# Laser Inter-Satellite Link Visibility and Topology Optimization for Mega Constellation
## 3 Methodology and Analytical Approach
### Chain Building Threshold Analysis  
The mega-constellation features relative motion between satellites, constant visibility changes between satellites in different orbits, and inter-satellite links are also opened and closed, resulting in a highly dynamic and time-varying spatial information network. In general, LISLs adopt the principle of continuous link building to establish an intersatellite network topology comprised of fixed pairs in spatial position relationships. Due to the fact that LISLs require pre-captured antennas, and the process of aligning antennas for correction takes a significant amount of time, fixed pairing can greatly reduce link switching and the time required for antenna alignment [20]. Based on the continuous regime of LISLs, it is prerequisite and essential to analyze the constellation network topology by evaluating the visibility relationship for each satellite in order to calculate the spatial position and inter-satellite distance of satellites in the constellation, as well as to select the appropriate satellites for link building between co-orbital and hetero-orbital links [21]. 1. Calculation of the distance of LISLs within the same orbit If there are satellites \(S_{1}\) and \(S_{2}\) in the same orbit then the distance between the two satellites \(d_{12}\) is shown in Equation (5). \[d_{12}=\sqrt{2}\times(R+h)\times\sqrt{1-cos\angle s_{1}os_{2}}\] (5) where \(d_{12}\) denotes the distance between satellites \(S_{1}\) and \(S_{2}\), \(R\) is the radius of the Earth, \(h\) is the orbital altitude of satellites \(S_{1}\) and \(S_{2}\), \(\gamma\) is half of the corresponding geocentric angle of satellites, and \(\angle s_{1}os_{2}\) is the angle between satellites \(S_{1}\) and \(S_{2}\). The radius of the Earth at its equator is approximately 6378 km, as shown in Figure 4, where the blue segment represents the Earth, the gray segment represents the atmosphere, and the yellow segment represents any satellite in the LEO layer. As a result of the previous analysis, it is evident that satellites in the same orbit remain relatively stationary throughout the operation, the link length formed is constant, and the azimuth and elevation angles do not change significantly, so it is relatively easy to establish LISLs. However, additional factors need to be taken into consideration regarding the satellite communication links, such as the satellites \(S_{1}\) and \(S_{3}\) in Figure 4 being at one time blocked by the Earth for communication links; the scattering  
Figure 3: Distribution of satellites in the Delta constellation. and absorption of various gases and other suspended particles in the Earth's atmosphere, and the influence of atmospheric turbulence, which can cause the laser communication links of the constellation to be attenuated, which may manifest as light intensity flicker, beam drift, and jitter [22]. Therefore, for co-orbital satellites to establish link connections, it is necessary for the conditions shown in Equation (6) to be met. \[(R+h)\times cos\gamma>(R+H) \tag{6}\]  
2. Distance calculation of LISLs between different orbits  
The angle formed between the two satellites and the center of the earth \(\theta\) can be calculated from the satellite's latitude and longitude position information, and the equation for the geocentric angle of satellites \(S_{1}\) and \(S_{2}\) is shown in (7). \[\theta=arcos[sin(\varphi_{1})\times sin(\varphi_{2})+cos(\varphi_{1})\times cos (\varphi_{2})\times cos(\lambda_{1}-\lambda_{2})] \tag{7}\]  
where \(\varphi_{1}\) is latitude and \(\lambda_{1}\) is longitude, the position information of the satellite \(S_{1}\) can be noted as (\(\varphi_{1}\), \(\lambda_{1}\)); the satellite geographic location coordinates formula is shown in (8)-(10).