## 4 System Geometrical Modelling  
In this section, the network geometry is described and analyzed. The SBs are located at height \(h_{s}\) from the ground surface (usually measured from sea level) as shown in Figure 2. If the aircraft flies at altitude \(h_{p}\), and the Earth's radius is \(R_{e}\), then the maximum range from the SB to the aircraft can be deduced as follows:  
Figure 1: Stratospheric basestations wireless network for aviation.  
Figure 2: A simple aircraft–stratospheric basestation (SB) communication link geometrical model.  
From triangle \(\Delta\)_pse_,  
\[\left(R_{e}+h_{p}\right)^{2}=d_{c}^{2}+\left(R_{e}+h_{s}\right)^{2}-2d_{c}(R_{e} +h_{s})\cos(\theta_{c}) \tag{1}\]  
or,  
\[d_{c}^{2}-2d_{c}(R_{e}+h_{s})\cos(\theta_{c})-\left(R_{e}+h_{p}\right)^{2}+ \ \left(R_{e}+h_{s}\right)^{2}=0 \tag{2}\]  
then,  
\[d_{c}=\frac{2(R_{e}+h_{s})\cos(\theta_{c})\pm\sqrt{\left(2(R_{e}+h_{s})\cos( \theta_{c})\right)^{2}-4\left(-\left(R_{e}+h_{p}\right)^{2}+\ \left(R_{e}+h_{s}\right)^{2}\right)}}{2} \tag{3}\]  
Logically, \(d_{c}\) has a solution which includes the negative sign of the square root. Then \(d_{c}\) is finally simplified and given by:  
\[d_{c}=\left(R_{e}+h_{s}\right)\left(1-\sqrt{1+\frac{\left(\left(R_{e}+h_{p} \right)^{2}-\ \left(R_{e}+h_{s}\right)^{2}\right)}{\left(\left(R_{e}+h_{s}\right)\cos( \theta_{c})\right)^{2}}}\right)\cos(\theta_{c}) \tag{4}\]  
where \(\theta_{c}\) is the pointing angle from the SB to the aircraft measured from the SB height \(h_{p}\) and is therefore given by:  
\[\theta_{c}=\cos^{-1}\left(\frac{d_{c}^{2}+\left(R_{e}+h_{s}\right)^{2}-\left( R_{e}+h_{p}\right)^{2}}{2d_{c}(R_{e}+h_{s})}\right) \tag{5}\]  
The communication range \(d_{c}\) lies between minimum and maximum values where the minimum value is given by:  
\[d_{cmin}=h_{s}-h_{p} \tag{6}\]  
While the maximum value of \(d_{c}\) can be deduced from Figure 2 as follows:  
\[d_{cmax}=\sqrt{d_{SBmax}^{2}+d_{Pmax}^{2}-2d_{SBmax}d_{Pmax}\cos\left(180^{ \circ}-2\alpha_{g}\right)} \tag{7}\]  
where \(\alpha_{g}\) is the ground elevation angle at the airport. \(d_{SBmax}\) and \(d_{Pmax}\) can be deduced as follows:  
\[\left(R_{e}+h_{s}\right)^{2}=R_{e}^{2}+d_{SBmax}^{2}-2R_{e}d_{SBmax}\cos\left( 90^{\circ}+\alpha_{g}\right) \tag{8}\]  
or  
\[d_{SBmax}^{2}+2R_{e}d_{SBmax}\sin(\alpha_{min})+R_{e}^{2}-\left(R_{e}+h_{s} \right)^{2}=0 \tag{9}\]  
then,  
\[d_{SBmax}=\frac{-2R_{e}\sin\left(\alpha_{g}\right)\pm\sqrt{\left(2R_{e}\sin( \alpha_{min})\right)^{2}-4\left(R_{e}^{2}-\left(R_{e}+h_{s}\right)^{2}\right)} }{2} \tag{10}\]  
or  
\[d_{SBmax}=-R_{e}\sin(\alpha_{min})+\sqrt{\left(R_{e}\sin\left(\alpha_{g} \right)\right)^{2}+h_{s}^{2}+2R_{e}h_{s}} \tag{11}\]  
Similarly, \(d_{Pmax}\) becomes:  
\[d_{Pmax}=-R_{e}\sin(\alpha_{min})+\sqrt{\left(R_{e}\sin\left(\alpha_{g} \right)\right)^{2}+h_{p}^{2}+2R_{e}h_{p}} \tag{12}\]  
As the aircraft is temporarily stayed at the airport during flight with onboard passengers and starts shortly for takeoff, so the minimum elevation angle can be considered less than the conventional \(5^{\circ}\) for satellite communications systems. This minimum elevation angle determines the coverage radius of the SB and the number of SBs in the network. At very low elevation angles, the slant distance between an aircraft and SB can be approximated by the sum of \(d_{SBmax}\) and \(d_{Pmax}\) which is important in discussing the atmospheric losses affecting the communication link between aircraft and SB.