Wavelike patterns in precessing elliptical rings for swarming systems

Acontinuum model for a swarm of devices is investigated with the devices moving along precessing elliptical Earthcentered orbits. Wavelike patterns in these precessing elliptical rings with peaks in swarm density are found that can be used to provide enhanced coverage for Earth observation and space science. Two orbital models are considered for the purpose of comparison: perturbed by J2 and solar radiation pressure, and perturbed by J2 and J3, respectively, each with a different frozen eccentricity. By removing osculating orbital elements, only the long-period orbit eccentricity and argument of perigee are chosen to derive closed-form solutions to the continuummodel for the swarm density. Zero-density lines in the swarm density are found, as well as infinite density at certain boundaries. Comparison between the analytic and numerical number density evolutions is made to yield the range of applicable eccentricity based on the maximum error tolerance, as well as the minimum number of swarm members required to approximate continuous evolution. Closed-form solutions are then derived to predict the number density of swarm devices for magnetic-tail measurement and Earth-observation applications.