## 2 Skywave Ionosphere Channel for Hypersonic Vehicles
As shown in Figure 2, the distance from the bottom of the ionosphere to the center of the Earth is \(R_{\rm b}=60+R_{0}=6431\) km. Given that the trajectory of the ray passing through the ionosphere at point A is (\(r_{j}\), \(\theta_{j}\), \(\varphi_{j}\)) and that the ray reaching the ground at position C (\(r_{o}\), \(\theta_{o}\), \(\varphi_{o}\)), the elevation angle \(\beta\) can be determined according to the geometric relationship shown in Figure 2:  
\[\beta_{u}=\arcsin\left(\frac{R_{\rm b}\cos\gamma_{j,o}-R_{0}}{\sqrt{R_{\rm b}^ {2}+R_{0}^{2}-2R_{\rm b}R_{0}\cos\gamma_{j,o}}}\right) \tag{13}\]  
The cosine of angle \(\gamma_{j,o}\) when the ray passes through the ionosphere and reaches the ground is  
\[\cos\gamma_{j,o}=\sin\theta_{j}\sin\theta_{o}\cos(\varphi_{j}-\varphi_{o})+ \cos\theta_{j}\cos\theta_{o} \tag{14}\]  
At this point, the complete skywave ionosphere channel fading (loss), \(h_{k}\) (\(t\)), can be calculated. It is clear from Equation (1) that in addition to \(h_{k}\) affecting the characteristics of the skywave ionosphere channel for a hypersonic vehicle, the time delay and Doppler of each path also play a significant role in shaping the channel characteristics. From the above deduction, it can be inferred that the k-th channel Doppler frequency and delay is \(f_{dk}\left(t\right)=-f\)-\(v\)-\(cos\theta_{k}\left(t\right)/c\) and \(\tau_{k}\left(t\right)=MP^{\prime}/c\), respectively. \(\theta_{k}(t)\) is the angle between the \(k\)-th wave vector \(k\) and the flight direction (vehicle vector velocity \(v\)), as shown in Figure 2. In the next section, we focus on analyzing parameters such as losses, delays, and Doppler effects, which influence the skywave ionosphere channel characteristics of hypersonic vehicles. The atmospheric ionosphere is generally located above 60 km, and the reflection height of a 14 MHz skywave is below 400 km. Therefore, we set the ionospheric height range to 60 km-600 km. The ionospheric model, used as the propagation medium for subsequent skywave trajectory simulations, employs the internationally recognized and well-established IRI-2020 (International Reference Ionosphere 2020) numerical model. This semi-empirical model is based on historical measurement data and includes electron density and collision frequency data across different latitudes, longitudes, and altitudes. The ionosphere has cloud-like structures of different ionization densities, referred to as ionospheric irregularities, which are small-scale structures with random temporal and spatial distributions, lacking universal applicability or sufficient empirical support. Hence, our study utilizes the IRI-2020 ionospheric model as the background ionosphere to quantitatively describe its spatial variation. Figure 2: Plan for ray ground reflection. The ray-tracing method employed in this study is based on Haselgrove's differential equations, a well-established numerical technique widely applied in skywave channel calculations [46; 47; 48]. To verify the accuracy of the ray-tracing algorithm, we conducted a historical comparative validation using a study by [PERSON] et al. [49] and selected the ionospheric IRI data model for a region (24.15E 45.8N to 26.15E 45.8N) at 10:10:00 UTC (Coordinated Universal Time) on 10 October 2012 for ray tracing. Figure 3 shows the ray trajectory at an elevation angle of 60\({}^{\circ}\)-70\({}^{\circ}\) and a frequency of 6 MHz. Figure 3a shows the ray-tracing trajectory results from reference [23], while Figure 3b shows the ray-tracing trajectory results from our algorithm, which are consistent with previous studies and indirectly demonstrate the effectiveness of our method. The 2D ray-tracing algorithm has been widely applied in shortwave communication studies [49; 50] and has shown strong agreement with empirical, measured data [51; 52]. One study [53] explicitly states the reliability of 2D ray tracing: \"The 2-D numerical ray tracing was considered adequate for this work as OTHR modeling done by [PERSON] et al.