We focus on stochastic geometry analysis of a low Earth orbit (LEO) narrowband terrestrial-satellite uplink with satellite base stations (SBSs) in a uniform constellation equipped with narrow Gaussian beams. The served and interfering omnidirectional user equipments (UEs) are distributed on the Earth's surface according to a homogeneous Poisson point process (HPPP) with Nakagami faded signals. This study presents a detailed but comprehensive mathematical analysis of several key metrics: the signal-to-interference ratio (SIR), the SIR meta distribution (MD), the signal-to-interference-plusnoise ratio (SINR), and the average throughput. Many results are presented in simple analytical and closed forms containing more insight than the expressions proposed in prior works. The results indicate an optimal UE density depending on the altitude, elevation angle, and the width of the antenna gain, maximizing the average throughput. However, this optimal density leads to a significant variance in the user experience regarding link quality (i.e., the users are not treated fairly).

We analyze a narrow-beam low earth orbit (NB LEO) terrestrial-satellite uplink from the perspective of the typical satellite base station. The user equipment (UEs) are located on the earth's surface according to a homogeneous Poisson point process (PPP). We present a simplified planar system model and focus on the point process of the approximate antenna gains of each UE in the satellite base station's antenna pattern and analyze the signal-to-interference ratio (SIR), the SIR meta distribution (MD), the signal-to-interference-plus-noise ratio (SINR) and the throughput. We present simple analytical results, some in closed form, for the performance metrics. We validate the approximate system model by comparing the results to the Monte Carlo simulated performance metrics using the more accurate system model incorporating the earth's curvature. The paper includes insightful results that help to understand the quantitative and qualitative nature of the NB LEO satellite uplink.