# Laser Inter-Satellite Link Visibility and Topology Optimization for Mega Constellation
## 3 Methodology and Analytical Approach  
### Numbering and Classification of Inter-LISLs  
In order to facilitate description, this paper unifies the orbits and satellites within the orbits of the [PERSON] constellation by naming the satellite nodes in the form of \(S(P_{i},M_{j})\), indicating the \(M_{j}\) satellite within the first \(P_{i}\) orbital plane. The orbit number \(P_{i}\) increases in clockwise direction, and the satellite number \(M_{j}\) increases in the same direction as the orbit. The orbits \(P_{i+1}\) and \(P_{i-1}\) are named as adjacent orbits of \(P_{i}\), since the satellites in the same orbit have the same orbital speed and direction, \(M_{j+1}\) and \(M_{j-1}\) are named as the forward and backward satellites of \(M_{j}\), respectively.  
The classification of LISLs for the mega-constellation is based on two different rubrics: whether the LISLs are in the same orbit, and whether the LISLs are permanently visible.  
1. Co-orbiting and hetero-orbiting LISLs  
In an orbital plane, satellites with the same orbital altitude all move in the same direction at identical speed. According to the positions of satellites within the constellation, LISLs within the constellation can be classified into two main types: (a) LISLs established between two satellites in the same orbital plane; (b) LISLs established between different orbital planes.  
Among them, the heterodyne LISLs can be further classified into the following three types:  
* Neighboring non-adjacent OP LISLs, which are between satellites in nearby instead of adjacent OPs;
* Crossing OP LISLs, which are between satellites in crossing OPs.  
As shown in Figure 2a, an example of adjacent and nearby OP LISLs, where the satellite \(S(1,1)\) has a LISL indicated by a yellow dashed line, and the satellites \(S(2,65)\) and \(S(24,35)\) are located in two nearby planes, \(P_{2}\) and \(P_{24}\), respectively; An example of crossing OP LISLs is shown in Figure 2b, where satellites \(S(12,32)\) and \(S(14,21)\) are located in orbital planes \(P_{12}\) and \(P_{14}\), respectively, both intersecting with \(P_{1}\) where \(S(1,1)\) is located.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Orbit Type** & **Orbital Altitude** & **Orbital Inclination** & **Number of Orbital Planes** & **Number of Satellites** & **Total** \\ \hline \multirow{4}{*}{LEO *} & 550 km & 53\({}^{\circ}\) & 66 & 24 & 1584 \\  & 540 km & 53.2\({}^{\circ}\) & 66 & 24 & 1584 \\  & 570 km & 70\({}^{\circ}\) & 36 & 20 & 720 \\  & 560 km & 97.5\({}^{\circ}\) & 6 & 58 & 348 \\  & 560 km & 97.5\({}^{\circ}\) & 4 & 43 & 172 \\  & 345.6 km & 53\({}^{\circ}\) & & 2547 \\ VLEO & 340.8 km & 48\({}^{\circ}\) & & 2748 \\  & 335.9 km & 42\({}^{\circ}\) & & 2493 \\ \hline \hline \end{tabular}  
* The above orbital data concerning Starlink Constellation Phase I are derived from the FCC reports and the detailed parameters of the Union of concerned scientists satellite database. [[https://www.ucsusa.org/nuclear-weapons//space-weapons/satellite-database](https://www.ucsusa.org/nuclear-weapons//space-weapons/satellite-database)]([https://www.ucsusa.org/nuclear-weapons//space-weapons/satellite-database](https://www.ucsusa.org/nuclear-weapons//space-weapons/satellite-database)) (accessed on 1 January 2022).  
\end{table}
Table 2: Modified Starlink Constellation Phase I configurations.