Analytical theory for spacecraft motion about Mercury

Abstract

In the framework of the elliptic restricted three-body problem we develop an analytical theory for spacecraft motion close to Mercury. Besides the perturbations due to the gravity of the Sun and Mercury and the eccentricity of Mercury's orbit around the Sun, i.e., the elliptic restricted three-body problem, the theory includes the effects of the oblateness and the possible latitudinal asymmetry of Mercury, and is valid for any eccentricity of the spacecraft's orbit. The initial Hamiltonian defines a non-autonomous but periodic dynamical system of two degrees of freedom. The mean motion of the spacecraft and the time are averaged using two successive Lie–Deprit transformations. The resulting Hamiltonian defines a one degree of freedom system and depends upon three essential parameters. When the latitudinal asymmetry coefficient vanishes the flow of this system is entirely analyzed through the discussion of the occurrence of its (relative) equilibria and bifurcations in accordance with the parameters the problem depends upon. Frozen orbits of the initial system together with their stability are obtained related to the relative equilibria. If the latitudinal asymmetry of Mercury is taken into account, the equatorial symmetry of the problem is broken and introduces important changes in the dynamics. A variety of tests show a very good agreement between averaged and non-averaged models, and the reliability of the theory is further checked by performing long-term integrations in ephemeris.