# Elevation Angle Characterization for LEO Satellites: First and Second Order Statistics
###### Abstract  
The elevation angle \(\theta\) is relevant for the Low Earth orbit (LEO) satellite communications since it is always changing its relative position with respect to fixed Earth stations (ES's), and this affects the link length and received power, \(P_{R}\). This article provides a new methodology to compute the probability density function (PDF) and cumulative distribution function (CDF) of the elevation angle, \(\theta\), for diverse ES locations. This methodology requires as input parameters an ES latitude, \(\phi\), an orbit inclination value, \(i\), and an orbit altitude, \(h\). The elevation angle is characterized through a well known random variable, which facilitates the computation of the first and second-order statistics, and helps to determine the expected value and measures of dispersion of the angle \(\theta\) for a particular ES location. The proposed methodology allows an easy and quick calculation of the elevation angle's CDF, facilitating comparisons against CDF's of more ES's located at different latitudes, and longitudes, \(\lambda\); as well as the comparisons of CDF's of the elevation angle produced by different orbits. Extensive simulation results are summarized in a small table, which allows computation of the elevation angle's CDF and PDF for multiple ES locations without requiring of simulations and statistical fitting. Finally, the proposed methodology is validated through an extensive error analysis that show the suitability of the obtained results to characterize the elevation angle. Low Earth orbit (LEO); elevation angle; satellite communications Article  
Elevation Angle Characterization for LEO Satellites: First and Second Order Statistics  
1 Tecnologico de Monterrey, School of Engineering and Science, Monterrey 64849, Mexico 1  
Aragon-Zavala, Aragon-  
have followed segmentation in best and worst-case of the elevation angle [6], instead of analyzing the short and long term behavior of the elevation angle as an analytical function. However, some emerging problems, such as efficient power management [7], related to LEO satellites have made evident the necessity of having a way to characterize the elevation angle as an analytical function. The observed elevation angle, \(\theta\), is always-varying for LEO satellites and can be described as a function of time, \(\theta(t)\), for intervals in which a satellite is visible. The calculation of \(\theta\) for a given position can be deterministically obtained with mathematical procedures developed in orbital mechanics books [8] and widely implemented in software for space dynamics simulations. However, it is important to note that even with accurate predictions of \(\theta\) (accounting for contacts in several days or months), the relations between one contact and another, as well as the long term behavior of the elevation angle can be hardly described without randomness. Thus, appropriate use of probability theory becomes a great tool to analyze the behavior of the elevation angle, \(\theta\), in the long term. The elevation angle, \(\theta\), is usually defined for an ES as the angle (above the local horizon) at which the satellite is visible, and within the interval of \(0^{\circ}\) to \(90^{\circ}\), \(\theta\in(0^{\circ},90^{\circ}]\), regardless of its azimuth. The minimum value of \(\theta\) at which communication is possible is often called \(\theta_{min}\) (subject to \(\theta_{min}\geq 0^{\circ}\)), similarly, the maximum value of \(\theta\) that can be observed from the ES is called \(\theta_{max}\) (subject to \(\theta_{min}\leq\theta_{max}\leq 90^{\circ}\)). We can define a random variable, r.v., to take possible values of \(\theta\), such that \(\theta\in(\theta_{min},\theta_{max}]\) as \(\Theta\). The probability density function, PDF, and cumulative distribution function, CDF, of the elevation angle, \(f_{\Theta}(\theta)\) and \(F_{\Theta}(\theta)\), respectively, are useful functions to characterize the LEO channel since the elevation angle affects the received power level, \(P_{R}\).