We summarize several recent findings on the size and shape of the region within which material can stably orbit an asteroid. If the asteroid (with assumed density 2.38 g/cm3) circles the Sun at 2.55 AU, co-planar prograde material will remain trapped whenever started on unperturbed circular orbits at less than about 220 RA (asteroid radii); co-planar retrograde particles are stable out twice as far. Our 3-D stability surface, which encloses several hundred numerically calculated orbits that start with various inclinations, is shaped like a sphere with its top and bottom sliced off; its dimensions scale like the Hill radius =(m/3)1/3R, where m is the asteroid-to-solar mass ratio and R is the asteroid's orbital radius. If the asteroid moves along an elliptical orbit, a fairly reliable indicator of the dimensions of the hazard zone is the size of its Hill sphere at the orbit's pericenter. Grains with radii less than a few mm will be lost through the action of radiation forces which can induce escape or cause collisions with the asteroid on time scales of a few years; interplanetary micrometeoroids produce collisional break-up of these particles in ~104 yrs. The effects of Jupiter and of asteroids that pass close to the target asteroid allow particles to diffuse from the system, again shrinking the hazard zone.

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