## 3 Threshold FOV of the Satellite Based on the Improved Observation Mode  
In order to describe the determined field of view more conveniently, we divided the FOV into the cross-orbit field of view (COFOV) and the along-orbit field of view (AOFOV) and determined their thresholds respectively. An important factor in determining whether a GEO target can be observed is whether the target can enter the FOV of the sensor. The size of sensor's FOV in turn determines the coverage range of the GEO belt by the observation satellite. In summary, when a satellite operating in a DDSSO observes the GEO belt under the improved observation mode, in order to meet the sensor coverage requirements, the sensor's COFOV must be greater than 15\({}^{\circ}\). In order to verify the correctness of the above theoretical derivation, we simulated the coverage rate of sensors in different FOVs. First, we set the COFOV to 15\({}^{\circ}\) and kept it unchanged, and simulated the changes of the sensor coverage with the observation days when the AOFOV ranged from 4\({}^{\circ}\) to 8\({}^{\circ}\). The simulation results are shown in Figure 5.  
Figure 4: The pointing area of the satellite’s sensor. (The white area in the figure represents the observation area before the improved observation mode is used, and the red area in the figure represents the observation area after the improved observation mode is used.) Figure 3: The celestial coordinate system with the center of the Earth as the origin. Figure 5 shows that the coverage rate of the observation satellite is less affected by the sensor's AOFOV, and the coverage rate difference under different AOFOVs hardly changes with the observation days. When the COFOV is fixed, the coverage rate does not change after the AOFOV reaches a certain value. When the AOFOV reaches 6\({}^{\circ}\), the coverage rate reaches a maximum value. Next, we set the sensor's AOFOV to 4\({}^{\circ}\) and changed the COFOV from 13\({}^{\circ}\) to 20\({}^{\circ}\), simulating the coverage rate of the observed satellite (see Figure 6). Figure 6 shows that with the increase in the COFOV, the coverage rate will increase, but the growth rate shows a gradually decreasing trend. The threshold COFOV is 15\({}^{\circ}\). When the COFOV is smaller than the threshold COFOV, the coverage rate increases rapidly. When the COFOV is larger than the threshold COFOV, the coverage rate increases slowly. According to the above simulation analysis, compared with the AOFOV, the COFOV is the main factor affecting the satellite coverage rate. Under the specific conditions of observing the GEO belt by a DDSSO satellite, it is reasonable to select the sensor's COFOV to be 15\({}^{\circ}\). Based on the characteristics of satellite observation patterns, we determined the threshold COFOV, and according to the simulation results, we determined that the coverage rate of the satellite can be guaranteed under this threshold, which proves the rationality of the COFOV threshold. Next, we focused on determining the threshold for AOFOV. Figure 5: Sensor coverage rate with cross-orbit FOV value of 15\({}^{\circ}\) and along-orbit FOV value from 4\({}^{\circ}\) to 8\({}^{\circ}\). Figure 6: Sensor coverage rate with cross-orbit FOV value from 13\({}^{\circ}\) to 20\({}^{\circ}\) and along-orbit FOV value of 4\({}^{\circ}\). Although the AOFOV of the sensor has little effect on the coverage rate of the observation satellite, it will directly affect the scanning width of the satellite to the GEO belt, which in turn affects the revisit period of the satellite to the GEO belt. Therefore, the AOFOV of the sensor needs to be constrained to meet the revisit period requirements of the observation satellite. Next, we used the relevant theory of orbit design to constrain the AOFOV of the spaceborne sensor. From the analysis in the previous section, the observation area formed by the sensor in the natural rendezvous mode is circular. Assuming that the GEO band is narrow, the GEO band can be viewed as a rectangle in the short term.