Amphitrite can be scaled for application to other asteroids using the same functional dependence as the Hill radius which is $r_H=(\mu/3)^{1/3}R$; for Amphitrite $r_H \sim 450 R_A$. We find empirically that initially circular prograde orbits remain bound out to a critical distance of about $r_H/2 = 225 R_A$, while initially circular retrograde orbits remain bound out to nearly twice that distance. Particle orbits that start out circular and are inclined with respect to the asteroid's orbital plane have critical distances between these two extremes. Note that our choice of initially circular orbits is arbitrary; different initial conditions would generally lead to different critical distances. This study explores the three-dimensional aspects of stability more thoroughly than previous studies. To first order, particles that are on stable orbits reside within a region that is approximately spherical for angles $\theta < 35\dg$ ($\theta$ is the latitude in a spherical coordinate system) such that $|z|< 285 R_A$, but is fairly flat and parallel to the $xy$ plane at $z=\pm~285 R_A$ over the poles. The radius of the spherical surface is roughly 480 $R_A$. These distances would be reduced if the model included the asteroid's orbital eccentricity or other perturbations such as those from Jupiter. This result does not address whether any mechanisms exist to populate such orbits.

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