<p>The low Earth orbit (LEO) satellite-terrestrial covert communication is one of the most promising research areas to provide security for seamless global communication. In this paper, we consider downlink LEO satellite-terrestrial covert communication networks, where the interference nodes follow a Poisson point process distributed in the terrestrial plane and the space-terrestrial fading is modeled by the Shadowed-Rician fading. The main concern is how to utilize the interference uncertainty and channel uncertainty to hide the communication behavior, thereby avoiding the detection of Willie and ensuring the satellite-terrestrial communication reliability in three distinct network models: (i) the satellite located at a fixed position, (ii) the satellite moving along its orbit, and (iii) the satellite randomly distributed in its visible orbit. We derive the analytical expressions for false alarm, missed detection, and connection probability for the three network models. To obtain the covert throughput, which comprehensively accounts for the covertness and reliability, we formulate an optimization problem and propose a one-dimensional search method to solve an equivalent problem. Finally, simulations validate the accuracy of the derived theoretical expressions and demonstrate the impact of various parameters on system performance.</p>