## 6 Simulation Results and Discussion
### Discussion  
Figure 16 presents a comparative analysis plot of four distinct scanning methods: spiral, raster, square spiral, and hexagonal. The mean acquisition time is plotted for increasing uncertainty areas across different \(\sigma\) values. These results provide insight into the efficiency of each scanning method in terms of mean acquisition time relative to the probability distribution of the receiver satellite's position. Among the four methods, the spiral, square spiral, and hexagonal scans start at the high-probability region and proceed outward toward areas of lower probability. This prioritization leads to reduced acquisition times as these scans effectively target the most probable areas first. In contrast, the raster scan initiates at a lower-probability region and progresses towards the high-probability region, which extends the overall acquisition time. Consequently, the raster scan demonstrates the longest acquisition time among the methods. Ranking the four scanning methods, the mean acquisition times follow this order: spiral \(<\) hexagonal \(<\) square spiral \(<\) raster. The square spiral and spiral scans both commence scanning from the center, quickly addressing high-probability areas, with the square spiral being more suited to square-shaped uncertainty regions. The hexagonal scan also begins at the center; however, due to its geometry, some scan points fall outside the square region, potentially reducing its  
Figure 16: Expected scan time for various scan paths for various receiver satellite position errors \(\sigma\). Expected scan time: spiral scan \(<\) hexagonal scan \(<\) square spiral scan \(<\) raster scan. (**a**) \(\sigma=10\) μrad. (**b**) \(\sigma=20\) μrad. (**c**) \(\sigma=40\) μrad. efficiency in fully covering the square uncertainty area. The raster scan, by starting from an edge, further delays coverage of these central, high-probability zones. In evaluating these metrics, the square spiral scan is shown to be particularly effective for square uncertainty areas, as it ensures complete and efficient coverage. The raster scan, while covering the entire area, is less efficient in prioritizing areas based on probability. The hexagonal scan may leave unscanned gaps along the edges due to alignment mismatches with square-shaped regions, and the spiral scan, due to its circular pattern, might leave portions of the square uncertainty area unscanned. However, since the spiral path aligns well with the possible positions of the target satellite within the uncertainty scan area, as illustrated for each path in Figure 17, the spiral scan method is more efficient in terms of both the least mean acquisition time and the number of scan points required to cover the uncertainty region with the highest probability of locating the target satellite. The scan points required by these four methods to cover the uncertainty area \(\theta_{\text{area}}=\pm 250\,\upmu\text{rad}\) (see Figure 17) differ. The square spiral and hexagonal scan methods also focus on the high-probability area; however, the overall number of scan points required is greater than that of the spiral scan method. In terms of mechanical implementation, the four scan paths--raster, square spiral, spiral, and hexagonal--require different approaches. The raster scan involves linear movements and straightforward directional changes. The square spiral scan combines straight-line movements with periodic directional shifts. The spiral scan requires smooth, continuous curved trajectories to maintain consistent motion. The hexagonal scan demands precise path calculations to ensure efficient coverage due to its angular and non-linear pattern. Each path has unique implementation requirements that influence how it is designed and executed. The availability of a power spectral density model for satellite and spacecraft vibrations is crucial for analyzing vibration sources and leveraging these insights in simulation studies.