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For low energy spacecraft trajectories such as multimoon orbiters for the Jupiter system, multiple gravity assists by moons could be used in conjunction with ballistic capture to drastically decrease fuel usage. In this paper, we investigate a special class of multiple gravity assists which can occur outside of the perturbing body's sphere of influence (the Hill sphere) and which is dynamically connected to orbits that get captured by the perturber and orbits which escape to infinity. Wedoi:10.1137/060663374 fatcat:s7fgibmtqbcjddwdrrysvaz5rq