## 3 BDS-3 Constellation Configuration and ISL Establishment Constraints
### Antenna Directivity Constraints  
The BDS-3 employs phased-array antennas for its ISLs. By controlling the phase relationships of the antenna elements, the system can achieve beam scanning over a wide spatial range, enabling effective signal transmission. The BDS-3 ISLs operate in the Ka-band, which features relatively narrow antenna beams. By adjusting the beam direction, the system can establish point-to-point signal transmission links. Furthermore, phased-array antennas have a maximum beam scanning angle. Therefore, when establishing ISLs, the antenna directivity constraints must be considered. Specifically, the satellite pair planning to establish a link must simultaneously fall within each other's beam scanning range to ensure effective link establishment.  
Figure 5 illustrates the antenna directivity constraints, where \(\beta_{i}\) and \(\beta_{j}\) represent the maximum beam scanning ranges of satellite \(i\) and satellite \(j\), respectively. If both satellites are within each other's visibility range, the following antenna directivity constraint conditions must be satisfied, as expressed in the equation below:  
\[\left\{\begin{array}{l}\alpha_{i}\leq\theta_{i}\leq\beta_{i}\\ \alpha_{j}\leq\theta_{j}\leq\beta_{j}\end{array}\right. \tag{4}\]  
Figure 5: Schematic diagram of antenna directivity constraint.  
Considering practical engineering constraints, the maximum beam scanning range for BDS-3 can be set to 60\({}^{\circ}\)[25]. Taking into account both satellite visibility and antenna directivity constraints, a theoretical analysis of the ISL antenna scanning range and the maximum communication distance can be conducted. For MEO-MEO type links and MEO-GEO/IGSO type links, the maximum communication distance can be calculated as follows:  
\[L_{\text{max}(MEO-MEO)}=2\sqrt{\left(R_{e}+H+H_{M}\right)^{2}-\left(R_{e}+H \right)^{2}} \tag{5}\]  
\[L_{\text{max}(MEO-GEO/IGSO)}=\sqrt{\left(R_{e}+H+H_{M}\right)^{2}-\left(R_{e}+ H\right)^{2}}+\sqrt{\left(R_{e}+H+H_{G}\right)^{2}-\left(R_{e}+H\right)^{2}} \tag{6}\]  
where \(L_{\text{max}(MEO-MEO)}\) represents the maximum inter-satellite distance for MEO-MEO type links, and \(L_{\text{max}(MEO-GEO/IGSO)}\) represents the maximum inter-satellite distance for MEO-GEO/IGSO type links. \(H_{M}\) denotes the theoretical orbital altitude of MEO satellites, and \(H_{G}\) denotes the theoretical orbital altitude of GEO/IGSO satellites. The Earth occlusion angles for MEO satellites and GEO/IGSO satellites can be calculated using the following equations:  
\[\alpha_{MEO}=\arcsin\frac{R_{e}+H}{R_{e}+H+H_{M}} \tag{7}\]  
\[\alpha_{GEO/IGSO}=\arcsin\frac{R_{e}+H}{R_{e}+H+H_{G}} \tag{8}\]  
where \(\alpha_{MEO}\) represents the Earth occlusion angle for MEO satellites, and \(\alpha_{GEO/IGSO}\) represents the Earth occlusion angle for GEO/IGSO satellites. Taking \(R_{e}\) = 6378 km, \(H\) = 1000 km, \(H_{M}\) = 21,528 km, and \(H_{G}\) = 35,786 km, and substituting these values into Equations (5)-(8), the maximum inter-satellite communication distance and the Earth occlusion angle can be calculated. The calculation results are shown in Table 5.  
Simulation results indicate that the maximum inter-satellite communication distance for MEO-MEO type links can reach 55,897.12 km, while for MEO-GEO/IGSO type links, it can reach 70,477.33 km. The Earth occlusion angles for MEO satellites and GEO/IGSO satellites are 14.79\({}^{\circ}\) and 9.84\({}^{\circ}\), respectively. In the next step of inter-satellite topology analysis, inter-satellite visibility and antenna directivity constraints should be considered. To ensure signal transmission quality with a sufficient margin, the beam scanning range for MEO satellites can be set to [15\({}^{\circ}\), 60\({}^{\circ}\)], while for GEO/IGSO satellites, it can be set to [10\({}^{\circ}\), 60\({}^{\circ}\)].