The increasing quality of service requirements and demand for satellite services necessitate higher data rates, spectral efficiency, and stability in satellite communication networks. Therefore, this paper investigates the non-orthogonal multiple access (NOMA) assisted satellite communication networks, where multiple users are uniformly distributed over a spherical hat according to the homogeneous Poisson point process (HPPP). Based on the characteristics of HPPP, we analyze the distance distributions of users. To evaluate the performance of the proposed networks, we first derive the closed-form expressions and approximated expressions of the outage probability (OP) for paired NOMA users. To obtain more insights into the proposed networks, the ergodic rate and diversity orders for paired NOMA users are also derived. Spectral efficiency is derived for NOMA and orthogonal multiple access (OMA) assisted satellite communication networks. Our analytical results demonstrate that the diversity order of the proposed networks is all one. Numerical results confirm that: 1) compared to OMA, the proposed NOMA-assisted satellite network demonstrates superior outage performance and spectral efficiency, especially in the higher power regimes; 2) fading factors have negligible effects on OP and spectral efficiency; and 3) under certain target rates for both near and far users, the outage performance of far users in NOMA-assisted satellite communication networks is superior to that of near users.