4.1 Test on 100 THE ASTEROID IDENTIFICATION PROBLEM 3.3 Normalization/weighting of the

4. Proposed algorithms and test

We have selected a number of algorithms to propose identifications. They are based on the computation of a number of ``distances'' in the elements space $X$, which depend not only upon the difference $\Delta=X_2-X_1$ but also upon the uncertainties, as expressed by the normal matrices $C_1$ and $C_2$. We use:

1.

The penalty function $d_6={\displaystyle m \over \displaystyle 2}\,\Delta Q_6$ for the linear identification of all the six equinoctial orbital elements.

2.

The penalty function $d_5={\displaystyle m \over \displaystyle 2}\,\Delta Q_5$ obtained by applying the same formulas of Section 2 to a set of only 5 equinoctial elements, excluding the mean longitude. This computation should be much less affected by high conditioning numbers and by strong nonlinearities.

3.

The penalty function $d_2={\displaystyle m \over \displaystyle 2}\,\Delta Q_2$ obtained by applying the same formulas of Section 2 to a set of only 2 elements: $p=\tan{\displaystyle I \over \displaystyle 2} \sin\Omega$ and $q=\tan{\displaystyle I \over \displaystyle 2} \cos\Omega$. This distance can be used as a preliminary filter, to select the candidate couples of orbits to be subjected to more complicated computations.

The only caution to be used is to compute, for a reduced identification of the subset $E$ of the orbital elements $X$, not the restriction $C_E$ of the normal matrix but the marginal normal matrix $C^E=\Gamma_E^{-1}$ obtained by inverting the corresponding portion of the covariance matrix, as discussed in Section 2.3.

The algorithm to propose identifications should be as follows. First, over all the candidate couples, the distance $d_2$ should be computed and tested against some control value $d_2< \varepsilon_2$. The logic is that the orbit plane is, in most cases, significantly better determined than the other orbital elements. Thus the corresponding marginal normal matrices are typically well conditioned, and the problems of numerical instability and nonlinearity are much less severe.

On the other hand, the use of this criterion is not enough to discriminate the strong identification candidates. If the control $d_2$ is kept too low, some real identifications could be missed; if it is too high, the number of proposed identifications would be huge.

Thus the following step in the algorithm is to compute, for the couples selected on the basis of the $d_2$ criterion, the $d_5$distance and to test it against another control $d_5< \varepsilon_5$. If this test is also positive, the $d_6$ distance is computed and tested $d_6<\varepsilon_6$. The problem is to determine values of the controls $\varepsilon_2,\varepsilon_5$ and $\varepsilon_6$, in such a way that real identifications are not discarded, and the number of proposed identifications is not too large. These ``distances'' are dimensionless, and they do not have a straightforward physical interpretation; thus it is not possible to decide a range of acceptable values of these controls a priori in such a way that they should be satisfied by all real identifications. These acceptable ranges would have to accommodate all the possible effects of the three difficulties discussed in Section 3, namely bad conditioning, nonlinearity, and problems in residual weighting.

Therefore we have adopted a criterion based upon experimental evidence: we have computed the values of $d_2$, $d_5$ and $d_6$ found in a test set of already identified orbits, as discussed in the following Subsection.

 

• 4.1 Test on 100 previously known cases

4.1 Test on 100 THE ASTEROID IDENTIFICATION PROBLEM 3.3 Normalization/weighting of the