# Elevation Angle Characterization for LEO Satellites: First and Second Order Statistics
## 2 Materials and Methods
### Analytical Characterization of the Elevation Angle  
The elevation angle was analytically characterized in [2] through its probability density function (PDF), \(f_{\Theta}\), which is defined as a marginal distribution from the joint PDF of \(\Theta\) and the maximum value of the elevation angle, \(\Theta_{max}\), as follows  
\[f_{\Theta}(\theta)=\int_{\theta}^{\theta_{M}}f_{\Theta,\Theta_{max}}(\theta, \theta_{max})d\theta_{max} \tag{5}\]  
Figure 5: Generalization of a three-state channel model as proposed by [24] considering state transitions based on elevation angle changes.  
Figure 6: Loo CDF of the received signal for two elevation angle values. The Loo triplet for \(\theta_{1}=10^{\circ}\) is \(\alpha=-105\) dB, \(\psi=5\) dB, and \(MP=-108\) dB; and \(\alpha=-95\) dB, \(\psi=5\) dB, and \(MP=-98\) dB for \(\theta_{2}=90^{\circ}\).  
where \(f_{\Theta,\Theta_{max}}\) is given by  
\[f_{\Theta,\Theta_{max}}(\theta,\theta_{max})=f_{\Theta|\Theta_{max}}(\theta| \theta_{max})f_{\Theta_{max}}(\theta_{max}) \tag{6}\]  
or equivalently  
\[f_{\Theta,\Theta_{max}}(\theta,\theta_{max})=\frac{G(\theta)\sin\gamma(\theta) }{\sqrt{\cos^{2}\gamma(\theta_{max})-\cos^{2}\gamma(\theta)}}\cdot\frac{f_{ \Theta_{max}}(\theta_{max})}{\int_{\theta_{min}}^{\theta_{M}}f_{\Theta_{max}}( x)\cos^{-1}\Bigl{(}\frac{\cos\gamma(\theta_{min})}{\cos\gamma(x)}\Bigr{)}dx} \tag{7}\]  
The integral dividing the right hand term is a constant, say \(C_{1}\), then, (7) can be rewritten as  
\[f_{\Theta,\Theta_{max}}(\theta,\theta_{max})=\frac{f_{\Theta_{max}}(\theta_{ max})G(\theta)\sin\gamma(\theta)}{C_{1}\sqrt{\cos^{2}\gamma(\theta_{max})-\cos^{2} \gamma(\theta)}} \tag{8}\]  
and the auxiliary functions \(\gamma(\cdot)\) and \(G(\cdot)\) require an extra parameter \(a\) defined as \(\alpha=r_{E}/r_{S}\), where \(r_{E}=6378\) km is the radius of the Earth; and \(r_{S}=r_{E}+h\) depends on the altitude \(h\) (from the Earth's surface) of the circular LEO orbit in kilometers. The functions \(\gamma(\cdot)\) and \(G(\cdot)\) are defined as follows  
\[\gamma(\theta)=\cos^{-1}(\alpha\cos\theta)-\theta \tag{9}\]  
\[G(\theta)=\frac{1+\alpha^{2}-2\alpha\cos\gamma(\theta)}{1-\alpha\cos\gamma( \theta)} \tag{10}\]  
The term \(f_{\Theta_{max}}(\theta_{max})\) is defined as follows  
\[f_{\Theta_{max}}(\theta_{max}) = \frac{G(\theta_{max})}{K_{2}}\cdot f_{\Phi}(\phi_{0}\,-\,\gamma( \theta_{max})),\quad\text{for}\quad\theta_{min}\quad\leq\quad\theta_{max} <\quad\theta_{c} \tag{11}\]  
\[f_{\Theta_{max}}(\theta_{max})=\frac{G(\theta_{max})}{K_{2}}\cdot[ f_{\Phi}(\phi_{0}-\gamma(\theta_{max})+\\ f_{\Phi}(\phi_{0}+\gamma(\theta_{max}))],\quad\text{for} \quad\theta_{c}<\theta_{max}<\theta_{M} \tag{12}\]  
where \(f_{\Phi}(\phi)\) is defined from the orbital inclination \(i\); and ES latitude, \(\phi\), as follows  
\[f_{\Phi}(\phi)=\frac{\cos\phi}{\pi\sqrt{\sin^{2}i-\sin^{2}\phi}},\quad\text{ for}\ |\phi|<i \tag{13}\]  
and \(\theta_{M}=\pi/2\), and \(K_{2}\) is given by  
\[K_{2}=\frac{1}{2}-\frac{1}{\pi}\sin^{-1}\biggl{(}\frac{\sin(\phi+\gamma( \theta_{min}))}{\sin(i)}\biggr{)} \tag{14}\]  
where \(\theta_{min}\) is the minimum elevation angle that will be considered and is in the range \(0^{\circ}<\theta_{min}<90^{\circ}\).  
Although results shown in [2] are highly accurate with respect to the actual elevation angle PDF, the analytical expression \(f_{\Theta}\) is cumbersome to obtain and evaluate. Furthermore, analytical expressions for second or higher order statistics, are a missing result in the literature to the best of our knowledge.