The gravitational effects of the Sun on a particle orbiting another massive body which itself moves on a circular path around the Sun have been studied extensively. Most recently, Hamilton and Burns (1991) characterized the size and shape of a stability zone around an asteroid on a circular heliocentric orbit within which material could remain bound for an extended period of time. We now consider two additional effects analytically and numerically: the asteroid's non-zero heliocentric eccentricity and solar radiation pressure. In both of these cases, our numerical integrations apply directly to a spherical asteroid, ``Amphitrite,'' with semimajor axis 2.55 $AU$, radius $R_A=100 km$, and density $2.38 g/cm^3$. For an asteroid on an eccentric orbit we argue, based on numerical integrations and analytical approximations, that the stability zone scales roughly as the size of the Hill sphere calculated at the asteroid's pericenter. This scaling holds for large values of eccentricity and allows results for one asteroid with a given mass, semimajor axis and eccentricity to be used for another with different values of these parameters. We compare predictions of the scaling law to numerical integrations for an ``Amphitrite'' with various orbital eccentricities and find good agreement for prograde orbits and for those with orbital planes nearly normal to the asteroid's heliocentric path, but not for retrograde orbits. We apply our results to the minor planet 951 Gaspra. We also determine that solar radiation pressure is a very efficient mechanism for removing relatively small particles from the circum-asteroidal zone. Radiation pressure acting on an orbiting grain can cause large oscillations in the grain's orbital eccentricity which in turn can lead to either escape from the system or impact with the asteroid. We find numerically that particles with radius 0.1 millimeter started on circular orbits escape from ``Amphitrite'' at all distances beyond 130 \RA. Grains of this size started anywhere between the asteroid's surface and 130 \RA are forced to crash into the minor planet. Smaller grains are even more severely affected; we find that all particles with radii ranging from $<1$ micron to tenths of millimeters are swept from the circum-asteroidal environment on timescales comparable to the asteroid's orbital period. The orbits of millimeter-sized grains are also strongly perturbed. Planar paths bound for twenty years are found to extend to only $\sim40\%$ of the critical distance found by Hamilton and Burns (1991); orbits with inclinations near $90\deg$ are somewhat more resilient. Particles larger than a few centimeters are only slightly affected by radiation pressure. These results can be applied to ``Gaspra'', an asteroid only one-thousandth as massive as ``Amphitrite'', by increasing all particle sizes by a factor of $\sim$ ten.

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