## 2 System Model of NGSO Satellites Equipped with Multi-Beam Antennas
### Analysis of Interference Based on Off-Axis Angle and Beam Distance
The relationship between \(\theta_{1}\), \(\theta_{2}\), and beam distance \(l\) is depicted in Figure 6a. With the beam distance varying within the range [100, 800] km, the off-axis angles \(\theta_{1}\) and \(\theta_{2}\) increase monotonically and almost linearly with respect to the beam distance \(l\). Satellites equipped with multi-beam antennas have a unique characteristic that not all beams keep pointing to the nadir of the Earth, but they may have a certain beam tilting angle instead. Here, we carry on an analysis of the variables mentioned above in the situation where the NGSO and GSO satellite both have certain tilting angles. Figures 4b and 5 show that \(\theta_{1}\) and \(\theta_{2}\) are solely influenced by the NGSO and GSO satellite antenna beam tilting angles \(\beta_{1}\) and \(\beta_{2}\), respectively. For the case of the NGSO satellite antenna beam with a tilting angle \(\beta_{1}\) which elongates the distance between the beam centers of the NGSO and GSO satellite antennas, we have the expressions of \(\theta_{01}\) and \(\theta_{02}\)  
\[\theta_{01}=\arcsin\left[\frac{R_{\rm e}+h}{R_{\rm e}}\cdot\sin(\beta_{1}) \right]-\beta_{1} \tag{17}\]  
\[\theta_{02}=\theta_{0}+\beta_{1}-\arcsin\left[\frac{R_{\rm e}+h}{R_{\rm e}} \cdot\sin(\beta_{1})\right] \tag{18}\]  
Then one can easily obtain the expression of \(\theta_{3}\)  
\[\theta_{3}=\arcsin\left[\frac{R_{\rm e}\cdot\sin(\theta_{02})}{\sqrt{R_{\rm e }^{2}+(R_{\rm e}+h)^{2}-2\cos(\theta_{02})\cdot R_{\rm e}(R_{\rm e}+h)}}\right] \tag{19}\]  
So the angles \(\theta_{1}\) and \(\theta_{2}\) can be expressed as  
Figure 3:0  
Figure 3:1  
Figure 3:2 The downlink scenarios where GSO satellite beams have tilting angle rendering the beam center farther away or closer to the other center. (**a**) Pointing of GSO satellite beams with tilting angle \(\beta_{2}\) making it farther away from the beam center of NGSO satellite; (**b**) Pointing of GSO satellite beams with tilting angle \(\beta_{2}\) making it closer to the beam center of NGSO satellite. For the situation that the tilting angle of the NGSO satellite decreases the beam distance, by replacing \(\beta_{1}\) with its corresponding negative number, one can obtain the corresponding expressions of \(\theta_{1}\) and \(\theta_{2}\). As regards the calculation of \(\theta_{2}\) with tilting angle \(\beta_{2}\), we notice that there always exists a \"tilting triangle\" comprised of the Earth center, the GSO satellite, and the GSO ground station, which would instigate the variation of \(\theta_{2}\). Let the variation resulting from the tilting angle \(\beta_{2}\) be \(\Delta\theta_{2}\). In the \"tilting triangle\", we have the expression of the supplementary angle of \(\Delta\theta_{2}\), i.e., \(\theta_{2}+\theta_{4}\), is expressed by  
Figure 3:3  
where \(h_{2}\) represents the orbital altitude of the GSO satellite, and thereafter we obtain  
Figure 3:4  
Analogously, the angle \(\theta_{4}\) can be derived as  
Figure 3:5  
For the GSO satellite antenna beam with a tilting angle, which renders the beam center becomes farther away or closer to that of the interfering satellite, we obtain expression of \(\theta_{1}\) and \(\theta_{2}\). For the former condition, \(\beta_{2}\) is taken positive and vice versa. Figure 3:6  
Figure 3:7  
Note that the tilting angles of both satellites have no essential influence on the relationship between the off-axis angle and beam distance. Whether the tilting angle is zero or not does not affect the variation trend. Figure 6a definitely shows that no matter what the satellite antenna beam tilting angles \(\beta_{1}\) and \(\beta_{2}\) are, the off-axis angles \(\theta_{1}\) and \(\theta_{2}\) always increase monotonically with respect to the beam distance \(l\). Variation of \(\beta_{2}\) will not lead to the change of \(\theta_{1}\), thus for variation of \(\theta_{1}\) the result with \(\beta_{2}\) is the same as that without any tilting angle at all.