Statistical method for deriving spatial and size distributions of sub-km main-belt asteroids from their sky motions

The importance of sub-kilometer-sized objects among the main-belt asteroids (MBAs) has been progressively recognized in relation to the origin of near-Earth asteroids and the formation of rubble-pile asteroids. However, exact orbit determinations of those individual objects are practically impossible, because both their discovery and subsequent follow-up observations require 8-10 m class telescopes, in which observation-time competition is severe. Therefore, we examine here instead a statistical method to deduce the spatial and size distributions of sub-km MBAs from only their apparent motion vectors on the sky. Assuming their near-opposition and near-ecliptic observations, we made simulations to estimate the accuracy of the semi-major axis (a) and inclination (I) obtained from their sky-motion vectors. The mean errors of a and I for each asteroid were found to be about 0.15 AU and 1degrees-2degrees (for the asteroids with I < 10degrees), respectively. Then, under a certain assumption, we calculated magnitudes and the size distribution of those computer-synthesized sub-km MBAs. Our statistical method could reproduce the slope of the cumulative size distribution for the original asteroid populations with errors of similar to 0.05-0.1. These values are small enough for our survey purposes using the Subaru Telescope.