2.1 First selection

The first step of the filtering procedure consists of a rough comparison of the available short arc observations with the predictions resulting from the available nominal orbit. This means we need two catalogs to start with. One catalog contains the nominal orbits, obtained by a least squares fit to the observations of arc 1. The second catalog contains the attributables, which are single observations computed in such a way that they can represent all of the observations of arc 2. Before giving a more precise description of the content of an attributable, we need to understand how it is going to be used, both in the first and in the second filtering stages. In the first step, the nominal orbits have to be propagated to the epoch of the attributables. The problem is that the number of attributables is the number of short arc discoveries. We use all the arcs containing at least two observations² not degraded in accuracy and with a time span not exceeding 8-10 days. Thus, the total number of attributables is usually more than 70,000 (e.g., in the April 2000 update, it was 74,656); moreover, they are unevenly distributed in time, much denser in recent times. Propagating each orbit to each one of these times would be very inefficient. Typically we perform about 5,000 integration steps for each orbit, and we would be forced either to perform integration steps shorter than the ones optimal for computational efficiency, or to use interpolation procedures which are also computationally expensive. Therefore in the attributable record we include an extrapolated observation for the nearest time which is an integer multiple of a fixed $\Delta t$ (we currently use 10 days). To obtain this, we first compute the straight lines in the $(t,\alpha)$ and in the $(t,\delta)$ planes best fitting to the data of arc 2. Then we define a central time t_m as the arithmetic mean of the observation times, neglecting the observations which are degraded in accuracy (with weights corresponding to RMS errors larger than 4 arc seconds). We then identify the time t_step multiple of $\Delta t$ and closest to t_m, and compute the value on the best fitting lines of $\alpha(t_{step}), \delta(t_{step})$. The use of a linear fit over a time span of up to $\Delta t/2=5$ days is inaccurate, but the first filtering stage uses a control d<R on the distance d between prediction and observation which is large, e.g., R=1.5 degrees (d is computed by the usual metric on the celestial sphere: $d^2=\Delta\delta^2+\cos^2\delta\,\Delta\alpha^2$). Of course, the use of such a loose control results in many false positive, but the area of the region within 1.5 degrees from a given position on the celestial sphere is only $A_1\simeq 7$ square degrees. Taking into account that most asteroid detections take place in a band around the ecliptic which has an area of the order of 10,000 square degrees, this explains the ratio between pairs examined and those passing the first filter. Thus the value of 1.5 degrees is appropriate to make the second stage of our procedure efficient. We have expended a significant effort to make the orbit computation efficient by optimizing the orbit propagation routines of the OrbFit software. Nevertheless, the first filter is a CPU intensive step of the attributions procedure since the orbit computation must be accurate and the attributables are spread over a total time span of about 100 years. As an example, during the May 2000 update we have used for this stage about 50 CPU hours (spread on three different computers). However, it is not necessary to search for attributions to all the orbits for all the attributables every month, as we did in April and May 2000; only the orbits changed since last month need to be tested on all attributables, and the new attributables need to be tested for all the orbits.