## 2 System and Theoretical Model
### Hybrid RF/FSO System Model  
Figure 2 demonstrates the asymmetric-based RF/FSO deep space communication system during superior conjunction. We adopt the dual-hop relay communication technology, where relay satellites orbiting the Earth first receive signals from Earth stations through RF and then transmit them via FSO.  
The RF link in the hybrid system obeys the Rayleigh distribution, and its probability density function (PDF) is expressed as:  
\[f_{\gamma_{1}}(\gamma_{1})=\frac{1}{\overline{\gamma_{1}}}\exp\biggl{(}-\frac{ \gamma_{1}}{\overline{\gamma_{1}}}\biggr{)}, \tag{8}\]  
where \(\gamma_{1}\) represents the instantaneous signal-to-noise ratio (SNR) of the RF link, and \(\overline{\gamma_{1}}\) represents the average SNR of the RF channel. Therefore, the CDF subject to Rayleigh distribution can be written as:  
\[F_{1}(\gamma_{1})=1-\exp\biggl{(}-\frac{\gamma_{1}}{\overline{\gamma_{1}}} \biggr{)}. \tag{9}\]  
On the other hand, in order to describe the weak to strong turbulence more accurately, we use the Gamma-Gamma distribution to describe the FSO channel fading while considering the pointing error. Thus, the PDF of the SNR is expressed as [11]:  
\[f_{\gamma_{2}}(\gamma_{2})=\frac{\xi^{2}}{2\gamma_{2}\Gamma(\varphi)\Gamma( \phi)}G_{1,3}^{3,0}\biggl{[}\varphi\phi\sqrt{\frac{\gamma_{2}}{\overline{ \gamma_{2}}}}\biggr{]}\begin{array}{c}\xi^{2}+1\\ \xi^{2},\varphi,\phi\end{array}\biggr{]}, \tag{10}\]  
where \(\gamma_{2}\) represents the instantaneous SNR of the FSO link and \(\overline{\gamma_{2}}\) represents the average SNR of the FSO link. \(\xi=\frac{\omega_{e}}{2\
u_{e}}\) is a pointing error parameter, which denotes the ratio of the equivalent beam waist radius and jitter standard deviation at the receiving end. \(G(\cdot)\) is the Meijer's G function as defined in [30]. \(\varphi\) and \(\phi\) are the large-scale scattering coefficient and the small-scale scattering coefficient, respectively, which are related to the beam type. For spherical waves, they can be expressed as [31]:  
Figure 2: Diagram of deep space RF/FSO communication system.  
\[\varphi =\left[\exp\left(\frac{0.49\sigma_{l}{}^{2}}{\left(1+0.18d^{2}+0.56 \sigma_{l}{}^{12/5}\right)^{7/6}}\right)-1\right]^{-1}, \tag{11}\] \[\phi =\left[\exp\left(\frac{0.51\sigma_{l}{}^{2}\left(1+0.69\sigma_{l}{ }^{12/5}\right)^{-5/6}}{\left(1+0.9d^{2}+0.62d^{2}\sigma_{l}{}^{12/5}\right)^{5 /6}}\right)-1\right]^{-1},\]  
where \(\sigma_{l}{}^{2}\) is the logarithmic intensity fluctuation variance, \(d=\left(k_{n}D_{m}{}^{2}/4L_{link}\right)^{1/2}\), and \(D_{m}\) is the receiver aperture. [PERSON] et al. proved that the variance of logarithmic intensity fluctuations and the variance of logarithmic amplitude fluctuations under weak turbulence are as follows [32]:  
\[\sigma_{l}{}^{2}=4\Big{\langle}\chi^{2}\Big{\rangle}. \tag{12}\]  
Therefore, the scintillation index for the Gamma-Gamma channel model is further written as:  
\[m^{2}=\frac{1}{\varphi}+\frac{1}{\phi}+\frac{1}{\varphi\phi}. \tag{13}\]  
Considering the complexity of the system, a fixed-gain amplify-and-forward (AF) relay technology is adopted, while an intensity modulation/direct detection (IM/DD) technique is used at the receiving end. Therefore, the end-to-end SNR can be expressed as \(\gamma=\frac{\gamma_{12}}{2+C}\) (Appendix A), where \(C\) is the fixed relay gain.  
The CDF of the hybrid RF/FSO system is expressed as [33]:  
\[F_{\gamma}(\gamma)=1-A_{1}\sqrt{\gamma}\exp\left(-\frac{\gamma}{\gamma_{1}} \right)\times C_{1,6}^{6,0}\left[B\gamma\left|\begin{array}{c}\frac{\xi_{2} ^{2}+1}{2}\\ \sigma_{1}\end{array}\right.\right], \tag{14}\]  
where \(A_{1}=\frac{2\left(\varphi+\varphi\right)\varphi\xi^{2}\sqrt{C}}{32\pi\sqrt{ \gamma_{1}}\sqrt{\gamma_{2}}\left(\varphi\right)\left(\varphi\right)}\), \(B=\frac{(\varphi\phi)^{2}C}{16\gamma_{1}\gamma_{2}}\), \(o_{1}=\frac{\xi^{2}-1}{2}\), \(\frac{\varphi-1}{2}\), \(\frac{\varphi}{2}\), \(\frac{\varphi-1}{2}\), \(\frac{\varphi}{2}\), \(-\frac{1}{2}\).