Journal of Optimization Theory and Applications

, Volume 157, Issue 3, pp 866–887

Periodic Image Trajectories in Earth–Moon Space


DOI: 10.1007/s10957-012-0220-5

Cite this article as:
Pontani, M. & Miele, A. J Optim Theory Appl (2013) 157: 866. doi:10.1007/s10957-012-0220-5


The problem of identifying orbits that enclose both the Earth and the Moon in a predictable way has theoretical relevance as well as practical implications. In the context of the restricted three-body problem with primaries in circular orbits, periodic trajectories exist and have the property that a third body (e.g. a spacecraft) can describe them indefinitely. Several approaches have been employed in the past for the purpose of identifying similar orbits. In this work the theorem of image trajectories, proven five decades ago, is employed for determining periodic image trajectories in Earth–Moon space. These trajectories exhibit two fundamental features: (i) counterclockwise departure from a perigee on the far side of the Earth, and (ii) counterclockwise arrival to a periselenum on the far side of the Moon. An extensive, systematic numerical search is performed, with the use of a modified Poincaré map, in conjunction with a numerical refinement process, and leads to a variety of periodic orbits, with various interesting features for possible future lunar missions.


Image trajectoriesPeriodic orbitsModified Poincaré mapsRestricted three-body problemLunar missions

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.University of Rome “La Sapienza”RomeItaly
  2. 2.Rice UniversityHoustonUSA