The survey model was developed starting from the study of the fragment cloud generated as described in Section 2 and from the information acquired on the fragmentation epoch. The idea is to perform a statistical scan of a region of the cloud selected by certain criteria. Hence, the fragments are propagated up to the observation night of interest to generate a series of points corresponding to the orbit of the fragments of the cloud (see Figure 5).

Figure 5 shows the simulated distribution of the fragments, spread across different altitudes. Given the mean geometry of the cloud and the positions of the observatories, all in the northern hemisphere, for this study and, thus, for the observation campaign, it was decided to select only the fragments placed in this hemisphere. From this first exclusion, the cloud was divided into sections or clusters (see Figure 6).

In Figure 6, the division into different sections is shown. After a series of iterations, to find a good distribution, the cloud was divided into 19 clusters for this study case. These are spaced 10° apart in RA, as shown in the right-hand graph, where the division in geocentric RA and Dec is depicted. In the left-hand graph, the division of the cloud into clusters is shown in the ECI reference frame, highlighting the centroids (black dots). This clustering is specific to the case under study and considers only the part of the cloud in the northern hemisphere, which is of interest because it is the part that was observable by the used observatory network.

Once the cluster is selected, to scan it, it is necessary to point to different regions within it using a specific pattern and then start over. For example, for a 10° × 10° cluster, it would be necessary to create a mosaic of 100 pointings for a system with 1° of FoV to complete it. Depending on the performance of the used observatory, this determines the time needed to complete the 10° × 10° cluster and restart the grid. If the time is longer than the dwell time, some objects that pass through a certain FoV of the grid might be missed. On the contrary, if the total time to complete the grid is less than the dwell time, the probability of spotting a fragment in a certain FoV of the grid is higher, because it has not crossed yet. Therefore, one of the factors considered in the survey strategy is the dwell time of the given cluster, which, together with the observatory performance, determines the maximum size of the grid that must be scanned. This size is certainly smaller than the cluster dimensions and could be arranged in different patterns. Figure 7 shows an example of a 3 × 2 mosaic of FoV and the pattern to scan this grid, constructed starting from the center of the cluster.

Two other important factors in choosing the target cluster, in relation to the observation site, are the average elevation of the cluster during the observation night of interest and the trend in the phase angle, which indicates the object’s visibility. The phase angle is the angle between the observatory, the object in orbit, and the Sun. The smaller this angle is, the more light is reflected by the object, and theoretically, the brighter the object [22]. In Figure 8, the trend in these two parameters is reported for the different clusters during the night.

The values shown in Figure 8 are the average values of the clusters, estimated relative to the cluster center, representing the general trend. As can be observed, some are automatically discarded because, for the entire duration of the night, they have an elevation below 0°, or in any case, below the observatory’s mask angle. An additional summary chart of the possible observable clusters and their trend during the observation night can be visualized through the polar plot in Figure 9, which also highlights the observatory’s mask angle. This chart can thus summarize the observation that was performed. From this chart, the best clusters to observe can be immediately identified, and comparing them with phase angle trends determines the optimal cluster to observe.

By combining elevation, phase angle, and dwell time (which should be maximized), a target cluster for the observation night is selected. At this stage, the strategy for scanning the cluster is chosen. It is possible to scan the selected cluster with various geometries; in this case, a mosaic grid of up to 6 FoVs was chosen to be scanned in various ways. These 6 FoVs were identified based on a maximum dwell time of around 180–190 s for the best observable cluster. Starting from this value and considering the sensor readout time, an exposure time of 6 s (calibrated on the mean altitude of the cloud), and the slewing performances of the mount, a minimum dwell time of 140 s is required. With this minimum time, it is possible to capture two frames for each of the 6 FoVs of the grid before restarting in time. Therefore, for the cluster selection, a minimum dwell time of 140 s was considered. In Figure 10, the estimated dwell time and the 3 × 2 (6 FoVs) grid for an observation night are reported as an example for each cluster. In Figure 10, it is also possible to see the different dimensions of the clusters based on their position. Here, a possible choice for an observable cluster is the 14 and the 15; the dwell time is greater than 140. After this selection, the elevation and phase angle graphs (Figure 8) are checked.

For the performed observational campaign, SCUDO, RESDOS, and the GAL HASSIN observatories have similar performances in acquiring images and slewing capabilities, so this fact allows them to observe these 6 FoVs for the best cluster, within the predetermined time, managing to take two frames of 6 s each for every field of the grid, for example, the one of the scheme of Figure 7. For the SCHIDMIT of the Asiago observatory, however, due to its slewing capabilities and readout time, it is not possible to execute this grid, so only the cluster center is observed. A grid of 6 FoVs is well suited for this study case, but the strategy can be generalized for other fragmentation events in different orbits and with different observatories’ performance, using grids with more or less FoV and patterns. The arrangement of the FoV grid to scan the cluster varies depending on the geometry used, in combining the FoVs on the same cluster for the various observatories. Three different macro-categories of the survey were implemented to arrange the grids for scanning the same and different clusters, which we shall call

The scheme of the three survey strategies is reported in Figure 11. The Single-cluster strategy is referred to as the scheme of survey used to scan the same cluster with multiple observatories. In this survey strategy, all the observatories are focused on a given target cluster chosen considering the best dwell time, elevation, and phase angle for all the observatories. The mosaic grid to scan the cluster is in the center, as shown in Figure 11. The pattern scheme to point the grid is shown in Figure 7. The pattern of the pointing starts from the first FoV on the top left and ends on the bottom right and then restarts again.

