On the Stability of Planar Librations of a Dumb-bell Satellite in an Elliptic Orbit

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Abstract
The planar librational motion of a dumb-bell satellite has been studied by several authors. In most cases the analysis has been restricted to circular or nearly circular orbits. In an elliptic orbit, the variation of the angular velocity of the satellite around the central body together with the change in the local gravity gradient provides the satellite with a mechanism for exchanging energy between the librational and orbital modes. In general, this imposes a limit on the orbit eccentricity for which stable (non-tumbling) librational motion is possible. This note investigates the bounds that must be placed on a disturbance applied to a slender gravity gradient stabilised satellite so that it will librate and not tumble. The determination of the stability boundary as presented here was arrived at by the numerical solution of the exact equation of motion, obtained using an IBM 7040 computer.

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