Low earth orbit (LEO) satellite communication systems have gained increasing attention as a crucial supplement to terrestrial wireless networks due to their extensive coverage area. This letter presents a novel system design for LEO satellite systems by leveraging stacked intelligent metasurface (SIM) technology. Specifically, the lightweight and energy-efficient SIM is mounted on a satellite to achieve multiuser beamforming directly in the electromagnetic wave domain, which substantially reduces the processing delay and computational load of the satellite compared to the traditional digital beamforming scheme. To overcome the challenges of obtaining instantaneous channel state information (CSI) at the transmitter and maximize the system's performance, a joint power allocation and SIM phase shift optimization problem for maximizing the ergodic sum rate is formulated based on statistical CSI, and an alternating optimization (AO) algorithm is customized to solve it efficiently. Additionally, a user grouping method based on channel correlation and an antenna selection algorithm are proposed to further improve the system performance. Simulation results demonstrate the effectiveness of the proposed SIM-based LEO satellite system design and statistical CSI-based AO algorithm.

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