## 3 Influence of Physical Properties of NS on Communication, Navigation, and TT&C Integration
### Atmospheric Loss in NS
#### 3.3.4 Influence of NS Plasma Sheath on Electromagnetic Wave  
The channel environment of NS vehicles is complex and changeable; to determine the type of the channel and design the communication, navigation, and TT&C system, it is necessary to have a qualitative understanding of the basic characteristics of the plasma sheath. When electromagnetic waves penetrate the plasma sheath, they will cause the attenuation and phase shift of the electromagnetic waves. If the plasma sheath (electron density, collision frequency, sheath thickness, etc.) is changing, the effects of attenuation and phase shift will change with time. From the perspective of signal processing, the plasma modulates the amplitude and angle of the electromagnetic signal. Therefore, the impact of the plasma sheath on the electromagnetic waves passing through it is mainly manifested in two main aspects: attenuation and phase shift. The forms of attenuation mainly include the reflection loss of electromagnetic waves passing through the plasma sheath, absorption loss, and distortion in refraction. Moreover, the plasma sheath will also increase the antenna noise temperature. When electromagnetic waves propagate in a stable plasma sheath, the sheath will cause a fixed phase shift of the electromagnetic waves. The attenuation coefficient k and the phase shift coefficient \(\mu\) are shown as follows [79],  
\[\kappa =\frac{\omega}{\sqrt{2c}}\sqrt{\frac{\omega_{p}^{2}}{\omega^{2}+v^{2} }-1+\sqrt{(1-\frac{\omega_{p}^{2}}{\omega^{2}+v^{2}})^{2}+(\frac{v}{\omega} \frac{\omega_{p}^{2}}{\omega^{2}+v^{2}})^{2}}} \tag{12}\] \[\mu =\frac{\omega}{\sqrt{2c}}\sqrt{1-\frac{\omega_{p}^{2}}{\omega^{2}+v^{2}}+ \sqrt{(1-\frac{\omega_{p}^{2}}{\omega^{2}+v^{2}})^{2}+(\frac{v}{\omega}\frac{ \omega_{p}^{2}}{\omega^{2}+v^{2}})^{2}}} \tag{13}\]  
where \(c\) is the speed of light; \(v\) is the flight speed of the aircraft; \(\omega\) = 2\(p\)\(f\) is the carrier angular frequency; \(\omega_{p}\) = 2\(p\)\(f_{p}\) is the angular frequency of the plasma, which depends only on the electron density and is calculated as follows [79],  
\[\omega_{p}=2\pi f_{p}\approx\sqrt{\frac{n_{e}e^{2}}{\varepsilon_{0}m_{e}}} \approx 9000\sqrt{n_{e}} \tag{14}\]  
where \(e\) = 1.60 \(\times\) 10\({}^{-19}\) C is the electron charge; \(m_{e}\) = 1.67 \(\times\) 10\({}^{-27}\) is the electronic mass; \(\varepsilon_{0}\) = 8.85 \(\times\) 10\({}^{-12}\) F/m is the dielectric constant in the vacuum; \(n_{e}\) is the electron density, the plasma can be divided into multiple thin layers along the direction of electric field propagation (z direction) according to the stratified method, the density of \(n_{e}\) in the space non-uniform plasma can be obtained by Gaussian distribution or Epstein distribution, as follows [79; 80; 81]:  
(1) Gaussian distribution  
\[n_{e}=\left\{ \begin{array}{ll}N_{\rm max}e^{[\varepsilon_{0}(z-z_{0})^{2}]} &(z_{1}\leq z\leq z_{0})\\ N_{\rm max}e^{[-\varepsilon_{0}(z-z_{0})^{2}]}&(z_{0}\leq z\leq z_{2})\end{array} \right. \tag{15}\]  
(2) Epstein distribution \[n_{e}=\left\{ \begin{array}{ll}N_{\max}\Big{\{}1+e^{[-\lambda_{01}(z-z_{0})]} \Big{\}}^{-1}&(z_{1}\leq z\leq z_{0})\\ N_{\max}\Big{\{}1+e^{[\lambda_{02}(z-z_{0})]}\Big{\}}^{-1}&(z_{0}\leq z\leq z_{2 })\end{array} \right. \tag{16}\]  
In Equations (15) and (16), \(N_{max}\) is the maximum electron density of the plasma; \(\sigma_{01}\) and \(\sigma_{02}\) are Gaussian distribution constants; \(\lambda_{01}\) and \(\lambda_{02}\) are Epstein distribution constants. \(z\) is the vertical distance from the measuring position to the surface of the aircraft; \(z_{0}\) is the piecewise point of Gaussian function and Epstein function. \(z_{2}-z_{1}\) is the thickness of the plasma. Figure 8 shows the simulation results of the attenuation coefficient, phase shift coefficient of the plasma sheath, and the electron density distribution model of the spatially inhomogeneous plasma, where the flight speed of the aircraft is 3 Mach.