## 2 Skywave Ionosphere Channel for Hypersonic Vehicles
For non-offset absorption, the attenuation coefficient of electromagnetic waves in the ionosphere is given by  
\[\alpha_{iok}=\omega\sqrt{\frac{\mu_{0}\epsilon_{0}}{2}\left[\sqrt{\epsilon_{r}^ {2}+\left(60\lambda\sigma\right)^{2}}-\epsilon_{r}\right]} \tag{3}\]  
where \(\omega\) is the electromagnetic wave angular frequency, \(\omega=2\pi f\), \(\lambda=f/c\), \(c\) is speed of light, \(\mu_{0}\) represents the magnetic permeability in a vacuum (\(4\pi\times 10^{-7}\) N\(\cdot\)A\({}^{-2}\)), \(\epsilon_{0}\) is the vacuum dielectric constant (8.854 \(\times\) 10\({}^{12}\) F/m), \(\epsilon_{r}\) is the equivalent relative dielectric constant, and \(\sigma\) is the conductivity of the plasma [43; 44]:  
\[\sigma=\frac{e^{2}N_{e}v_{en}}{m_{e}(\omega^{2}+v_{en}^{2})} \tag{4}\]  
\[\epsilon_{r}=1-\frac{e^{2}N_{e}}{m_{e}\epsilon_{0}(\omega^{2}+v_{en}^{2})} \tag{5}\]  
where \(\lambda\) is the electromagnetic wavelength, \(e\) is the electron charge (1.6 \(\times\) 10\({}^{-19}\) C), \(N_{e}\) is the electron density of the plasma (electrons/m\({}^{3}\)), \(m_{e}\) is the electron mass (9.1 \(\times\) 10\({}^{-31}\) kg), \(v_{en}\) is the plasma collision frequency in radians (rad/s), \(v_{en}=2\pi v_{e}\), and \(v_{e}\) is the plasma collision frequency (Hz). The total attenuation (absorption loss) of radio waves propagating through the ionosphere is given by  
\[a_{iok}(t)=e^{-\int a_{iok}(t)dl(t)} \tag{6}\]  
To calculate the total distance \(l(t)\) of skywave propagation through the ionosphere, it is essential to determine the position of the path as it passes through the ionosphere. The calculation of the propagation path of radio waves in ray tracing is carried out incrementally, starting from the initial values of a set of given variables in the Haselgrove equation [45]. In the spherical coordinate system, the component of the wave vector in spherical coordinates is defined as \(\mathbf{a}=(k_{r},k_{\theta},k_{\theta})\), where \(k_{r}\) is vertical to the upward direction of the ground, \(k_{\varphi}\) is the east direction, and \(k_{\theta}\) is the south direction. Due to the fact that collision effects primarily cause energy absorption and have minimal impact on the propagation path of signals, only the ray path is considered in the calculations, and collision effects are ignored [46]. Without considering the geomagnetic field, the Haselgrove differential equation, which is based on the group path \(P^{\prime}\) of skywave propagation, is solved as follows [47]:  
\[\left\{ \begin{array}{l}\frac{df}{dP^{\prime}}=\frac{c}{\omega}k_{r}\\ \frac{d\theta}{dP^{\prime}}=\frac{c}{\omega\sigma}k_{\theta}\\ \frac{d\theta}{dP^{\prime}}=\frac{c}{\omega\sin\theta}k_{\varphi}\\ \frac{dk_{r}}{dP^{\prime}}=-\frac{\omega\omega}{2c}\frac{\omega}{\omega\theta }+k_{\theta}^{2}\frac{c}{\omega}+k_{\varphi}^{2}\frac{c}{\omega}\\ \frac{dk_{\theta}}{dP}=\frac{1}{\left(-\frac{\omega}{2c}\frac{\omega}{\omega \theta}\right)}-k_{r}k_{\theta}\frac{c}{\omega}+k_{\theta}^{2}\frac{c}{\omega} \cot\theta)\\ \frac{dk_{\varphi}}{dP^{\prime}}=\frac{1}{r\sin\theta}(-\frac{\omega}{2c}\frac {\omega}{\omega\theta}-k_{r}k_{\theta}\frac{c}{\omega}\sin\theta+\frac{c}{ \omega}k_{\theta}k_{\varphi}\cos\theta)\end{array} \right. \tag{7}\]where (\(r\), \(\theta\), \(\varphi\)) is the spherical coordinate of the skywave ray path, and \(c\) is the speed of light, where  
\[X=\frac{\omega_{p}^{2}}{\omega^{2}}=\frac{N_{e}e^{2}}{m_{e}\varepsilon_{0} \omega^{2}}=80.8\frac{N_{e}(r,\theta,\varphi)}{f^{2}} \tag{8}\]  
The \(\partial X/\partial r\), \(\partial X/\partial\theta\), and \(\partial X/\partial\varphi\) values depend on specific ionospheric plasma parameters [48]:  
\[