## 3 Enhanced Mathematical Model with Atmospheric and Environmental Effects
### Performance Analysis of Atmospheric Attenuation in 5G-Satellite Integrated Networks  
Figure 2 provides key insights into the influence of frequency, altitude, and distance on atmospheric attenuation in 5G-satellite integrated networks. This understanding is crucial for optimizing the network design, particularly in high-frequency 5G bands, where signal loss due to atmospheric effects can be substantial. The plot illustrates that as frequency increases from 24 GHz to 32 GHz, atmospheric attenuation grows considerably, particularly at longer distances. This behaviour is due to the proportional relationship between attenuation coefficients and frequency, given by  
\[\alpha(f)\propto f \tag{19}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value(s)** \\ \hline
**Satellite Altitudes** & 300 km, 500 km, 700 km, 1000 km \\ \hline
**Frequency Bands** & 24 GHz, 28 GHz, 32 GHz \\ \hline
**Bandwidths** & 10 MHz, 20 MHz, 50 MHz \\ \hline
**Distance Range** & 0–3000 km \\ \hline
**Transmitter Height (\(h_{t}\))** & 30 m \\ \hline
**Receiver Heights (\(h_{r}\))** & 1.5 m (Low Rx), 100 m (High Rx) \\ \hline
**Effective Earth Radius (\(R_{e}\))** & 6371 km \\ \hline
**Dielectric Constant (\(e_{r}\))** & 15 \\ \hline
**Noise Power Spectral Density (\(N_{0}\))** & \(-174\) dBm/Hz \\ \hline
**SNR Thresholds** & 5 dB, 10 dB, 15 dB \\ \hline
**Atmospheric Attenuation Factors (\(A_{\text{atm}}\))** & Calculated as \(\alpha_{\text{dry}}(f)+\alpha_{\text{wet}}(f)\) \\ \hline
**Dry Atmospheric Absorption (\(\alpha_{\text{dry}}(f)\))** & Frequency-dependent, per ITU-R P.676 \\ \hline
**Wet Atmospheric Absorption (\(\alpha_{\text{wet}}(f)\))** & Frequency-dependent, per ITU-R P.676 \\ \hline
**Rainfall Intensity (\(R\))** & 0–50 mm/h \\ \hline
**Rain Attenuation Coefficient (\(\gamma_{r}\))** & \(kR^{a}\) (parameters from ITU-R P.838) \\ \hline
**Rician \(K\)-factor** & 3, 6, 10 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation parameters for enhanced 5G–satellite propagation model. Figure 2: Atmospheric attenuation vs. distance for different frequencies and satellite altitudes. This implies that, for a fixed distance \(d\), higher-frequency signals (such as those at 32 GHz) experience greater atmospheric attenuation than lower-frequency signals (e.g., 24 GHz). This trend is clearly visible in the plot, where the attenuation curves for 32 GHz are consistently higher than those for 24 GHz across all altitudes, highlighting the frequency-dependent nature of atmospheric losses. Another critical factor influencing atmospheric attenuation is the altitude of the satellite. The plot includes data for four distinct satellite altitudes: 300 km, 500 km, 700 km, and 1000 km. Lower altitudes are associated with higher attenuation, especially over long distances. This is because lower altitudes imply a denser atmospheric path, increasing the probability of molecular absorption and scattering. For a satellite at altitude \(h\), the effective path length through the atmosphere \(d_{\rm eff}\) reduces as \(h\) increases, leading to the following expression:  
\[A_{\rm atm}\approx\left(\alpha_{\rm dry}(f)+\alpha_{\rm wet}(f)\right)\cdot d _{\rm eff} \tag{20}\]  
where \(d_{\rm eff}\) is shorter for higher altitudes, thereby reducing cumulative attenuation. Consequently, satellites positioned at 1000 km experience lower attenuation compared to those at 300 km, as seen in the plot. Figure 2 further reveals a nearly linear increase in attenuation with distance for a fixed frequency and altitude, which is consistent with the linear dependence on \(d\). However, the combination of high frequency (such as 32 GHz) and low altitude (e.g., 300 km) results in a substantial amplification of attenuation, suggesting an almost multiplicative effect of frequency and altitude on atmospheric losses. At shorter distances (less than 500 km), the variation in attenuation across different altitudes is minimal. However, as the distance extends, the separation between curves becomes more pronounced.