Mars-Phobos/Deimos libration points revisited

The study of libration-point location in the Mars-Phobos and Mars-Deimos three-body systems is a relatively new endeavour. Although the study of libration points is centuries old, calculating the locations of libration points and their dynamical substitutes commenced only after advances in perturbation theories have been made. Thus far, most of the studies on libration-point locations were mainly theoretical, usually without treating a specific planetary system. In particular, only a handful of studies included calculations of the libration-point location in a realistic Martian dynamical environment; all of them focused only on the Mars-Phobos system. In this paper, the locations of the L1 and L2 libration points in the vicinity of Phobos and Deimos are calculated, while considering the oblateness of Mars, the non-spherical gravity fields of Phobos and Deimos, the orbital eccentricity of Phobos, and the gravitational perturbation due to the Sun. The averaged effect of the Sun on the libration-point locations is also studied. The resulting libration-point locations differ from their classical values by up to 0.88 km in the case of Phobos and 0.45 km in the case of Deimos. It is shown that the displacement of libration points L1 and L2 in the vicinity of Deimos is mostly in the xz plane of the rotating frame, and that the libration-point motion under the gravitational perturbation of the Sun creates a three-dimensional figure-eight-shaped curve that collapses into an ellipse-like shape every Martian equinox. When the averaged gravitational effect of the Sun is considered, the libration-point locations are calculated for multiple dates along the Martian year.