5. New Identifications 4. Proposed algorithms and 4. Proposed algorithms and

4.1 Test on 100 previously known cases

To test the proposed algorithm, and also to tune the control values of the different distances to be used to propose identifications, we have selected a set of already identified orbits with the following criteria:

1.

the asteroids have been observed at exactly two oppositions;

2.

for each opposition, there is an arc of at least 6 days with at least 5 observations;

3.

the semi-major axis is less than 6 AU (no trans-Neptunians, for which different criteria should be used).

  

We have used a set of 100 examples, satisfying the above conditions, for which observational data are available from the Minor Planet Center Extended Computer Service. For all, we have recomputed the best fit orbit and the normal/covariance matrices for each one of the two arcs containing observations from a single opposition.

These nominal solutions, and the corresponding matrices, are computed for the epoch of the last observation in each arc. Then we have propagated all the orbital elements, and all the normal and covariance matrices, to a common epoch (we have used $t_0 =2\,447\,000.5\; JD$; for the dependence of the results upon this date, see Section 5). This propagation was done with accurate numerical solutions of the $N$-body equations of motion, and of their variational equations, while the normal and covariance matrices were both propagated by means of equations (4), with no further inversions after the single one performed at the observations epoch.

With all the state vectors and normal/covariance matrices referred to the same epoch, the algorithms of Section 2 can be applied, and we have used them according to the same procedure outlined at the beginning of this Section, namely by computing $d_2$, $d_5$ and $d_6$ for each test couple, formed by two observed arcs known to be of the same asteroid.
  

The results of the computation of our three metrics are shown in the histograms of Figures 2-4. In Figure 2 we show the results for the inclination-only distance $d_2$; it is apparent that this distance has always a low value, $\leq 10$ in $89\%$ of the test cases, $\leq 30$ in $97\%$, with a maximum value of $80.4$. This is, however, a loose criterion which can be used only as a preliminary filter, since in a large catalog with tens of thousands of orbits it would be satisfied by millions of couples.

The distance $d_5$ being small is obviously a much stronger constraint on the couples of orbits to be identified. Indeed it has a small value for many of the actually identified orbits: $\leq 10$ in $59\%$ of the test cases, $\leq 30$ in $73\%$. It has, however, as shown in figure 3, a moderate value in a number of cases: $30\leq d_5\leq 1\,000$ in $18\%$ of the cases, and an even larger value in $9\%$. The distance $d_6$ is the one associated with the fully linear identification algorithm. Its value is always larger than that of $d_5$ (this is just an example of the property $K_E\leq K$ proven in Section 2.3). Nevertheless, the value of $d_6$ is low in most cases: $\leq 30$ in $70\%$ of our test sample; a low to moderate value $\leq1\,000$ covers $86\%$ of the cases, as shown in Figure 4.

  

A final test is based upon equation (2), proposing two different formulas to compute the matrix $C$, therefore $K$, therefore the identification metrics. These two formulas would give identical results in exact arithmetic, but are susceptible to give discordant results when the normal matrices have very large conditioning numbers. The alternative values for both $d_2$ and $d_5$ are essentially identical (differences less than $0.01$) in all cases, while for $d_6$ the difference are larger than $1/10$ of the value of $d_6$ in $13\%$ of the cases. This is not unexpected: the full covariance matrix (also the normal matrix) has a conditioning number growing with the time elapsed after the observations, as it can be seen already from the 2-body approximation of Paper I, Section 4.1; the reduced $5\times 5$ matrix does not have this property. Anyway the difference between the two values is not really important in any of the test cases.

Another important test of our identification algorithm is the following. The theory outlined in Section 2 does not only provide a minimum identification penalty, but also a first guess for the orbit resulting from the identification. The full linear algorithm computes a complete set of orbital elements ($X_0$ in the notations used in Section 2.1) which is the best solution in the linear approximation, and should be used as starting value for an iterative differential correction procedure, including all the observations from both arcs.

On the contrary, the restricted identification procedure provides only a first guess for the orbital elements in the selected subspace ($E_0$ in the notations used in Section 2.2). For the reasons already discussed, this procedure does not provide a first guess for the elements not included in the vector $E$. Thus, the identification based only upon the two elements $p,q$ does not provide a useful first guess, and the procedure based upon five orbital elements provides a first guess for all the elements but the mean longitude $\lambda$.

A simple minded procedure could be to devise some first guess $\lambda_0$ by a procedure which does not take into account the uncertainty of the two separate solutions for the two arcs, e.g. $\lambda_0$ could be just the mean of the two values resulting from the two separate arc solutions for the same epoch $t_0$ (note that the average of two angles is not the average of their principal values, but needs to be done with some care to avoid a mistake by $2\pi/2$).

As it could have been expected, this method to compute a first guess is not very successful: the differential correction iterative procedure converges to the identification orbit for only $80\%$ of the cases. But it can be valuable in cases where the $d_6$ is not useful due to very large uncertainty in $\lambda$. In fact, if the marginal uncertainty in $\lambda$ is more than one revolution the $d_6$computation can actually return more than one value, because the difference in $\lambda$ can contain integer multiples of $2\pi$. In these cases we use the result with the lowest $d_6$, but the reliability of the test in these cases is dubious.

Then the real test of the quality of the linear identification algorithm is to try to achieve convergence of the iterative differential corrections procedure, using as starting point the $X_0$ set of orbital elements suggested by the algorithm as best solution in the linear approximation. We have performed this test, and found convergence to the identification orbit in $99\%$ of the cases.

  

The results of these tests on the convergence of differential corrections are summarized in Figure 5; the crosses indicate the cases in which the convergence failed, when starting from the $E_0$ guess; in all these cases, however, the full algorithm starting from $X_0$ succeeded, with only one exception, which is indicated by the cross at the top right of the Figure, with values of both $d_5$ and $d_6$ above $30\,000$. Even in this isolated case, the differential corrections algorithm converged when the first guess was the set of interpolated elements obtained with the algorithm proposed in [Sansaturio et al. 1996].

5. New Identifications 4. Proposed algorithms and 4. Proposed algorithms and