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### Abstract

Satellite constellations are families of orbits selected to provide useful coverage patterns for telecommunications, Earth observation and navigation services. Such constellations are often assembled from families of circular orbits, which ensures a uniform spacing between satellites in each circular ring. However, there is a large class of elliptical orbits which are of practical interest including Molniya-like orbits and so-called Magic orbits [1,2]. Constellations of satellites using such elliptical orbits will then exhibit a time varying spacing between satellites as the orbital angular velocity experienced by each satellites varies around the elliptical ring.

While current constellations use relatively modest numbers of satellites, future microspacecraft [3] or ‘smart dust’ type devices [4,5] may enable constellations with extremely large numbers of nodes. In this Note a continuum approach is used to model the dynamics of such constellations. A continuity equation is formed to describe the evolution of the number density of nodes as a function of both true anomaly and time. For small eccentricities, the continuity equation can be solved analytically to provide closed-form solutions which describe the evolution of the constellation for some initial distribution of nodes. The closed-form solutions can then be used to investigate pattern formation in elliptical rings. Wave-like patterns are found which circulate around the elliptical ring, with peaks in density which can in principle be used to provide enhanced coverage. A similar continuum approach with a continuity equation has been used in previous studies to develop closed-form solutions which model the time evolution of the radial distribution of constellations of microspacecraft under the action of air drag [6,7].

While current constellations use relatively modest numbers of satellites, future microspacecraft [3] or ‘smart dust’ type devices [4,5] may enable constellations with extremely large numbers of nodes. In this Note a continuum approach is used to model the dynamics of such constellations. A continuity equation is formed to describe the evolution of the number density of nodes as a function of both true anomaly and time. For small eccentricities, the continuity equation can be solved analytically to provide closed-form solutions which describe the evolution of the constellation for some initial distribution of nodes. The closed-form solutions can then be used to investigate pattern formation in elliptical rings. Wave-like patterns are found which circulate around the elliptical ring, with peaks in density which can in principle be used to provide enhanced coverage. A similar continuum approach with a continuity equation has been used in previous studies to develop closed-form solutions which model the time evolution of the radial distribution of constellations of microspacecraft under the action of air drag [6,7].

Original language | English |
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Pages (from-to) | 1767-1771 |

Journal | Journal of Guidance, Control and Dynamics |

Volume | 36 |

Issue number | 6 |

Early online date | 15 Oct 2013 |

DOIs | |

Publication status | Published - Nov 2013 |

### Keywords

- satellite constellations
- pattern formation
- elliptical orbit
- micro spacecraft

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## Projects

- 1 Finished

## VISIONSPACE - VISIONARY SPACE SYSTEMS: ORBITAL DYNAMICS AT EXTREMES OF SPACECRAFT LENGTH SCALE (ERC ADVANCED GRANT)

McInnes, C.

European Commission - FP7 - European Research Council

1/02/09 → 30/09/14

Project: Research

## Cite this

McInnes, C., & Colombo, C. (2013). Wave-like patterns in an elliptical satellite ring.

*Journal of Guidance, Control and Dynamics*,*36*(6), 1767-1771. https://doi.org/10.2514/1.55956