## 2 System Model of NGSO Satellites Equipped with Multi-Beam Antennas
### Analysis of Interference Based on Off-Axis Angle and Beam Distance
As defined in Equation (13), the variation of \(F_{\rm u}\) with respect to the beam distance \(l\) is depicted in Figure 6b. From Equation (2), it is known that the function \(F_{\rm u}\) directly reflects the adameasurement of the interference \(l\) received by the GSO satellite ground stations. It shows that when the beam distance reaches close to nearly 350 km, the value of the function \(F_{\rm u}\) expressed in dB declines to half the value when the beam centers of both satellites coincide. The parameter \(F_{\rm u}\) is the sum of two gains, and the off-axis angles \(\theta_{1}\) and \(\theta_{2}\) both increase monotonically with respect to the beam distance \(l\). Therefore, the variation model of \(F_{\rm u}\) with respect to the beam distance \(l\) would be similar to that of antenna gain with respect to the off-axis angle. This is why we can observe some sudden changes occurring for the values of \(F_{\rm u}\), as previously shown in Figure 3:8  
Here it is worth pointing out that for different beams with distinct tilting angles, the identical distance between the beam centers may correspond to different off-axis angles, which is straightly related to the level of interference received by the GSO ground stations. As illustrated in Figure 7a, for convenience we suppose that the NGSO satellite beams with off-axis angles \(\alpha_{1}\), \(\alpha_{2}\), and \(\alpha_{3}\) are named beam 1, beam 2, and beam 3, respectively. We speculate that the three beams of the NGSO satellite are all illuminated. One points to the center of the Earth, while the others have a tilting angle \(\gamma\). For the three scenarios, the mathematical relationship between the beam distance \(l\) and off-axis angles \(\alpha_{1}\), \(\alpha_{2}\), and \(\alpha_{3}\) can be expressed as  
Figure 3:9  
\[F_{\rm u}=G_{\rm ii}(\theta_{1})\ {\rm dBi}\ +G_{\rm r}(\theta_{2})\ {\rm dBi} \tag{13}\]0  
\[F_{\rm u}=G_{\rm ii}(\theta_{1})\ {\rm dBi}\ +G_{\rm r}(\theta_{2})\ {\rm dBi} \tag{13}\]1  
When the tilting angle is taken as \(6^{\circ}\), \(12^{\circ}\), and \(18^{\circ}\), the relationship between the off-axis angle and beam distance is pictured in Figure 7b. It is recognizable that the difference between the off-axis angles is very slight with different tilting angles \(\gamma\) for beam 3. Furthermore, compared with the scenario for beam 1, the difference between them is also negligible. However, the increment of tilting angle \(\gamma\) brings about great changes of off-axis angle for beam 2. When the tilting angle \(\gamma\) becomes very large, a significant difference in off-axis angles occurs. Considering the most severe scenario with \(\gamma=18^{\circ}\) for beam 2, the difference between the off-axis angles can reach nearly \(10^{\circ}\) when the beam distance is over 600 km. \[F_{\rm u}=G_{\rm ii}(\theta_{1})\ {\rm dBi}\ +G_{\rm r}(\theta_{2})\ {\rm dBi} \tag{13}\]2 Relationship between off-axis angles \(\theta_{1}\), \(\theta_{2}\), and \(F_{\rm u}\) and beam distance \(l\), with different tilting angles \(\beta_{1}\) and \(\beta_{2}\). (**a**) Relationship between variations of \(\theta_{1}\), \(\theta_{2}\), and beam distance \(l\) with different value of tilting angle \(\beta_{1}\) and \(\beta_{2}\); (**b**) Relationship between \(F_{\rm u}\) in dB, \(F_{\rm u}\) in number, and beam distance \(l\) with different value of tilting angle \(\beta_{1}\) and \(\beta_{2}\).