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Next: 2. The filtering procedure Up: THE ASTEROID IDENTIFICATION PROBLEM Previous: THE ASTEROID IDENTIFICATION PROBLEM

1. Introduction

The general problem of asteroid identification aims at detecting cases in which the same physical object has been observed in two arcs, separated in time, without an a priori knowledge that they were the same. The algorithms being used, by our group as well as by others, to achieve identifications are many and very different. The main distinction is whether the possible identifications are selected by proximity in the observations space or by proximity in the phase space of orbital elements. Since the algorithms in these two cases are very different, we prefer to use distinct words, following the definitions given in the first paper of this series [Milani 1999] and corresponding to the distinction between different identification procedures in [Marsden 1985]. An orbit identification is a case in which the observational data of both arcs are sufficient to separately solve for two orbits, one for each arc; then the input data are two sets of orbital elements, and the identifications are proposed on the basis of some measure of similarity of the orbits (which is not just the size of the difference in the orbital elements; [Milani et al. 2000a]). In contrast, a linkage joins together two observed arcs, each one over a too short time span and/or with too few observations to allow for a meaningful orbit determination, in such a way that the two together are enough to compute an orbit. The third case is mixed: an attribution is obtained by comparing an amount of observational data, insufficient to compute a reliable orbit for one arc, to the orbit already computed for the other arc. As a matter of principle, this goal is obtained by predicting observations, based upon the orbit which is known for one arc, and then comparing with the observed data for a second arc. It is important to realize that attributions sometimes are very simple, and sometimes are very difficult. When the time span of the better observed arc is very long, covering several apparitions, the predictions of the observations can be very accurate and the conjectured attribution is obtained by a simple comparison: the data either are or are not consistent with the prediction, within the expected observational errors. We find this kind of situation when, for example, observations are attributed to numbered asteroid orbits. In contrast, the shorter the time span of the arc used to compute an orbit, the larger the uncertainty of the predictions. For short arc orbits the comparison between predicted and observed values cannot be done meaningfully unless the prediction uncertainty is accounted for in the comparison algorithm. If all the asteroids detected (or at least the vast majority of them) were followed up for a time span long enough to compute a reasonable orbit, then orbit identification would be a more effective algorithm because it indeed exploits more information; in intuitive terms, the comparison taking place in a six dimensional space can provide tighter constraints than a comparison in a two dimensional space (the celestial sphere). Unfortunately the majority of asteroid discoveries are not followed up after either one or just a few nights. This fact results from priority being given only to discovery observations, as opposed to observations allowing recovery and good orbit determination, in giving credit to the observers. It is also a consequence of the pressure to preferentially follow up unusual objects. Let us take as an example the dataset of asteroid astrometric observations made public by the Minor Planet Center (MPC) on April 18, 2000. Out of 135,435 provisional designations (corresponding to independent discoveries of asteroids still unnumbered) there were 14,416 with observed arc shorter than 12 hours (pre-1992 ``one night stands''), 19,062 longer than 12 but shorter than 36 hours (observations in two consecutive nights), and 18,435 longer than 36 hours and shorter than 60. The reasons for a larger number of two nighters has to do with the criterion used by the MPC to assign a new designation, which was modified in 1992 to the effect that a one night stand is no longer provided with a provisional designation. Essentially none of the one night stands since 1992 are currently made available by the MPC for outside analysis. Although we have successfully applied an orbit identification algorithm even to orbits based upon observed arcs of 2-4 days, for roughly three out of four of the designations with arcs between 2 and 4 days we are not currently computing any orbit at all1. Thus, limiting the identification algorithms to the orbit identification case would represent a significant decrease of the opportunities of identification. As long as the residuals (observations, conjectured to belong to the same object, minus prediction) are small enough, the linear formalism of normal and covariance matrices provides a rigorous and comparatively efficient algorithm to assess the likelihood of an attribution. Nonlinear methods are available to deal with cases in which the residuals are well outside the linearity region, e.g., the semilinear method of [Milani 1999]. Indeed, these methods have been successfully used when the problem is to investigate the suitability of a specific attribution, as it is the case when the observations belong to a targeted recovery of the same object. Much more challenging is to propose attributions, that is to start from a large catalog of orbits and a large catalog of short arc observations and to select a reasonable number of couplings between them, to be confirmed by a more detailed analysis of each case. Using again the data from the April 2000 monthly update, we have computed 67,324 unnumbered asteroid orbits, and have generated 74,656 attributables (this term is better explained in Section 2) that might be linked to the unnumbered orbits. This means we should, in principle, perform about $5\times 10^9$ comparisons. This large number of pairs to be tested implies that there are two problems to be considered distinctly. On one hand we would like to define rigorous algorithms to filter out, among the huge number of orbit-attributable pairs, the ones that could correspond to the same object. On the other hand, the computational efficiency of the filtering procedure is critical to apply the algorithms with reasonable computing resources. This paper contains the definition of a procedure for selecting orbit-attributable pairs and reports on the practical tests performed in about one year of operation. The main feature of this procedure is that it consists of a three stage filtering, with the pairs passing the first filter further analyzed with the second filter and finally the ones passed by the second filter confirmed by accurate least squares fit. Thus, unlike the procedure for orbit identifications of [Milani et al. 2000a], the three stages are qualitatively different, and they require different strategies for optimization. Orbit identification algorithms are discussed in [Milani et al. 2000a], but there is one interaction with attributions that needs to be discussed here: after an identification has been found, with any one of the two algorithms, this new orbit can be used to search for attributions to it. Since in most cases the orbit resulting from an identification is multi-opposition, the uncertainty will be much smaller and additional attributions, if they indeed exist in the observations archives, are comparatively easy to add. However, this scheme requires the implementation of a recursive procedure with considerable efficiency problems. We discuss here the extent to which the entire sequence of procedures for orbit identification, attribution, and attribution to identification can be automated. This paper is organized as follows: Section 2 describes the three stage filtering procedure, Section 3 discusses the results of the application of this procedure for about one year and analyzes the properties of the identifications found in this way. Section 4 contains an overall appraisal of the results, a comparison with the results obtained by other groups and some indications on the open problems to be solved to further increase the efficiency and the productivity of searches for identifications.
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Next: 2. The filtering procedure Up: THE ASTEROID IDENTIFICATION PROBLEM Previous: THE ASTEROID IDENTIFICATION PROBLEM
Andrea Milani
2001-12-31