Eddy currents applied to de-tumbling of space debris: Analysis and validation of approximate proposed methods

The space debris population has greatly increased over the last few decades. Active debris removal (ADR) methods may become necessary to remove those objects in orbit that pose the biggest collision risk. Those ADR methods that require contact with the target show complications if the target is rotating at high speeds. Observed rotations can be higher than 60 deg/s combined with precession and nutation motions. 'Natural' rotational damping in upper stages has been observed for some space debris objects. This phenomenon occurs due to the eddy currents induced by the Earth׳s magnetic field in the predominantly conductive materials of these man made rotating objects. Existing solutions for the analysis of eddy currents require time-consuming finite element models to solve a Poisson equation throughout the volume. The first part of this paper presents a new method to compute the eddy current torque based on the computation of a new tensor called the 'Magnetic Tensor'. The general theory to compute this tensor by Finite Element Method is given as well as a particular frame model. This last model enables an explicit formula to be determined to evaluate the magnetic tensor. Analytical solutions for the spherical shell, the open cylinder and flat plates are given for the magnetic tensor and the eddy current torque model is validated with existing published work. The second part of the paper presents an active de-tumbling method for space debris objects based on eddy currents. The braking method that is proposed has the advantage of avoiding any kind of mechanical contact with the target. The space debris object is subjected to an enhanced magnetic field created from a chaser spacecraft which has one or more deployable structures with an electromagnetic coil at its end. The braking time and the possible orientation change in the rotation axis is analysed for a metallic spherical shell considering different ratios of conductive versus non-conductive material. The paper finalises with a case study based on the de-tumbling of an Ariane-4 Upper Stage H10 under the effect of the gravity gradient and a preliminary analysis of the non-uniformity of the magnetic field is presented.