## 2 Methods
### Feature of Density Distribution  
The second step is to extract the feature of density distribution since the design of the multi-beam GEO data relay satellite requires the knowledge of the LEO statistics and density distribution. As shown in Figure 3, the characteristics of the object distribution in the FOV of the GEO data relay satellite are extracted with \(2^{\circ}\) uniform spot beams. Figure 3a illustrates LEO satellite occupation for a \(\pm 11^{\circ}\) FOV subdivided into hexagonal cells of \(2^{\circ}\) diameter. The number in the center of each cell indicates the transient LEO satellites in the coverage of the certain beam. The blue dash line sketches the contour of Earth's surface. Figure 3b shows the statistics of the cell occupation. It is obvious that the number of satellites per cell is larger in the peripheral FOV compared with the central \(\pm 8.5^{\circ}\) FOV. The area surpassing the contour of earth surface offers more space to satellites in adjacent orbits with line-of-sight propagation toward the GEO data relay satellite. There are eight cells in which the number of satellites exceeds 16. Four of them lie around the intersection of the orbit panels with an inclination of \(88^{\circ}\), and the other four cells are around the corners, where the orbit panels with an inclination of \(55^{\circ}\) overlap. For most of the area within the \(\pm 8.5^{\circ}\) FOV, the number of satellites per cell is less than 5.  
Figure 2: Flow chart of proposed methods.  
To further feature the details, the diameter of the cell decreases from 2\({}^{\circ}\) to 1\({}^{\circ}\). Figure 4a illustrates the LEO satellite occupation for a \(\pm\)11\({}^{\circ}\) FOV subdivided into hexagonal cells of 1\({}^{\circ}\) diameter. Figure 4b shows the statistics of the cell occupation. There exist hollow cells within the \(\pm\)8.5\({}^{\circ}\) FOV. The satellites also tend to concentrate around the intersection of orbit panels with inclinations of 88\({}^{\circ}\) and 55\({}^{\circ}\).  
To build up the data set for analysis, a finer cell is necessary to depict the distribution more accurately and help advance the precision of the output beam width and beam centers. As shown in Figure 5, the characteristics of the object distribution in the FOV of the GEO data relay satellite are extracted with 0.5\({}^{\circ}\) uniform hexagonal cells along continuous three moments with a one-second interval to mitigate the impact of randomness.  
Figure 4: Characteristics of the object distribution in FOV of GEO data relay satellite with 1\({}^{\circ}\) spot beams: (**a**) LEO satellite occupation of spot beams. The number in the center of each cell indicates the transient LEO satellites in the coverage of a certain beam. The blue dash line sketches the contour of Earth’s surface; (**b**) statistics of the cell occupation.  
Figure 3: Characteristics of the object distribution in FOV of GEO data relay satellite with 2\({}^{\circ}\) spot beams: (**a**) LEO satellite occupation of spot beams. The number in the center of each cell indicates the transient LEO satellites in the coverage of a certain beam. The blue dash line sketches the contour of Earth’s surface; (**b**) statistics of the cell occupation.