Asymptotic Solution for the Current Profile of Passive Bare Electrodynamic Tethers

Abstract

A relatively high-accuracy analytical solution for the current and potential profile along a passive bare electrodynamic tether is provided using perturbation theory. An ad hoc nondimensional formulation of the governing local bias and orbital motion limited current collection equations allows one to approach the problem with a perturbation technique in which a parameter, epsilon, quantifies the influence of ohmic effects on the final solution. For the case of small ohmic effects an approximate solution is obtained with a third-order expansion. Conversely, the case of dominant ohmic effects is treated based on an extension of the exact analytical solution available for the particular case of zero load and negligible potential drop at the cathodic end of the tether. After computing the analytical current and potential profile the maximum and average current, the Lorentz force and torque, as well as the optimum load impedance for maximum power generation are obtained. When compared with the exact, numerically-computed solution an accuracy of better than 5 % is achieved for the computation of the average current across the full parameter space. The error with respect to the generated power becomes negligible when the load impedance is set to the optimum value, while it can grow to a maximum of about 30% for the less relevant case in which the load impedance of the power generation system is badly mismatched. The results, which are valid for a general rectilinear passive electrodynamic tether with constant cross section satisfying orbital motion limited theory and irrespective of the particular orbit configuration, will be of aid in the design and analysis of space missions involving bare electrodynamic tethers.