## 2 Optical Uplink Channel Model
### Closed-Form Expression for the Channel Model  
The comprehensive channel impairments in deep space optical uplinks encompass turbulence-induced beam scintillation (atmospheric \(h_{t}\), coronal \(h_{s}\)), pointing errors \(h_{p}\), AOA fluctuations \(h_{a}\), link attenuation \(h_{c}\), and background noise \(P_{b}\). While these parameters are critical for channel modeling, their synergistic impacts remain scarcely addressed in the existing literature. To bridge this gap, we present a novel closed-form analytical model (Equation (2)) that integrates all degradation mechanisms. The methodological framework for deriving the PDF of the composite channel coefficient \(h\) is as follows. First, let \(h_{2}=h_{t}h_{s}\) represent the beam scintillation caused by atmospheric turbulence and coronal turbulence, the cumulative distribution function (CDF) of \(h_{2}\) is  
\[F_{h_{2}}(h_{2})=\int_{0}^{\infty}\int_{0}^{\infty}f_{g}(h_{t})f_{g}(h_{s})dh_ {t}dh_{s}. \tag{12}\]  
The PDF of \(h_{2}\) is written as \(f_{h_{2}}(h_{2})\), which is the derivative of the CDF of \(h_{2}\). Hence, with \(h_{t}=h_{2}/h_{s}\), \(f_{h_{2}}(h_{2})\) is derived as  
\[f_{h_{2}}(h_{2})=\frac{\partial}{\partial h_{2}}\int_{0}^{\infty}\int_{0}^{ \infty}f_{g}\left(\frac{h_{2}}{h_{s}}\right)f_{g}(h_{s})d\frac{h_{2}}{h_{s}}dh_ {s}=\int_{0}^{\infty}\frac{1}{h_{s}}f_{g}\left(\frac{h_{2}}{h_{s}}\right)f_{g}( h_{s})dh_{s}. \tag{13}\]  
According to (3), (13) is rewritten as  
\[f_{h_{2}}(h_{2})=\frac{4(\alpha_{t}\beta_{t})\frac{\alpha_{x}+\beta_{t}}{2}( \alpha_{s}\beta_{s})\frac{\alpha_{y}+\beta_{t}}{2}\frac{\alpha_{z}+\beta_{z}}{ 2}-1}{\Gamma(\alpha_{t})\Gamma(\beta_{t})\Gamma(\alpha_{s})\Gamma(\beta_{s})} \int_{0}^{\infty}\frac{\mathrm{K}_{\alpha_{t}-\beta_{t}}\left(2\sqrt{\alpha_ {t}\beta_{t}h_{2}/h_{s}}\right)\mathrm{K}_{\alpha_{t}-\beta_{s}}\left(2\sqrt{ \alpha_{s}\beta_{s}h_{s}}\right)}{h_{s}^{1-\frac{\alpha_{z}+\beta_{z}-\beta_{z }-\beta_{z}}{2}}}dh_{s}, \tag{14}\]  
\[\text{where }K_{\
u}(x)=\frac{1}{2}\hat{\mathsf{G}}_{0,2}^{2,0}\left[\frac{x^{2}}{4} \right]_{\
u/2,-\
u/2}\right]\text{, and }\hat{\mathsf{G}}_{P,d}^{m,n}[\cdot]\text{ is the Meijer's G function. Employing (07.34.21.0011.01) in [47], the result of (14) is  
\[f_{h_{2}}(h_{2})=\frac{\alpha_{t}\beta_{t}\alpha_{s}\beta_{s}}{h_{2}\Gamma( \alpha_{t})\Gamma(\beta_{t})\Gamma(\alpha_{s})\Gamma(\beta_{s})}\hat{\mathsf{G }}_{0,4}^{4,0}\left[\alpha_{t}\beta_{t}\alpha_{s}\beta_{s}h_{2}\right]_{\alpha _{t}-1,\beta_{t}-1,\alpha_{s}-1,\beta_{s}-1}\right]. \tag{15}\]  
The specific derivation can be found in Appendix A.2. Next, let \(h_{3}=h_{2}h_{p}\) represent the joint effect of turbulence-induced scintillation and pointing errors. Combining the PDF expressions in (6) and (15), the PDF of \(h_{3}\) is derived as  
\[f_{h_{3}}(h_{3})=\frac{\mathrm{C}_{\alpha\beta}\mathrm{C}_{P1}\mathrm{C}_{P2}^ {2}}{\Gamma(\alpha_{t})\Gamma(\beta_{t})\Gamma(\alpha_{s})\Gamma(\beta_{s})} \hat{\mathsf{G}}_{1,5}^{5,0}\left[\mathrm{C}_{\alpha\beta}\mathrm{C}_{P1}h_{3} \right] \begin{matrix}\mathrm{C}_{p2}^{2}\\ \mathrm{C}_{p2}^{2}-1,\alpha_{t}-1,\beta_{t}-1,\alpha_{s}-1,\beta_{s}-1 \end{matrix} \text{,} \tag{16}\]  
\[\text{where }\mathrm{C}_{\alpha\beta}=\alpha_{t}\beta_{t}\alpha_{s}\beta_{s}\text{, }C_{P1}=w_{b}^{2}/2d_{r}^{2}\text{, and }C_{p2}=w_{b}/2\sigma_{p}.\]  
Finally, we take into account AOA fluctuations and link attenuation, i.e., \(h=h_{3}h_{a}h_{c}\), where \(h_{a}\) is modeled as (7), and \(h_{c}\) is a constant when link length, divergence angle, and weather condition are given. Therefore, the PDF of \(h\) is derived as \[f_{h}(h) =\frac{C_{\alpha\beta}C_{p1}C_{p2}^{2}}{h_{c}\Gamma(\alpha_{t}) \Gamma(\beta_{t})\Gamma(\alpha_{c})\Gamma(\beta_{c})}\left(1-\exp\left(-\frac{ \Omega_{F}^{2}}{2\sigma_{\theta}^{2}}\right)\right)\hat{C}_{1,5}^{5,0}\left[ \frac{C_{\alpha\beta}C_{p1}}{h_{c}}h\right| \begin{array}{c}C_{p2}^{2}\\ C_{p2}^{2}-1,\alpha_{t}-1,\beta_{t}-1,\alpha_{c}-1,\beta_{c}-1\end{array}