  Quantifying long-term statistical properties of satellite trajectories
typically entails time-consuming trajectory propagation. We present a fast,
ergodic\cite{Arnold} method of analytically estimating these for
$J_2-$perturbed elliptical orbits, broadly agreeing with trajectory
propagation-derived results. We extend the approach in Graven and Lo (2019) to
estimate: (1) Satellite-ground station coverage with limited satellite field of
view and ground station elevation angle with numerically optimized formulae,
and (2) long-term averages of general functions of satellite position. This
method is fast enough to facilitate real-time, interactive tools for satellite
constellation and network design, with an approximate $1000\times$ GPU speedup.
