The evolution of a system consisting of an inviscid polytrope and a point mass in initial orbits which are either elliptical, parabolic or hyperbolic is studied. For each type of central orbit several orbits were followed to determine the effect of periastron separations less than or greater than the Roche distance for circular orbits. Two numerical methods were used to allow a treatment of both very weak perturbations and perturbations sufficiently strong to disrupt the polytrope. The results have been applied to the Fabian–Pringle–Rees capture mechanism, to Roche’s hypothesis of the origin of Saturn’s rings and to Woolfson's tidal model of the origin of the solar system.