## 4 Analysis
### Simulation Experiments for the Optimal Satellite Orbit in the Common-View Mode between Two Stations
These distributions are relatively concentrated and closely aligned with the global optimum. The standard deviations for the improved simulated annealing algorithm and the particle swarm algorithm are 0.45486 and 0.62972, respectively. Under the experimental conditions outlined previously, the optimal satellite orbit between two stations was optimized using the improved simulated annealing algorithm, particle swarm optimization, and the conventional iterative traverse method. The results obtained are summarized in Tables 1 and 2. Table 1 shows a high level of precision between the results produced by the improved simulated annealing algorithm and those from the conventional iterative traversal method. It also achieves a similar level of accuracy compared to the particle swarm algorithm, with a discrepancy of no more than 2 s in the optimal common-view duration. Table 2 shows that increasing the orbit radius leads to longer computational times for all three methods, but the improved simulated annealing algorithm operates more than 20 times faster than the conventional iterative traverse method. Additionally, compared to the particle swarm algorithm, the improved simulated annealing algorithm significantly reduces the computation time required to reach the optimal solution, resulting in decreased computational costs, enhanced efficiency, and improved performance. Figure 9 displays the optimized satellite trajectories between two SLR stations at an orbit radius of 8000 km, obtained through the optimization process using the improved simulated annealing algorithm. Figure 8: Comparison of outcomes from thirty simulations conducted using the improved simulated annealing algorithm and the particle swarm algorithm. (**a**) Common-view time resulting from the thirty simulations of both algorithms; (**b**) histograms depicting the distribution of distinct outcomes generated by the algorithms. Selection of the Jason-3 and Ajisai satellites was based on existing Time Transfer by Laser Link missions and potential future LTT tasks. For the same timestamp of 1 August 2023, 00:00:00, utilizing their satellite orbit predictions (CPF), the common-view times between the Changchun and Wuhan stations within a 10-day timeframe were computed. An improved simulated annealing algorithm was employed to optimize the satellite orbits in the common-view mode at the same orbital altitude and date. The results of this optimization, compared to the existing satellite orbits, are presented in Table 3. Jason-3, \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{\
\begin{tabular}{c} **Different** \\ **SLR Stations** \\ \end{tabular} } & \multicolumn{2}{c}{**Improved Simulated Annealing**} & \multicolumn{2}{c}{**Particle Swarm Optimization**} & \multicolumn{2}{c}{
\begin{tabular}{c} **Conventional Iterative** \\ **Traverse Method** \\ \end{tabular} } \\ \cline{2-7}  & \(Time_{com}\)**(S)** & \((i_{com}\),\(\Omega_{com})\) & \(Time_{com}\)**(S)** & \((i_{com}\),\(\Omega_{com})\) & \(Time_{com}\)**(S)** & \((i_{com}\),\(\Omega_{com})\) \\ \hline CC WH & & & & & \\ (R = 7000 km) & 354 & 27.5481) & 354 & (47.9924, & \\ CC WH & & & & & \\ (R = 8000 km) & 930 & (48.0299, & & & \\ (R = 8000 km) & 27.5674) & 931 & 27.6326) & 930 & (48.16, 27.55) \\ CC WH & & & & & & \\ (R = 9000 km) & 1463 & (48.0008, & & & \\ (R = 9000 km) & 27.6434) & 1463 & (48.0612, & & \\ CC WH & & & & & \\ (R = 10,000 km) & 2009 & (48.0461, & & \\ (R = 10,000 km) & 27.6291) & 2009 & (48.0852, & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Optimization results of the improved simulated annealing, the particle swarm optimization algorithms, and the conventional iterative traverse method.