## 4 Results
### Scenario A  
Scenario A is similar to Starlink's Phase 1 V2 constellation [6]. The optimal \(F\) is equal to 13 as quoted in Table 3. To ensure connectivity across the network as discussed in Section 2.3, we first need to estimate the distances between a satellite and its adjacent neighbors. Figure 7 (left, right) show the distances of the upper- and lower-orbit adjacent satellites for satellite \((k,m)=(0,0)\). For this constellation, the distance between two consecutive satellites in the same orbit is \(r_{\rm c}\approx 659.3\) km. We observe that, in both cases, the distances between the satellite and its upper- and lower-orbit neighbors fluctuate significantly within one orbital period, ranging from \(R_{\rm min}\approx 132.9\) km to \(R_{\rm max}\approx 1645.6\) km. These distance fluctuations exhibit a periodic behavior as the central satellite moves through the upper and lower halves of its trajectory around the Earth. Due to the uniformity of the constellation, these fluctuations are practically the same for the upper- and lower-orbit neighbors, \(\mathsf{U}_{i}\) and \(\mathsf{L}_{i}\), respectively, differing only by an initial time displacement. The figures indicate the varying conditions for inter-orbit links: while intra-orbit link distances remain constant at \(r_{\rm c}\), the inter-orbit link length varies from about one fifth of \(r_{\rm c}\) to more than twice that. As mentioned in Section 2.2, the \(F\) value is used to determine the allocation of satellites within a constellation, ensuring a minimum safe distance between them. To achieve the \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **A** & **B** & **C** \\ \hline \(N_{\rm sat}\) & 66 & 33 & 132 \\ \(N_{\rm orb}\) & 24 & 12 & 48 \\ \(N\) & 1584 & 396 & 6336 \\ \(i\) & \(53^{\circ}\) & \(53^{\circ}\) & \(53^{\circ}\) \\ \(a\) [Km] & 550 & 550 & 550 \\ \(F\) & 13 & 6 & 19 \\ \(r_{\rm c}\) [Km] & 659.3 & 1317.1 & 329.7 \\ \(R_{\rm max}\) [Km] & 1645.6 & 3241.9 & 802.4 \\ \(R_{\rm min}\) [Km] & 132.9 & 231.5 & 17.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Scenarios considered. Figure 7: Distance variations between satellite \((0,0)\) and its adjacent neighbors: (**left**) in the upper orbit and (**right**) in the lower orbit, observed over one orbital period. The intra-orbit link length between consecutive satellites remains constant throughout. safest possible configuration, we analyze how the minimum distance varies with changes in the \(F\) value, as illustrated in Figure 8. Using the Constellation class from Pyminisat, as referenced in Appendix A, we incorporate constellation dynamics to determine the optimal \(F\) value. The optimal \(F\) value is the one that maximizes the minimum distance between satellites, ensuring the most efficient and secure allocation. In Figures 9 and 10, we show the variations in path loss \(L_{\text{PS}}\) and channel gain \(L_{\text{ch}}\), respectively, for the links between the central satellite and both the upper- and lower-orbit neighbors. Both \(L_{\text{PS}}\) and \(L_{\text{ch}}\) follow an inverse square variation (\(\propto\)\(R^{-2}\)) with respect to the corresponding inter-satellite distance \(R\). For the set of parameters in Tables 1 and 2, the transmitter and receiver gains are \(C_{\text{T}}\) = 120 dB and \(G_{\text{R}}\) = 104.2 dB, respectively, while the transmitter and receiver pointing losses are \(L_{\text{T}}\) = \(-4.3\) dB and \(L_{\text{R}}\) = \(-0.11\) dB, respectively. High transmitter and receiver gains are necessary to compensate for the path losses shown in Figure 9. This is a clear advantage of optical inter-satellite links (OILs), which can utilize highly collimated beams to optimize the link budget. Next, we discuss the transmitter power \(P_{\text{T}}\) required to maintain the links between all adjacent satellites of Figure 4. Using the link budget model of Section 3 and assuming a target data rate of \(R_{\text{b}}=10\) Gbps, we readily obtain the results depicted in Figure 11 for the upper and lower orbital neighbors.