# Optimal Progressive Pitch for OneWeb Constellation with Seamless Coverage
## 2 Problem Modeling
### Beam Layout  
In this paper, we focus on the polar orbit constellation. The OneWeb satellite has \(N\) user beams where \(N\) equals 16. Each beam is denoted as \(b_{i}\), \(i\in\{1,2,\cdots,N\}\), and its shape is highly elliptical. The coverage pattern is shown in Figure 1, where the major and minor axis of the elliptical beam are along the east-west and north-south directions, respectively. The coverage of Figure 1 is a simplified version of a spherical cap in Earth's surface, and the flattened coverage in this figure is used to specify how the beams are deployed, such as the frequency plan, the layout of beams and their elevation angles. The beams of the same color in the satellite coverage have the same frequency. The set of frequencies is \(\mathbf{F}\), where \(\mathbf{F}=\{f_{1},f_{2},\cdot f_{m},\cdot f_{M}\}\) and \(F\) is 8 in the OneWeb system. \(N\) is an integer multiple of \(M\), and the number of beams of the same frequency is \(\frac{N}{M}\). It is assumed that the satellite is located in the northern hemisphere and moves southward. \(SSP\) represents the subsatellite point, where \(BC_{i}\) denotes the center of the beam \(b_{i}\), and \(CC\) denotes the center of the satellite coverage. The angle between \(SSP\) and \(CC\) as observed from the satellite is the pitch angle \(\chi\). For the \(BC_{i}\), the elevation angle is \(\chi+(2i-N-1)\mu_{b}\). The antenna gain of the beam \(b_{i}\) with respect to the point \(E\) is determined by the off-axis angles in the major and minor axis directions. To describe the antenna gain, off-axis angles \(\mu_{i}\) in the minor axis and \(\
u_{i}\) in the major axis are defined. The antenna gain of beam \(b_{i}\) at point \(E\) is a function of \(\mu_{i}\) and \(\
u_{i}\), denoted as \(G_{t}(\mu_{i},\
u_{i})\). \begin{table}
\begin{tabular}{c l} \hline \hline
**Notation** & **Description** \\ \hline \(\chi\) & The pitch angle of the LEO satellite. \\ \(\alpha_{i}\) & Elevation angle corresponding to the boresight of the beam \(b_{i}\). \\ \(\mu_{i}\) & The off-axis angle in the minor-axis of the beam \(b_{i}\). \\ \(\
u_{i}\) & The off-axis angle in the major-axis of the beam \(b_{i}\). \\ \(\mu_{b}\) & The one-half 3 dB beamwidth in the minor axis of the highly elliptical beam \\ \(\
u_{b}\) & The one-half 3 dB beamwidth in the major axis of the highly elliptical beam \\ \(G_{t}(\mu,\
u)\) & The antenna gain of LEO satellite transmitting antenna. \\ \(p_{b}^{t}\) & Input power of the LEO satellite transmitting antenna. \\ \(S_{f}\) & Beam set with frequency being\(f\). \\ \(G_{\tau}(\varphi)\) & Antenna gain of the GSO earth station with the off-axis angle \(\varphi\). \\ \(\psi_{es}\) & Latitude of GSO earth station. \\ \(\sigma\) & Longitude difference between GSO satelite and NGSO satellite. \\ \(K\) & The number of closed beams. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Notations used in this paper. Figure 1: The coverage of LEO satellite with the pitch angle being \(\chi\). The antenna model recommended by ITU-R S.1528 [31] is used to model the satellite radiation pattern for NGSO satellite operating in the fixed-satellite service below 30 GHz. According to ITU-R S.1528, the logarithm of \(G_{n}(\Psi)\) is given by  
\[[G_{n}(\Psi)]=\left\{ \begin{array}{cc}G_{max}-3(\Psi/\Psi_{b})^{\alpha}&0<\Psi \leq a\Psi_{b}\\ G_{max}+L_{N}+25\log(z)&a\Psi_{b}<\Psi\leq 0.5b\Psi_{b}\\ G_{max}+L_{N}&0.5b\Psi_{b}<\Psi\leq b\Psi_{b}\\ X-25\log(\Psi)&b\Psi_{b}<\Psi\leq Y\\ L_{F}&Y<\Psi\leq 90^{\circ}\\ L_{B}&90^{\circ}<\Psi\leq 180^{\circ}\end{array} \right., \tag{1}\]  
where \(X=G_{max}+L_{N}+25\log(b\Psi_{b})\) and \(Y=10^{0.04(G_{max}+L_{N}-L_{F})}\), \(G_{max}\) is the maximum gain in the mainlobe (dBi) and \(\Psi_{b}\) is the one-half the 3 dB beamwidth. The far-out side-lobe level, the back-lobe level and the main beam and near-in side-lobe mask cross point below peak gain are denoted as \(L_{F}\), \(L_{B}\) and \(L_{N}\), separately. For \(L_{N}=-25\), the values of \(a\), \(b\) and \(\alpha\) are \(2.58\sqrt{1-0.6\log z}\), 6.32 and 1.5.