## 3 Optimal Beamwidth Maximizing Uplink Coverage Probability Scheme
### Uplink Coverage Probability Analysis Model  
This section uses mathematical reasoning to build the uplink coverage probability analysis model. The uplink coverage probability analysis model calculates the uplink coverage probability with different time \(t\) (different elevation angle), equivalent antenna aperture \(m\), and target SINR \(\gamma\). Since the elevation angle and footprint are different within each beam, the uplink coverage probability of the satellite should be calculated beam by  
Figure 2: The illustration of Equations (12)–(15). beam. We calculate the uplink coverage probability \(p(t,m,\gamma)\) of the satellite by the uplink coverage probability of each beam with weighting as follows:  
\[p(t,m,\gamma)=\sum_{f\in\mathcal{F}}\sum_{l\in\mathcal{L}_{f}}w_{l}\mathbb{P}( \xi_{s,l}(t,m)\geq\gamma) \tag{17}\]  
\[\text{s.t. }|\mathcal{F}|=1,\ t\in\{\mathcal{T}\ |\ \frac{\pi}{6}\leq\varepsilon_{l}(t) \leq\frac{\pi}{2}\} \tag{18}\]  
where \(w_{l}\) is the ratio of the coverage area of beam \(l\). Since the FSPL and elevation angle result in minor changes within a beam, our model uses the location of the beam center to present the FSPL and elevation angle to reduce the computation complexity as follows:  
\[\mathbb{P}(\xi_{s,l}(t,m)\geq\gamma)\] \[=\mathbb{P}(-\kappa_{\epsilon_{s,l}}-\zeta_{\epsilon_{s,l}}+G( \varphi_{l}^{s}(t),m)\geq\alpha_{s,l}(t)) \tag{19}\] \[\simeq\mathbb{P}(-\kappa_{\epsilon_{l}}-\zeta_{\epsilon_{l}}+G( \varphi_{l}^{s}(t),m)\geq\alpha_{l}(t))\]  
\[\alpha_{l}(t)=\gamma-P_{t}+L_{l}(t)+W_{l}(t) \tag{20}\]  
\[W_{l}(t)=10\log_{10}(10^{N/10}+I_{l}(t)) \tag{21}\]  
where \(W_{l}(t)\) is the interference-plus-noise power of beam \(l\) at time \(t\), and \(I_{l}(t)\) is the expect interference power received in beam \(l\) at time \(t\). Similarly, our model uses the location of the interference beam center to present the FSPL and elevation angle. Since shadow fading follows zero-mean Gaussian distribution, it is neglectable for the expected interference. Moreover, since the clutter is zero for LOS users, the LOS probability should be considered. Considering the LOS probability and received antenna gain, the expected interference power in linear form is calculated as  
\[\bar{I}_{l}(t)=\sum_{k\in\mathcal{L}_{f}\backslash l}p_{LOS,\epsilon_{k}}10^{ (P_{l}-L_{k}+G(\varphi_{k,l}(t),m))/10}+p_{NLOS,\epsilon_{k}}10^{(P_{l}-L_{k} +G(\varphi_{k,l}(t),m))/10} \tag{22}\]  
where \(\varphi_{k,l}(t)\) denotes the angular difference between two beams. The angular difference can be derived similarly to (16) as follows:  
\[\varphi_{k,l}(t)=\arccos\frac{||\mathbf{x}_{sat}(t)-\mathbf{v}_{k}(t)||^{2}+ ||\mathbf{x}_{sat}(t)-\mathbf{v}_{l}(t)||^{2}-||\mathbf{v}_{k}(t)-\mathbf{v }_{l}(t)||^{2}}{2||\mathbf{x}_{sat}(t)-\mathbf{v}_{k}(t)||\cdot||\mathbf{x}_{ sat}(t)-\mathbf{v}_{l}(t)||}. \tag{23}\]  
To calculate the uplink coverage probability in (19), the cumulative distribution function (CDF) for clutter loss, shadow fading, and antenna gain needs to be discussed. Clutter loss and shadow fading distributions follow LOS probability and Gaussian distribution, which can be calculated with low complexity. However, the adjacent beam spacing for each beam is different, so it is complex to calculate the probability of serving antenna gain. Thus, we simplify the model by assuming the adjacent beam spacings of each beam are the same value \(\hat{\varphi_{l}}\). \(\hat{\varphi_{l}}\) equals to the average of adjacent beam spacings. With the assumed adjacent beam spacing \(\hat{\varphi_{l}}\) in the hexagonal layout, the receive antenna gain \(G(\varphi_{l}^{s}(t),m)\) from user \(s\) to serving beam \(l\) is calculated by assuming \(0\leq\varphi_{l}^{s}(t)\leq\frac{\hat{\varphi_{l}}}{\sqrt{3}}\), and the probability of receive antenna gain is determined as follows (as shown in Figure 3):  
\[p_{\hat{\varphi_{l}}}(\varphi)=