Asteroid orbits using phase‐space volumes of variation

We present a statistical orbit computation technique for asteroids with transitional observational data, that is, a moderate number of data points spanning a moderate observational time interval. With the help of local least‐squares solutions in the phase space of the orbital elements, we map the volume of variation as a function of one or more of the elements. We sample the resulting volume using a Monte Carlo technique and, with proper weights for the sample orbital elements, characterize the six‐dimensional orbital‐element probability density function. The volume‐of‐variation (VOV) technique complements the statistical ranging technique for asteroids with exiguous observational data (short time intervals and/or small numbers of observations) and the least‐squares technique for extensive observational data. We show that, asymptotically, results using the new technique agree closely with those from ranging and least squares. We apply the technique to the near‐Earth object 2004 HA 39 , the main‐belt object 2004 QR and the transneptunian object 2002 CX 224 recently observed at the Nordic Optical Telescope on La Palma, illustrating the potential of the technique in ephemeris prediction. The VOV technique helps us assess the phase transition in orbital‐element probability densities, that is, the non‐linear collapse of wide orbital‐element distributions to narrow localized ones. For the three objects above, the transition takes place for observational time intervals of the order of 10 h, 5 d and 10 months, respectively, emphasizing the significance of the orbital‐arc fraction covered by the observations.