# Performance Modeling Framework for IoT-over-Satellite Using Shared Radio Spectrum
## 3 IoT-over-Satellite Performance Modeling
### Orbital Mechanics and Geometric Model  
In this evaluation framework, we use the SGP4 algorithm to calculate orbital velocity and location vectors of the LEO satellite. The orbital data is then fed into the geometric model to estimate the slant distance (\(d\)), the elevation angle of the measurement point (\(\theta_{e}\)), and the depression angle of the satellite (\(\theta_{d}\)). The SGP4 model is an old algorithm developed in the 1970s, obtained by simplifying the Air Force General Perturbation theory (AFGP4) and using the [PERSON]'s gravitational model [28; 29]. The SGP4 model takes satellite orbital elements from a set of parameters called the Two Line Element (TLE). A TLE entry of a particular satellite consists of a list of orbital elements including [10]; satellite code, classification, international desiccator, epoch year, day, hour, the mean velocity of the satellite, mean acceleration of the satellite, inclination, right ascension of the ascending node, argument of perigee, mean anomaly, mean motion, and revolution number at epoch. Since the SGP4 model has an error around 1-3 km at epoch per day, TLE sets are regularly updated by the NASA and NORAD to mitigate possible errors. The Cartesian coordinates (The utilized frame of reference is called the True Equator Mean Equinox Earth-Centered Inertial (TEME ECI). The origin of this frame is at the Earth's center of mas, and it does not rotate with the Earth movement, the \(x\)-\(y\) plane is the true Equator, and the mean Equinox is the \(z\)-axis) of the satellite point at a time interval can be calculated from the output of the SGP4 as follows [28],  
\[\left[\begin{array}{c}x\left(t\right),\;y\left(t\right),\;z\left(t\right) \end{array}\right]^{\top}=\mathbf{A}\left[\cos E\left(t\right)-e\right]+\mathbf{B}, \sin E\left(t\right), \tag{1}\]  
where  
\[\mathbf{A} = \left[\begin{array}{c}\cos g\cos\lambda-\sin g\sin\lambda\cos I \\ a\left[\sin g\cos\lambda\cos I+\cos g\sin\lambda\right]\\ a\sin g\sin\lambda\end{array}\right],\] \[\mathbf{B} = \left[\begin{array}{c}-a\sqrt{(1-e^{2})}\left[\sin g\cos\lambda+ \cos g\sin\lambda\cos I\right]\\ a\sqrt{(1-e^{2})}\left[\cos g\cos\lambda\cos I-\sin g\sin\lambda\right]\\ a\sqrt{(1-e^{2})}\cos g\sin\lambda.\end{array}\right] \tag{2}\]  
Here \(a\) is the semi-major axis of the satellite orbit, \(e\) is its eccentricity, \(g\) is the argument of the periapsis, \(\lambda\) is the longitude of ascending node, \(I\) is the inclination, and \(E\) is the eccentric anomaly. An illustration of the satellite orbital parameter is depicted in Figure 2.  
We convert the True Equator Mean Equinox (TEME) Cartesian coordinates of the satellite into latitude and longitude, referring to the World Geodetic System 1984 (WGS84) frame. Then, we can  
Figure 2: Illustration of the key satellite orbital parameter used in the SGP4 model where \(E\) is the eccentric anomaly of the satellite at time t.  
obtain the slant range distance (\(d\)), elevation angle (\(\theta_{e}\)), and azimuth angle from the measurement points. Also, the depression angle (\(\theta_{d}\)) of the antenna which can be calculated by the cosine law [10],  
\[\theta_{d}\left(t\right)=\cos^{-1}\left[\frac{d\left(t\right)^{2}+\left(h+R \left(\phi,\lambda\right)\right)^{2}-R\left(\phi_{o},\lambda_{o}\right)^{2}}{2 \times d(t)\left(h+R\left(\phi,\lambda\right)\right)}\right]\, \tag{3}\]  
where \(R(\phi\), \(\lambda)\) is the true radius of the Earth at the subsatellite point, and \((\phi,\lambda)\) is the latitude and longitude of the subsatellite point, \(R(\phi_{o},\lambda_{o})\) is the true radius of the Earth at the observation point, where \((\phi_{o},\lambda_{o})\) is the latitude and longitude of the observation point, and \(h\) is the altitude of the satellite, as depicted in Figure 3. According to these parameters, we calculate the satellite channel the path loss, air absorption, and excessive path loss between the satellite and the observation point as further detailed in the below subsection.