Precise Distances for Main-Belt Asteroids in Only Two Nights

We present a method for calculating precise distances to asteroids using only two nights of data from a single location --- far too little for an orbit --- by exploiting the angular reflex motion of the asteroids due to Earths axial rotation. We refer to this as the rotational reflex velocity method. While the concept is simple and well-known, it has not been previously exploited for surveys of main-belt asteroids. We offer a mathematical development, estimates of the errors of the approximation, and a demonstration using a sample of 197 asteroids observed for two nights with a small, 0.9-meter telescope. This demonstration used digital tracking to enhance detection sensitivity for faint asteroids, but our distance determination works with any detection method. Forty-eight asteroids in our sample had known orbits prior to our observations, and for these we demonstrate a mean fractional error of only 1.6% between the distances we calculate and those given in ephemerides from the Minor Planet Center. In contrast to our two-night results, distance determination by fitting approximate orbits requires observations spanning 7--10 nights. Once an asteroids distance is known, its absolute magnitude and size (given a statistically-estimated albedo) may immediately be calculated. Our method will therefore greatly enhance the efficiency with which 4-meter and larger telescopes can probe the size distribution of small (e.g. 100 meter) main belt asteroids. This distribution remains poorly known, yet encodes information about the collisional evolution of the asteroid belt --- and hence the history of the Solar System.