From the test performed in Section 4 we can only conclude that, of the already known identifications, could have been found by using the algorithm of linear identification (as described in Section 2.2) to provide a first guess, followed by standard iterative differential corrections. Even the remaining could have been found at the cost of performing two attempts of differential corrections, using an alternative (low computational cost) algorithm to compute the first guess. Thus, there is good evidence that we have found a workable algorithm to propose orbit identifications.
The question is, how effective such an algorithm is in finding new, not previously proposed identifications. The results we have obtained in the March and April 1999 runs, essentially 150 `true' orbit identifications, are impressive and at the same time inadequate. More than 100 of these were not based on recent data. They are impressive in that we have recovered 150 lost asteroids, and this by pure computation, without using a telescope; moreover, we have removed from the catalogues 300 poor orbits, for which telescope recovery would have been time consuming, in some cases almost impossible. But they are inadequate to the size of the problem in that the number of orbits has only decreased by less than . Thus we can conclude that yes, the algorithm of this paper is effective in detecting orbit identifications, which not only were unknown, but in many cases did escape to all the other methods in use for identification. We have also to conclude that no, the method of this paper is not the final solution of the orbit identification problem, because many more are certainly yet to be found.
The conclusion of this paper should therefore be positive: this method should be used because it is very effective, especially in that it uses efficiently both computing and telescope resources. It needs to be stressed that this efficiency would be even more significant if this method was used as a matter of routine on all new discoveries for which an orbit can be computed (by least square fit of all the elements). In this case the computational complexity would be linear in the number of known orbits, not quadratic as it happens in an initial catalog cleanup with a new method. The immediate availability of the identified orbit, as a result of a fully automated procedure, without waiting for the identification diggers' hard work, allows to immediately exploit it for attributions and close approach analysis; dedicated astrometric follow up may become unnecessary, and the reduced uncertainty makes telescope recovery much easier, in case it is needed.
The conclusions about the global problem of asteroid identification are not so simple, and although they are beyond the scope of this paper, we need to anticipate some of them, at least as much as it is necessary to understand the direction of our future work.
One possible way to attack the problem, that is to find many more identifications, is to take better into account the nonlinearity of the identification problem. There is a well known way to approximate a linear function better than a single linear approximation: to use the differential, namely the local linear approximation, at many different points. This can be achieved by combining the multiple solutions algorithm of [Paper I], Section 5, with the linear algorithm of this paper, Section 2.
Another direction is to look for identifications of the attribution type, that is to perform matching of the observations on the celestial sphere rather than of the elements in the phase space. It is indeed the case that today many more identifications are found by attribution, rather than by orbit identification, and this by many people, including ourselves. Attribution methods are thus promising, but they also have their problems, especially in the difficulty of confirming the identification without additional observations.
In the near future we plan to work following these two approaches, and also by combining both of them.
The software to compute all the identification metrics and the first guess for identification, as discussed in this paper, has been included in the free software system ORBFIT. The most recent version of ORBFIT (currently 1.9.0) as well as online documentation can be obtained on the WWW at http://copernico.dm.unipi.it/neo/. It is also available by anonymous ftp from the server copernico.dm.unipi.it, in the directory pub/orbfit.
Acknowledgements: We thank K. Muinonen for giving the original impulse to this paper by suggesting to combine two confidence ellipsoids. Muinonen favours the probabilistic interpretation of the least squares principle (as opposed to the optimisation interpretation used in this paper), but many of the equations are the same. We thank E. Bowell for providing not only the public version of the Lowell Observatory Asteroid Orbit file, but also the unpublished covariance data, and even versions of the file specially adapted for our purposes. The ORBFIT free software is maintained by a consortium led by A. Milani, M. Carpino, Z. Knezevic and G.B. Valsecchi. This research has been supported by the European Community grant CHRX-CT-94-0445.