arXiv:2607.11620v2 Announce Type: replace Abstract: This paper investigates uplink coverage probability, i.e., the SIR distribution, and SIR order statistics at a low Earth orbit satellite base station (LEO BS) with narrow Gaussian antenna beams. The user equipments (UEs) connected to the satellite radio access network are distributed according to a Poisson point process (PPP) on the Earth's surface. We propose a novel framework for stochastic geometry analysis of the SIR using a Gaussian mixture shadowing model, derive tractable expressions for the mean, variance, and probability density function, and analyze the SIR distributions for the three strongest UEs under successive interference cancellation (SIC). Furthermore, we propose a gamma distribution model for the interference distribution. We show that SIC enables each LEO BS to serve multiple UEs per beam cell, improving user fairness while preserving the network throughput. The observations imply design recommendations for next-generation non-orthogonal multiple access (NOMA) LEO networks. The analytical insights are validated against Monte Carlo simulations.