## 1 Introduction  
The demand for high-throughput satellite communication (SatCom) and enhanced coverage for areas inaccessible to terrestrial and nonterrestrial networks has been increasing rapidly, driven mainly by the Internet of Things and upcoming 6G applications, such as drones, vehicles, and terrestrial/nonterrestrial sensors [1; 2; 3]. Accordingly, reconfigurable intelligent surfaces (RISs) have been investigated to support this growing demand for SatCom systems [4; 5; 6; 7]. RISs intelligently manipulate the amplitude, phase, and frequency of the incident signals and reflect them to a target destination using passive reflecting elements [8; 9; 10]. In particular, an RIS provides the capability to control the frequency of the incident electromagnetic wave, allowing a controlled frequency shift to the incident RF signal [11]. As RISs are composed of massive reflecting elements, typically constructed from passive components like inductors, capacitors, and resistors, they have the capability to control the phase of the incident signal without requiring additional energy consumption. In this regard, the utilization of an unmanned aerial vehicle (UAV) equipped with RIS has garnered considerable attention in recent years because most UAVs are battery-limited and must be operated energy-efficiently to be power-sustainable in the network [12; 13]. A UAV equipped with an RIS can complement existing SatCom systems and enhance the network capacity by covering inaccessible terrestrial/nonterrestrial areas and providing additional capacity to satisfy the steep increase in data requirements [14]. Prior studies [15; 16; 17] have investigated the challenges of SatCom systems in which a UAV operates as a mobile relay between the satellite and users. In [15], a relay structure for a UAV between the satellites and the ground is examined. A blind beam-tracking approach for SatCom systems was proposed in [16] when a UAV is equipped with a large-scale antenna array and operates as a relay between the satellite and terrestrial networks. Another study [17] analyzed the performance of SatCom systems in which a UAV is equipped with a multi-antenna array and operates as a moving decode-and-forward relay. However, in [15; 16; 17], a UAV was considered as a mobile active relay that consumes additional energy due to the weight of active elements and signal processing, which is an impractical assumption because UAVs are battery-limited and must be operated energy-efficiently. Also, the authors in [7; 18] analyzed the performance of SatCom systems with a UAV equipped with an RIS. However, these works focus on a scenario where all users share the same frequency and consider only the unicast SatCom system. In contrast, this paper examines a scenario where the earth station (ES) allocates different frequencies to each user and analyzes the performance of both unicast and multicast systems. Meanwhile, in [4], the authors proposed a joint optimization problem that considers the beamforming weights at the satellite and the phase shift at the RIS. In this work, the authors proposed an alternative optimization scheme because the optimized beamforming weight vectors at the satellite and BS and phase shifters at the RIS are coupled. In [5], the authors focused on designing an RIS-assisted SatCom system for Internet of Things (IoT) networks. The primary objective of their study was to improve energy efficiency within the RIS-assisted SatCom system. Furthermore, in [6], the authors tackled the challenge of maximizing the channel capacity in an RIS-assisted SatCom system. They formulated a joint optimization problem that involves power allocation and RIS phase shift to achieve this goal and proposed a sub-optimal algorithm because the problem was formulated as nonconvex due to coupled variables.