# Optimal Progressive Pitch for OneWeb Constellation with Seamless Coverage
## 2 Problem Modeling
### Inline Interference Model  
The interference scenario is illustrated in Figure 2, where the GSO earth station, LEO satellite and GSO satellite are located on a straight line. The east-west boundaries of the coverage region of beam \(b_{i}\) are marked by points A and B, respectively. The plane ABL is the cross section where the major axis of elliptical beam \(b_{i}\) is located, and point C is the center of beam \(b_{i}\). The elevation and azimuth angles are \(\alpha_{i}\) and 0, respectively, where the elevation angle \(\alpha_{i}\) is expressed as \(\alpha_{i}=-(N-2i+1)\mu_{b}+\chi\).  
The off-axis angles of point D with respect to the major-axis and minor-axis directions of beam \(b_{i}\) are denoted as \(v_{i}\) and \(\mu_{i}\) respectively. The LEO satellite is located in the YOZ plane, and the longitude difference is \(\sigma\) between the GSO satellite and it. The EPFD of point D is shown as below  
\[p_{f}=\sum_{b_{i}\in S_{f}}\frac{p_{b}^{t}G_{t}(\mu_{i},v_{i})}{4\pi d^{2}}\leq \frac{p_{b}^{t}G_{t}(\mu_{f,min},0)\Big{|}S_{f}\Big{|}}{4\pi h_{n}^{2}}. \tag{4}\]  
Given the latitude \(\psi\), longitude difference \(\sigma\) and pitch angle \(\chi\) of the NGSO satellite, refer to Appendix B to calculate \(\mu_{i}\) and \(v_{i}\) of \(b_{i}\). Suppose the central frequency of \(b_{i}\) is \(f\), \(S_{f}\) is the set of beams with frequency \(f\), and \(\Big{|}S_{f}\Big{|}\) represents the number of beams with central frequency \(f\), which equals \(\frac{N}{M}\).  
In the Inequality (4), \(\mu_{f,min}\) is \(min_{b_{j}\in S_{f},|\sigma|<\sigma_{max}}\mu_{j}\), and \(d\) represents the distance between the NGSO satellite and the GSO earth station. The input power of the reference bandwidth is denoted as \(p_{t}^{b}\), which equals to \(10^{EIRP-10 log_{10}(\frac{B_{w}}{B_{ref}})}\), where \(B_{w}\) is the bandwidth, and \(B_{ref}\) is the reference bandwidth regarding the caculation of EPFD. Then, we can have the inequality  
\[\begin{split} epfd_{f}&=10\log_{10}\left(p_{f}\frac {G_{r}(0)}{G_{r,\max}}\right)\leq EIRP-10\log_{10}(\frac{B_{w}}{B_{ref}})+10 \log_{10}(\Big{|}S_{f}\Big{|})\\ &+10\log_{10}(\frac{G_{t}(\mu_{f,min},0)}{G_{t}(0,0)})-10\log_{10 }(4\pi h_{n}^{2}),\end{split} \tag{5}\]  
where \(G_{r,max}\) denotes the maximum antenna gain of the GSO earth station, and \(G_{r,max}=G_{r}(0)\).  
To make the downlink EPFD satisfy the corresponding limits, it is ensured that the right-hand side of (5) is smaller than \(epfd_{th}\). Therefore, the off-axis angle in minor axis of the beams should be less than \(\mu_{th}\). According to Equation (6), the off-axis angle threshold (\(\mu_{th}\)) can be obtained. In order to circumvent interference from LEO satellites to the GSO  
Figure 2: Downlink inline interference from NGSO satellite to GSO earth station.  
satellite system, the off-axis angle of the minor axis of all activated beams of LEO satellites should be less than the threshold.  
\[10\log_{10}\left(\frac{G_{n}(\mu_{th})}{G_{n}(0)}\right)=epfd_{th}+10\log_{10} \left(4\pi h_{n}^{2}\frac{B_{w}}{B_{ref}}\right)-EIRP-10\log_{10}\left(\left|S_{f }\right|\right) \tag{6}\]