On stability problems of the space elevator

Abstract

A long dumb-bell satellite in the radial configuration represents a simple model of a Space Elevator. The dumb-bell moves with the angular velocity of the planet, but not relative to it. To calculate the stability of such a relative equilibrium we used the reduced energy-momentum method. We considered dumb-bell satellite models with rigid and elastic, but massless, rods. Figure 1 shows the model with the three DOFs r, ϑ and ϕ. A rigid dumb-bell satellite has a radial (ϕ = 0) and a tangential (ϕ = π/2) relative equilibrium position in the orbit. It is well known that the tangential one is unstable and the radial one is stable below a certain critical length that is for β > β crit = r/a ≈ 3.146 [1]. Integration of the equations of motion for a long satellite (β < β crit = 3.125), with a small deviation from the radial equilibrium position towards the planet, gives the unstable motion of the dumb-bell which is shown in Fig. 2a. When the disturbed radius is larger than the equilibrium position, the dumb-bell moves away from the planet in a helical path (Fig. 2b). We need an additional mass at the geostationary orbit to reduce the altitude of the inner mass beyond the critical value β crit , in order to stabilize the radial equilibrium position. When the inner mass of the dumb-bell touches the surface of the earth, we end up with a simple Space Elevator. For a terrestrial dumb-bell satellite a mass at the geostationary height (r gs ≈ 42164 km) of approximately 6200 times the endmass m = 1 kg is necessary to stabilize the radial configuration. Rigid dumb-bell Elastic dumb-bellNext we assumed two endmasses m, each 1 kg, connected with a mass at the geostationary orbit m s by means of massless, linear elastic springs. This model has just the interesting radial equilibrium position (ϕ = 0). With the definition of the spring stiffness c = EA/l 0 and a realistic choice of the cross-section A = 2.10 −9 m 2 we investigated the stability of the elastic dumb-bell satellite for different materials given in Tab. 1. Because of the non-linear gravitation field we have different unloaded spring lengths for the inner l 0i and outer l 0o spring. With the cyclic variable ϑ and the DOFs r i , r o and r gs (=radii of the three masses) we obtained the amended potentialwith the momentum μ 0 for the motion along the group orbit and the mass ratio α = m s /m. The first and second derivatives of Eqn. 1, with respect to the non-cyclic variables, lead to a system of equations for the equilibrium positions and their stability. Evaluation of these equations was done with a homotopy procedure with the