The Single-cluster barrier strategy is selected only as the best cluster for all the observatories. As compared to the Single-cluster strategy, the scanning grid is placed in different positions of the cluster to try to cover all the cloud section parts and have an extended total FoV. As in the Multi-cluster barrier strategy, the grid is disposed orthogonally to the main cluster direction to create a barrier. The idea is to try to cover all the cluster sections and spot also the fragments at the side of the cloud. This strategy is useful several months after the fragmentation epoch due to the expansion of the cloud: The use of an observatory network is fundamental to cover the whole given cluster. The GAL HASSIN telescope is placed at the bottom grid part. The central grid is scanned by RESDOS and ASIAGO, and the upper part of the cluster by SCUDO. In Figure 11, the cluster and the assigned observatory are reported.

The performances of the three listed survey strategies are reported in Section 4.2, together with the comparison with the fragment cloud simulation. Once the optimal cluster to observe is selected and the scanning strategy is chosen, the telescope pointings are generated and scheduled. The Cartesian coordinates of the grid to scan are converted into topocentric RA and Dec coordinates that are used by the telescope. The frames acquired with the generated survey strategy are in sidereal tracking, so in the collected images, the object results as a tracklet. Then, the stars are dots. The sidereal tracking acquisition is bound to the survey. The purpose of the observation strategy is to statistically search for uncataloged objects related to the fragmentation event with a stare method, i.e., point the telescope in a certain region of the sky vault and acquire a certain number of frames, two or more, and try to detect an object that crosses in the FoV of the system.

The images acquired with the survey strategy have been analyzed and processed to find the unknown, i.e., the not-cataloged observed objects, and to extrapolate the measurements in terms of topocentric RA and Dec coordinates and magnitude. The unknowns seen in the used survey strategy are the probable fragments’ target of the campaign. For the implementation of the correlation of measurements from the surveys, the identification of the objects in the acquired frames is the first step. Then, the measurement extrapolation and the comparison with the list of TLEs for all objects in the NORAD catalog updated at the observation epoch. The process begins with measurement acquisition, for which object detection is applied to all frames using the You Only Look Once (YOLO) model, a machine learning algorithm capable of detecting objects at first glance [23]. The model is based on a pre-trained network with a dataset of over 800 frames containing objects from different orbital regimes, in sidereal tracking acquisition from different observatories, with different sensors. The properties of the bounding boxes containing the objects are saved, and then, the centroids of the objects are calculated with deterministic algorithms [24]. In the correlation procedure to identify the uncataloged measurements, the first step involves the one-point propagation of the NORAD catalog compared with the image center. The catalog is propagated to the middle time, the start of the exposure, plus half exposure time, which corresponds to the tracklet center. A pre-filtering is then performed by excluding objects out of the FoV by comparing the angle between the lines of sight with a threshold radius of FoV/2 degrees around the center. The cataloged objects inside the threshold are filtered again by a second propagation to find the streak properties and compared with the detected ones to perform the correlation. Figure 12 shows the comparison method between the measurements of the streak properties and the propagations. All detections are compared with all propagations, matching the detected inclinations (θ), the streak’s length (L), and the distance between the streak’s centers (D) with the propagated ones to assign a probability percentage. The correct object is assigned based on this probability. The not-assigned objects are the search unknown and the probable fragments.

The acquired unknown measurements are characterized by the estimation of the characteristic length (Lc). The characteristic lengths are estimated by approximation as a Lambertian sphere, as proposed in [25] and compared with the Lc of the simulated fragments visible in the same period of observation for each observatory. The Lc of the fragments generated by simulation is shown in Figure 13, the average being 20.7 cm. The dimensions of the simulated fragments are included between 10 and 100 cm.

There are many considerations that must be considered on the estimated characteristic lengths computed from the approximation of the Lambertian sphere. For the observed unknown objects, i.e., the fragments, the considered approximation could not be representative of the shape and the reflectivity. Let us consider the theoretical magnitude for a Lambertian Sphere [25]:

where $m_{obj}$ is the magnitude of the object; $m_{sun}$ is the reference magnitude of the Sun; $\alpha =2/3$ for a Lambertian sphere; $d$ is the diameter of the sphere; $R$ is the distance between the observer and the object; ${\rho }_{bond}$ is the bond albedo, i.e., the reflected fraction of the total radiation incident on the object (0.2 for the aluminum); and $p\left(\theta \right)$ is the phase function. Equation (1) depends on the distance R; the greater this distance, the less bright the object will appear, resulting in a higher magnitude. The dimension d, which is the diameter of the sphere, or, more generally, the Lc, is constrained by these two parameters. Once the estimated magnitude is fixed based on the measurements of uncataloged objects, the distance R is taken as the average distance of the observed cluster at the time of the measurement. The resulting dimension may not correspond to what is expected from the simulated fragments. The shape, and therefore, the reflectivity of the unknown observed object may not be approximated to that of a sphere, leading to an expected magnitude less than the theoretical one at the distance, implying a larger object. Alternatively, the assumed distance may not be representative of the object’s actual orbit, as it could occupy an orbital regime significantly different from the estimated cloud’s average. In cases in which the estimated Lc is in the order of several meters, it is reasonable to consider these objects as not belonging to the fragmentation. It is possible to assume that they occupy orbital regimes far from those of the cloud.