2.2 The checking algorithm

In the second filtering stage we propagate the orbit to the exact time of the attributable, rather than the rounded time used in the first stage. As the time of the attributable we use the central time t_m defined above. The attributable contains the values of $\alpha(t_m)$ and $\delta(t_m)$ as derived from the linear fit, but also the slopes of the same fit, that is the proper motions $\dot{\alpha}(t_m)$ and $\dot{\delta}(t_m)$, which provide the rate of motion on the plane tangent to the sky. This information allows us to perform the comparison with the predictions in a four dimensional space. The orbit is propagated with the full variational equations, in such a way that the covariance of the predictions can be computed (see [Milani 1999], Section 3.1). Let $\widetilde{\Gamma}_{obs}$ be the $2\times 2$ covariance matrix of the predicted angles $(\alpha,\delta)$ and $\widetilde{C}_{obs}=\widetilde{\Gamma}_{obs}^{-1}$ the corresponding normal matrix. Since it is better at this stage to use a prudent estimate, we assume that the observation errors could be comparatively large, e.g., with an RMS $\sigma_{obs}=2$ arc seconds. Then, the covariance matrix expressing this assumed uncertainty in the $(\alpha,\delta)$ plane is

$$\Gamma_{obs}=\left[\begin{array}{cc}{\sigma^2_{obs}/\cos^2 \delta}&{0}\\ {0}&{\sigma^2_{obs}}\end{array}\right] \ ;$$

let $C_{obs}=\Gamma_{obs}^{-1}$ be the corresponding normal matrix. Then, the likelihood that the predicted and the attributable observations are indeed the same can be expressed by the 2-dimensional penalty [Milani et al. 2000a]

$$\Delta Q_{obs}=\left[\begin{array}{c}{\alpha(t_m)-\alpha}\\ ... ...(t_m)-\alpha}\\ {\delta(t_m)-\delta}\end{array}\right] \ .$$

Since the proper motion data are also available, let $\widetilde{\Gamma}_{vel}$ be the covariance matrix of the predicted proper motions $(\dot{\alpha},\dot{\delta})$, and $\widetilde{C}_{vel}=\widetilde{\Gamma}_{vel}^{-1}$ the normal matrix. The uncertainty of the proper motions as estimated from observations depends upon the length $\Delta t$ of the observed arc: $\sigma_{vel}=\sqrt{2}\,\sigma_{obs}/\Delta t$. Then the covariance matrix expressing this assumed uncertainty in the $(\dot{\alpha},\dot{\delta})$ plane is

$$\Gamma_{vel}=\left[\begin{array}{cc}{\sigma^2_{vel}/\cos^2\delta}&{0}\\ {0}&{\sigma^2_{vel}}\end{array}\right]$$

and, with $C_{vel}=\Gamma_{vel}^{-1}$, we can use the same 2-dimensional penalty:

$$\Delta Q_{vel}=\left[\begin{array}{c}{\dot\alpha(t_m)-\dot\al... ...lpha}\\ {\dot\delta(t_m)-\dot\delta}\end{array}\right] \ .$$

The two penalties have to be combined to assess the likelihood of the attribution: we use

$$\sigma=\sqrt{\Delta Q_{obs}+ \Delta Q_{vel}}$$

because each penalty is an increase in the value of the target function, which is related to the square of the residuals (in this case the residuals are the difference between the prediction for the given orbit and the observation of the given attributable). It would be possible to use a full 4-dimensional penalty, taking also into account the correlation between predicted angles and proper motions, but this does not appear to be necessary. The computational cost of the second stage is much less than that of the first one, essentially because the number of pairs orbit-attributable to be tested has been decreased by three orders of magnitude by the first filter. As an example, during the May 2000 update we have used about 7 CPU hours. The geometrical meaning of this method is the following. The observation as given in the attributable has an unknown error; the real one could be anywhere within a distance of the order of $\sigma_{obs}$, along great circles on the sphere (the factor $\cos^2\delta$ correctly accounts for the metric on the sphere); this is analytically described by the normal matrix C_obs. The prediction is in turn uncertain, its confidence region being the ellipse defined by the normal matrix $\widetilde{C}_{obs}$. We are looking for intersections of the two confidence regions. This situation is mathematically the same as the identification problem in the elements space, only in two dimensions. The same argument applies to proper motions.

  

Figure: Histogram of the number of attributions, submitted by us and published by the MPC, as a function of the control parameter $\sigma$ of the checking algorithm.

It is not easy to decide a priori the control value $\sigma$ to be used for confirming the proposed attribution and passing it to the final differential correction procedure. Figure 1 shows the values of $\sigma$ for the attributions which have been accepted by the MPC. Note that during the April update we have ourselves selected for differential corrections all the cases with $\sigma<20$, but only $\simeq 25\%$ of these have $\sigma\geq 8$. Among the pairs later passing the third filter, only $2\%$ had $\sigma\geq 8$. This means the control value can be kept very low, and we actually plan to use a lower value in the future. The number of good cases that could be missed by decreasing the control value would be small: as shown in Figure 1, the number of published attributions with $\sigma>15$ is just 5 (including one case out of scale in the plot, with $\sigma\simeq 25$). From Table 1 we can infer that the fact of passing the second filtering stage with a low value of $\sigma$ has a good predictive value for attributions to multi-opposition and medium arc ($>\,20$ days) orbits; that is, a significant fraction of the pairs passing the second filter are also passing the third filter. However, for shorter arcs the second filter is not so effective and a very significant computational effort has to be used in the third stage; this can be understood as follows. For short arc orbits, the confidence boundary of an observation prediction, many years after the asteroid has been lost, is typically several degrees long. Then the ratio between the area in which the attributables are passed by the first filter ( $A_1\simeq \pi\,R^2\simeq 7$ square degrees) and the area A₂ of the confidence region acceptable for the second filter is roughly

$${\displaystyle A_1 \over \displaystyle A_2}\simeq {\displaystyle \pi\, R^2 \over \displaystyle 2\, R\, \sigma_w\, \sigma}\ ,$$

where $\sigma_w$ is the width of the confidence region, which can be computed as the square root of the lower eigenvalue of $\Gamma_{obs}$. For R=1.5 degrees and a control value of the second filter $\sigma=20$

$${\displaystyle A_1 \over \displaystyle A_2}\simeq {\displaystyle 7\,' \over \displaystyle \sigma_w}\ .$$

This simple order of magnitude computation shows that for a width of the confidence region of the order of several arcmin the second filter becomes ineffective. If the available computing resources are not enough, a decrease in the control value of $\sigma$ is an acceptable compromise, in particular for short arc orbits: it would result in a significant decrease of the computational load of the third stage, with the possible loss of a small fraction of the real attributions detected.