  This study investigates the problem of areostationary orbits around Mars in
the three-dimensional space. Areostationary orbits are expected to be used to
establish a future telecommunication network for the exploration of Mars.
However, no artificial satellites have been placed in these orbits thus far. In
this paper, the characteristics of the Martian gravity field are presented, and
areostationary points and their linear stability are calculated. By taking
linearized solutions in the planar case as the initial guesses and utilizing
the Levenberg-Marquardt method, families of periodic orbits around
areostationary points are shown to exist. Short-period orbits and long-period
orbits are found around linearly stable areostationary points, and only
short-period orbits are found around unstable areostationary points. Vertical
periodic orbits around both linearly stable and unstable areostationary points
are also examined. Satellites in these periodic orbits could depart from
areostationary points by a few degrees in longitude, which would facilitate
observation of the Martian topography. Based on the eigenvalues of the
monodromy matrix, the evolution of the stability index of periodic orbits is
determined. Finally, heteroclinic orbits connecting the two unstable
areostationary points are found, providing the possibility for orbital transfer
with minimal energy consumption.
