  Recently, stochastic geometry has been applied to provide tractable
performance analysis for low earth orbit (LEO) satellite networks. However,
existing works mainly focus on analyzing the ``coverage probability'', which
provides limited information. To provide more insights, this paper provides a
more fine grained analysis on LEO satellite networks modeled by a homogeneous
Poisson point process (HPPP). Specifically, the distribution and moments of the
conditional coverage probability given the point process are studied. The
developed analytical results can provide characterizations on LEO satellite
networks, which are not available in existing literature, such as ``user
fairness'' and ``what fraction of users can achieve a given transmission
reliability ''. Simulation results are provided to verify the developed
analysis. Numerical results show that, in a dense satellite network,
{\color{black}it is} beneficial to deploy satellites at low altitude, for the
sake of both coverage probability and user fairness.
