1. Introduction

A large fraction of the asteroids observed so far are to be considered lost: they cannot be recovered by pointing the telescope at the predicted position only. The catalogs of asteroid orbits are therefore polluted by large numbers of low accuracy orbits that cannot be easily improved by observation. Two of these inaccurate orbits can belong to the same physical object, and indeed identifications are regularly found when the orbital elements are close; an unknown number of real identifications have not been found in the existing catalogs because the orbital elements as computed are far apart. This situation requires some action to be taken because the number of low quality orbits is indeed increasing with time, notwithstanding the efforts of many dedicated people in the maintenance of catalogs, and it results in a significant waste of the precious resources of telescope and observer time.

The completion of the task requires an improvement in both the observation capability and the theoretical understanding of the problem. To contribute to the solution from the theoretical side, this paper belongs to a series dedicated to the asteroid identification problem, begun with [Milani 1999], hereafter referred to as [Paper I].

In general, the asteroid identification problem deals with separate sets of astrometric observations, which are assumed to form two observed arcs, each one belonging to one object; what is not known is whether the two arcs refer to the same object. The problem can take very different forms, depending upon the amount and distribution in time of the available observations for each arc, and upon the time interval between the two arcs; a rigorous terminology to distinguish the different cases has been introduced in [Paper I]. This paper deals with one specific case, the orbit identification problem: it occurs when each one of the two arcs has been observed well enough to allow for the computation of a complete orbit, by which we mean an orbit with all six orbital elements solved for by a least squares fit to the observations. It needs to be stressed that most asteroid identifications performed by observers and orbit computation centers do not belong to this case, but to either the attribution or to the linkage ones. This is because a large fraction of the available observations belong to very short arcs, lasting only very few nights, and even a single night, and in these cases complete orbits either do not exist, or are essentially undetermined, in a sense which will be better explained in Section 3.1. Even a few orbit identifications are, however, important achievements: each one of them not only removes two low quality orbits from the catalogs, but also provides a good one, which can in turn be useful in many ways, including the attribution of other short arcs, the prediction of close approaches [Milani and Valsecchi 1999], and the recovery.

Two facts are the main sources of difficulty in finding a rigorous and effective algorithm for orbit identifications. First, the number of observed asteroids is very large (and increasing fast). Even taking into account only the complete orbits there are far too many couples to subject each one to rigorous testing. In Section 5 we have used a catalog of more than $35\,000$ orbits for unnumbered objects, so the algorithm to propose identifications should test on the order of $6\times10^8$ couples. Despite the power of modern computers, this implies that the computations performed on each couple need to be very simple. All the algorithms we propose here involve the computation of some distances $d_i(N_1,N_2)$ between two orbits identified by the identifiers $N_1, N_2$, and then the couples satisfying the simple condition $d_i(N_1,N_2)<\varepsilon_i$ are selected for further investigation. These selection criteria can be applied as a cascade of filters, each selecting fewer and fewer couples until a small number of strong proposals are submitted to the final test of the least squares fit of the orbit to the observations of both arcs.

The second fact generating difficulties is that even a complete orbit, i.e., the solution of a least squares fit to all the observations of a given arc, does not indicate the only possible orbit for the observed object; there is around each solution a confidence region where the true orbit could be without significantly increasing the size of the observation residuals. Thus it is not enough to compute some distance, defined by a suitable metric, between the nominal least squares solutions; the distance between two orbits must be computed taking into account the uncertainty of both orbits. As an example, if a well determined orbit is inside the confidence region of a poorly determined orbit, this couple is a strong candidate for orbit identification, independent of the size of the difference between the two nominal orbits.

In this paper we propose a rigorous and computationally efficient algorithm to propose identification by means of distances between couples of orbits taking into account the linear approximation of the uncertainty of both orbits, as expressed by their normal and covariance matrices, and by their geometric counterparts, the confidence ellipsoids. This method has been tested both on already known identifications, which have been reproduced, and in searching for new ones, of which a significant number has been found.

This paper is organized as follows. In Section 2 we introduce the idea of a penalty associated with the identification, taking into account the linearized orbit uncertainty. In Section 3 we discuss the difficulties, and necessary cautions, arising from the weakly determined orbits, from nonlinear propagation of the uncertainty, and from poor knowledge of the error statistics. In Section 4 we propose a sequence of filtering stages, based upon some different identification penalties, and test them on a sample of already identified orbits. In Section 5 we describe results obtained by applying these tests to an actual asteroid orbit catalog that we have computed based on the recently available global dataset of asteroid observations. Finally, in Section 6 we draw some conclusions about the value of our methods to actually find new identifications, but also on the need for further work.