## 4 Numerical Results  
In this section, the numerical simulation results are given to investigate and analyze the impact of system parameters. Moreover, the SR fading channels parameters are shown in Table 1[37]. The parameters of the considered system are given in Table 2[40]. Figure 2 depicts \(\mu\) versus \(\lambda\) for different \(P_{C}\). It is obvious that the \(\mu\) can be minimized by optimizing \(\lambda\). Combined with Equations (21) and (22), it is easy to know that \((1-\varepsilon)FA(\lambda)\) is a minus function of \(\lambda\), and \(\varepsilon MD(\lambda)\) is an incremental function with respect to \(\lambda\). Moreover, \(\mu\) is a continuous function of \(\lambda\). Therefore, the optimal solution of \(\lambda\) named \(\lambda_{opt}\) belongs to \([\beta,\alpha]\) which has been verified by the simulation results in this figure. Finally, we can have \(\mu_{opt}=\varepsilon MD\big{(}\lambda_{opt}\big{)}+(1-\varepsilon)FA\big{(} \lambda_{opt}\big{)}\). Figure 3 shows \(\mu_{opt}\) versus \(P_{C}\) for the different number of relays. In [21], the authors gave the covert constraint, namely, \(\mu_{opt}\geq\min\{\varepsilon,1-\varepsilon\}-\delta\), where \(\delta\geq 0\). Therefore, the maximum value of \(\mu_{opt}\) is \(\rho\), which equals 0.5. Moreover, it is obvious that as the power used for covert communication \(P_{C}\) gradually increases, the value of \(\mu_{opt}\) decreases. It is because the enhanced power radiation used for covert communication is helpful for the detection of the satellite. All in all, the value of \(P_{C}\) is determined by the covert constraint. Finally, we can also see that the increased number of relays leads to a decrease in \(\mu_{opt}\), which indicates that the gain generated by the relay selection is beneficial in eliminating the uncertainty of detection at \(S\). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Shadowing** & \(m_{j}\) & \(b_{j}\) & \(\Omega_{j}\) \\ \hline Frequent heavy & \(1\) & \(0.063\) & \(0.0007\) \\ shadowing (FHS) & \(5\) & \(0.251\) & \(0.279\) \\ \hline Infrequent light & \(10\) & \(0.158\) & \(1.29\) \\ shadowing (ILS) & \(10\) & \(0.158\) & \(1.29\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Channel parameters. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Satellite Orbit & GEO \\ \hline Carrier Frequency & \(f=18\) GHz \\ \hline \(3\) dB angle & \(\hat{\theta}=0.4^{\circ}\) \\ \hline Maximal Beam Gain & \(G_{MS}=48\) dB \\ \hline Maximal Receive Antenna Gain & \(G_{MR}=4\) dB \\ \hline \hline \end{tabular}
\end{table} 
Table 2: System parameters. Figure 2: The EDP versus \(\lambda\) for the different \(P_{C}\). Figure 4 plots ACR versus \(P_{C}\) for the different number of relays. It is shown that there exists a value of \(P_{C}\) that maximizes ACR, namely, ACR can be optimized by adjusting \(P_{C}\). Furthermore, it is evident that as the fading of the satellite-terrestrial relay links worsens, it inevitably leads to a decrease in ACR because the degree of fading determines the proportion of on and off time for the overall system communication link. Finally, we can also see that an increase in the number of relays contributes to the rise in ACR, implying a diversity gain in relay selection. Figure 5 depicts the ACR versus \(r_{SU}\) for the different number of relays. First, it is easy to see that the increase in \(r_{SU}\) is detrimental to the covert communication because the enhancement in the decoding threshold increases the difficulty of keeping the communication link connected. In addition, similar to the previous conclusion, the rise in the number of relays favors the increase in the rate of covert communication, which is evident when \(r_{SU}\) is small. Finally, as in the previous analysis, the ACR decreases when the satellite-terrestrial  
Figure 4: The ACR versus \(P_{C}\) for the different number of relays. Figure 3: The optimal EDP versus \(P_{C}\) for the different number of relays. -relay links' fading deteriorates.