## 2 Topological Analysis of Mega-Constellations
In the context of Starlink's laser links, communication performance is primarily influenced by atmospheric molecules and aerosols, which notably affect satellite communications at altitudes approximately 50 km above the Earth's surface. Utilizing the Earth's radius (\(R\)) as 6378.14 km, we derive the maximum in-layer link distance for each satellite using Equation (6), yielding an approximate value of 5168.419 km. This  
Figure 3: Geometric structure of orbital and inter-satellite links. indicates that any satellite within this specified distance range would be visible and capable of establishing a connection. Figure 1:0  
Through calculations, it is determined that, on average, each satellite can establish connections with 16 neighboring satellites in adjacent orbits on one side. Figure 4 illustrates the calculated maximum distance from each satellite to the primary reference satellite, as well as the maximum pitch angle, azimuthal maximum angular velocity, and maximum angular acceleration values. In order to minimize the distance between adjacent satellites and to reduce the angular velocity and angular acceleration between them, as shown in Figure 4, the trajectories of satellites 5 to 11 are relatively close to meeting these requirements. Table 2 lists the relevant data between these satellites and the selected main satellite, designated as'satellite 10554'. Based on official SpaceX data, the laser system on Starlink satellites exhibits a maximum azimuth angular velocity range of approximately 0.1 to 0.5 degrees per second. To optimize inter-satellite connectivity, minimizing the relative distance, angular velocity, and acceleration is paramount. The seventh satellite, particularly linking the main satellite with satellite 10454, emerges as the optimal choice, demonstrating the efficacy of connecting satellites with \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Satellite Number** & **Maximum Distance** & \begin{tabular}{c} **Maximum Pitch** \\ **Angular Velocity** \\ \end{tabular} & \
\begin{tabular}{c} **Maximum Azimuth** \\ **Angular Velocity** \\ \end{tabular} &
\begin{tabular}{c} **Maximum Azimuth** \\ **Angular Acceleration** \\ \end{tabular} \\ \hline
5 & 2498.5076 & 0.0021 & 0.0481 & 0.0016 \\
6 & 2002.8720 & 0.0028 & 0.0688 & 0.0020 \\
7 & 1620.0576 & 0.0040 & 0.1267 & 0.0028 \\
8 & 1446.5249 & 0.0062 & 0.8587 & 0.0255 \\
9 & 1555.5025 & 0.0044 & 0.1527 & 0.0030 \\
10 & 1898.1008 & 0.0030 & 0.0701 & 0.0022 \\
11 & 2373.7009 & 0.0022 & 0.0450 & 0.0013 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Maximum inter-satellite link distance, angular velocity, and angular acceleration of visible satellites in adjacent orbits for important satellites. Figure 4: Distance, angular velocity, and angular acceleration of visible satellites in adjacent orbits. identical numerical designations in adjacent orbits. This strategy yields a maximum inter-satellite distance of 1620.06 km, an azimuth angular velocity of 0.1267 deg/s, and an elevation angular velocity of 0.00402 deg/s. Our previous analysis identified the most suitable satellite for each satellite to connect to in its adjacent orbit, demonstrating the rationale behind our connection choices and providing a basis for subsequent connection strategies. The following section provides a detailed description of how the entire constellation is interconnected into a cohesive network using fixed links. In the constellation configuration, each satellite maintains a single fixed link to a neighboring orbit satellite. To establish a comprehensive network, an alternating connectivity approach is employed as follows: odd-numbered orbital tracks see even-numbered satellites connecting to their same-numbered counterparts on the right-side orbit, while odd-numbered satellites link to those on the left-side orbit with corresponding numbers. Conversely, for even-numbered orbits, even-numbered satellites connect to their same-numbered counterparts on the left orbit, and odd-numbered satellites establish links with their counterparts on the right orbit.