Suppressing chaotic oscillations of a tether anchored to the Phobos surface under the L1 libration point

The paper deals with the problem of the chaotic behaviour of a tethered system anchored on the Phobos surface directly under the L1 collinear libration point. Two gravitational forces of Mars and Phobos, plus a centrifugal force due to the rotation of the Mars-Phobos system, act on the tether. These forces vary with time due to the small eccentricity of the Mars-Phobos orbit. The basic assumptions are formulated in terms of a planar elliptic restricted three-body problem. The motion equations in Nechvile's variables are derived in polar coordinates relative to the anchor point of the tether. The motion of the tethered system is divided into perturbed and unperturbed when the eccentricity of the Mars-Phobos orbit is zero. The points of unstable equilibrium of the tether are found, which together with periodic perturbations associated with small eccentricity are the cause of chaotic behaviour of the tether. To suppress chaos, the control coefficient is chosen using a known tether length control law. The Melnikov method is used to prove the chaotic nature of the tether and to find approximate the control coefficient needed to suppress the chaos. Verification of the obtained calculations is performed by means of Poincaré portraits for the basic nonlinear tether equation. The results of this study can be used for new Phobos exploration missions using an anchored tethered system and other future missions to study small planetary satellites in the Solar System.