## 1 Introduction
However, those studies [4; 5; 6] did not fully address the practical limitations associated with estimating individual channels between the satellite and the RIS, as well as between the RIS and the UE, which remain significant challenges in practical SatCom scenarios. Therefore, it is essential to analyze the performance bounds achievable when utilizing only the practically available information in an RIS-assisted SatCom system. The main contribution of this work lies in conducting an asymptotic analysis of the data rate within a SatCom system assisted by RIS. Also, the asymptotic analysis takes into consideration the practical RIS environment and its limitations. To address the aforementioned limitations, we present an asymptotic analysis of an RIS-assisted SatCom system that employs a UAV equipped with an RIS. In this system, the UAV acts as a mobile reflector between a satellite and users, while relying solely on the practically available information. In particular, we first derive the asymptotic upper bound of the achievable rate and propose a passive beamformer that can asymptotically achieve the optimality by considering the presence of numerous elements on the RIS. We then derive the closed-form expressions for the ergodic achievable rate and outage probability using the proposed passive beamforming technique. Our analyses can accurately determine the performance of the SatCom system without requiring extensive simulations. Leveraging the derived asymptotic results, we extend our results to cover an additional scenario, such as an RIS-assisted SatCom system that enables a UAV to serve multiple user equipments (UEs) simultaneously. From the simulation results, we can observe that the proposed scheme asymptotically achieves optimal performance and also achieves its upper bound, verifying its asymptotic optimality. The remainder of this paper is organized as follows. The system model is described in Section 2. The asymptotic performance and outage probability are discussed in Section 3, and the performance of a SatCom system with an RIS serving multiple UEs is analyzed in Section 4. The simulation results are presented in Section 5 and the concluding remarks are presented in Section 6. Notations: Throughout the paper, matrices are represented by boldface uppercase symbols, while vectors are denoted by boldface lowercase symbols, and \(\mathbf{I}_{N}\) and \(\mathbf{1}_{M\times N}\) are, respectively, a size-\(N\) identity matrix and an \(M\times N\) matrix of ones. The transpose, Hermitian transpose, and conjugate operators are \(\left(\cdot\right)^{\mathsf{T}}\left(\cdot\right)^{\mathsf{H}}\), and \(\left(\cdot\right)^{*}\), respectively. The operator \(\operatorname{diag}(x_{1},\cdots,x_{n})\) is a diagonal operator that constructs a diagonal matrix from a vector, \([x_{1},\cdots,x_{n}]^{\mathsf{T}}\). The phase and amplitude of a complex number, \(y\), are denoted by \(\angle y\) and \(\left|y\right|\), respectively, and \(\bmod\) is the modulo operation. \(\mathcal{O}(\cdot)\) is the big O notation, and \(\operatorname{E}[\cdot]\) and \(\operatorname{Var}[\cdot]\) denote, respectively, expectation and variance operators. \(\mathcal{CN}\left(m,\sigma^{2}\right)\) denotes a complex Gaussian distribution with mean \(m\) and variance \(\sigma^{2}\), and \(\mathcal{LN}\left(m,\sigma^{2}\right)\) and \(\mathcal{L}^{2}\mathcal{N}\left(m,\sigma^{2}\right)\) denote a lognormal distribution and double-lognormal distribution, respectively.