Dynamics of tethered satellites in the vicinity of the Lagrangian point $L_{2}$ of the Earth–Moon system

Abstract

This paper analyzes the dynamical evolution of satellites formed by two masses connected by a cable—tethered satellites. We derive the Lagrangian equations of motion in the neighborhood of the collinear equilibrium points, especially for the $L_{2}$, of the restricted problem of three bodies. The rigid body configuration is expanded in Legendre polynomials up to fourth degree. We present some numerical simulations of the influence of the parameters such as cable length, mass ratio and initial conditions in the behavior of the tethered satellites. The equation for the collinear equilibrium point is derived and numerically solved. The evolution of the equilibria with the variation of the cable length as a parameter is studied. We also present a discussion of the linear stability around these equilibria. Based on this analysis calculate some unstable Lyapunov orbits associated to these equilibrium points. We found periodic orbits in which the tether travels parallel to itself without involving the angular motion. The numerical applications are focused on the Earth–Moon system. However, the general character of the equations allows applications to the $L_{1}$ equilibrium and obviously to systems other than the Earth–Moon.