The coverage of elliptical orbits using ergodic theory

One of the key performance metrics for satellite constellations is the statistics of the visibility periods between the satellites and points on the ground. Associated with this are other desirable communications statistics such as data through-put, link qualities, etc. Typically, the computation of coverage statistics requires the propagation of the trajectories. For orbits with non-repeating ground tracks, this may require orbit propagation for tens of years per spacecraft. Lo (1994) proposed an approach using ergodic theory which replaced the need to compute the statistics from integrated trajectories by a definite integral over the circular region of the elevation mask of a point on the ground. The effects of J2 due to the non-spherical shape of the Earth are included in the definite integral. The definite integral can be implemented in Excel for quick trade studies. But the simple geometric methods used to derive the integral for circular orbits cannot be readily extended to elliptical orbits. In This work a new algorithm using differential geometry enables us to extend this theory to elliptical orbits.