# Optimal Progressive Pitch for OneWeb Constellation with Seamless Coverage
## Appendix A EPFD Analysis for the Orbit Plane  
In the LEO satellite constellation, in addition to the EPFD contribution from satellites in the same orbit, it is also necessary to consider satellites in adjacent orbits. To simplify the analysis process, we mainly consider the interference from satellites in the same orbit to the GSO earth station, and the interference scenario is shown in Figure 11, where the red ellipses denote represent the beams with same frequency. The contribution to EPFD of a particular GSO earth station is dominated by the nearby satellites. Therefore, in order to evaluate the interference of different satellites in the LEO constellation to the GSO satellite system, we propose an analytical model of EPFD. In Figure 11, the GSO earth station beam is pointed at the GSO satellite and also receives co-frequency interference from the LEO satellite. The elevation angle of the GSO earth station with respect to the \(j\)-th LEO satellite is denoted as \(\delta_{j}\), and the set of beams with frequency \(f_{m}\) is denoted as \(S_{f_{m}}\). For the \(j\)-th satellite, the latitude and attitude pitch angle are \(\psi_{j}\) and \(\chi_{j}\), respectively. The downlink interference is determined by the aggregation of the interference links from the LEO satellites. For frequency \(f_{m}\), the aggregate EPFD at the GSO earth stations can be expressed as  
\[epfd^{f_{m}}=10\log_{10}\left(\sum_{b_{i}\in S_{f_{m}},j\in J}p_{j,b_{i}}\frac {G_{r}\big{(}\varphi_{j}\big{)}}{G_{r,\max}}\right), \tag{11}\]  
where \(J\) denotes the set of LEO satellites in the same orbit. The \(p_{j,b_{i}}\) and \(\varphi_{j}\) are the power flux density from \(b_{i}\) of the \(j\)-th LEO satellite and the off-axis angle corresponding to the \(j\)-th satellite. The distance between the LEO satellite and the GSO earth station is \(d_{j}\), and the expression of \(p_{j,b_{i}}\) is shown as  
\[p_{j,b_{i}}=\frac{p_{b}^{t}G_{t}\big{(}\mu_{ij},0\big{)}}{4\pi d_{j}^{2}}, \tag{10}\]  
where \(p_{b}^{t}\) is the input power of the LEO satellite transmitting antenna, and \(\mu_{ij}\) is the off-axis angle in the minor axis of \(b_{i}\) of the \(j\)-th NGSO satellite corresponding to the GSO earth station. According to \(eirp\) (\(eirp=10^{EIRP-10 log_{10}(\frac{B_{w}}{B_{wrf}})}\)), \(p_{b}^{t}\) can be expressed as \(eirp/G_{t}(0,0)\), where \(B_{w}\) and \(B_{ref}\) are the bandwidth of user beam and the reference bandwidth (40 kHz). The maximum EPFD of the GSO earth station is \(epfd_{\max}=\max_{f_{i}}epfd^{f_{i}}\), and the corresponding frequency is \(f_{\max}=\arg\max_{f_{i}}epfd^{f_{i}}\). In order to analyze the contribution of a single LEO satellite to the downlink EPFD, we define the EPFD of the \(j\)-th satellite, and its expression is shown as  
\[epfd_{j}=10\log_{10}\left(p_{jf_{\max}}\frac{G_{r}\big{(}\varphi_{j}\big{)}}{G _{r,\max}}\right). \tag{11}\]  
In order to explore the distribution of adjacent satellites to EPFD, we analyze the contribution of three adjacent satellites to EPFD. According to the relevant parameters of the OneWeb system shown in Table 2, the simulation results are obtained as shown in Figure 10.  
According to the ITU-RS 1428 [34], the antenna model of the GSO earth station with the given antenna aperture to wavelength ratio (\(20<\frac{D}{\lambda}<25\)) is represented as  
\[[G_{r}(\psi)]=\left\{ \begin{array}{ccc}G_{max}-2.5\times 10^{-3}(\frac{D}{ \lambda}\varphi)^{2}&0<\varphi<\varphi_{m}\\ 29-25 log_{10}(95\frac{\lambda}{D})&\varphi_{m}<\varphi<95\frac{\lambda}{D}\\ 29-25 log_{10}\varphi&95\frac{\lambda}{D}<\varphi<33.1^{\circ}\\ -9&33.1^{\circ}<\varphi<80^{\circ}\\ -5&80^{\circ}<\varphi<180^{\circ}\\ \end{array} \right., \tag{12}\]where \(G_{max}\) represents the maximum gain of the GSO earth station which equals to \(20 log_{10}(\frac{D}{A})\), and \(\varphi_{m}\) is \(\frac{20\lambda}{D}\sqrt{G_{max}-29+25\log(95\frac{\lambda}{D})}\). The logarithm of \(G_{r}(\varphi)\) is written as \([G_{r}(\varphi)]\).