Analytical expression for distant retrograde orbits around a small natural satellite

An analytical expression for distant retrograde orbits (DROs) is obtained in this study. Owing to the fact that a planar DRO is a closed orbit and can be expressed as an approximately elliptical orbit, respective geometries and periods of DROs are analytically calculated. A switching point, where various properties of planar DROs change abruptly with an increase in the orbital radius, is determined. The Mars–Deimos system is taken as a case study in this work. The proposed method can be applied to cases where the Hill’s approximation of the restricted three-body problem is valid. Numerical calculations are performed to validate the proposed method.