## 4 Analysis
### Simulation Experiments for the Optimal Satellite Orbit in the Common-View Mode between Two Stations  
Based on the research conducted on the minimum satellite orbit altitude concerning various inter-station distances, designing an optimal satellite orbit to maximize the common-view time among multiple sites becomes feasible. Initially, the focus was on determining an optimal satellite trajectory to enhance mutual visibility between two ground-based laser stations, and the stations selected for common view were the Changchun and Wuhan ground laser stations. This process involves utilizing the aforementioned system model and employing conventional iterative traversal methods, particle swarm algorithms, and improved simulated annealing algorithms to determine the optimal satellite orbit between two stations in a common-view model. Comparative experiments underscore the effectiveness and advantages of the improved simulated annealing algorithm. Parameters for the three methods were consistent across the data selection settings The Changchun(CC)[\(-2674.387\), \(3757.189\), \(4391.508\)] and Wuhan(WH)[\(-2279.756\), \(5004.737\), \(3219.791\)] SLR field stations were selected as an example to fulfill the experiment. The initial observation moment was set at UTC time on 1 August 2023, 00:00:00. The satellite's orbit radius is established considering the minimum altitude necessary for orbits at different distances between stations, accounting for the satellite in-orbit flight life. The satellite orbit radii were set as 7000 km, 8000 km, 9000 km, and 10,000 km for different distances between the stations. Based on the experimental conditions described, the study generated outcomes that optimized the satellite orbit between two stations in the common-view model, utilizing three different methods. The conventional iterative traverse method was performed as the following procedure:  
1. The optimal orbit inclination in circular orbits involving SLR stations and satellites is determined through geometric analysis. This analysis operates under the assumption that the longest common-view duration between two SLR stations takes place when the plane of the circular orbit is perpendicular to the connecting line (L1 to L2), which is regarded as the angular momentum of the circular orbit. 2. Compute the common-view time of different ascending node right ascension at a specified orbital altitude and inclination angle. 3. Investigate the orbital altitudes impact on the duration of common view. 4. Determine the optimal orbit configuration to facilitate common view between the two stations. Through the conventional iterative traverse method, the following results were obtained: At an orbital inclination of \(48.16^{\circ}\) for the ground stations in Changchun and Wuhan, an optimal ascending node right ascension of \(27^{\circ}\) was identified. Upon fixing the orbital inclination, adjusting the orbit altitude produced the optimal right ascension of the ascending node within an error range of \(\pm 3^{\circ}\), illustrated in Figure 5a. Similarly, setting the right ascension of the ascending node while varying the orbit altitude led to the optimal orbital inclination within an error range of \(\pm 1^{\circ}\), outlined in Figure 5b. Figure 5a indicates a common-view time curve displaying two distinctive peaks. These peaks correspond to orbital planes during the satellite trajectories: the first peak aligns with the maximum common-view time when the line connecting the ground stations is perpendicular to the satellite orbital plane (Figure 5c), while the second peak occurs when this line is nearly parallel to the satellite orbital plane (Figure 5d). The optimal satellite orbit, which corresponds to its longest common-view duration, aligns with the assumption that when the line between the two stations roughly represents the orbital plane's angular momentum, the orbit plane coincides with the optimal satellite orbit plane within the common-view model.