A computer assisted method is applied to rigorously prove the existence of chaotic behaviour in a differential equation modelling a dumbbell satellite moving on a Keplerian ellipse. The problem is described by a second order time periodic differential equation. The satellite is composed of two equal point masses connected by a rigid massless bar. It is assumed that the motion takes place in a plane. The method applied has earlier been developed in \textit{D. Stoffer} and \textit{K. J. Palmer} [Nonlinearity 12, 1683-1698 (1999; Zbl 0988.37041)] and \textit{D. Stoffer} and \textit{U. Kirchgraber} [Verification of chaotic behaviour in the planar restricted three body problem (preprint, 1999)]. In the paper, the key results of the scheme are stated and the procedure by which the authors establish a shadowing result is briefly described. The scheme is then applied to the dumbbell satellite model. The continuous system is replaced by a periodic discrete dynamical system for which the two periodic pseudo orbits are computed. The so-called validated bounds for some estimates needed are also calculated by computer methods. Thus the main result on shadowing in the dumbbell satellite model is obtained.