Anatomy of the Constant Radial Thrust Problem

Abstract

N this paper, we intend to investigate the solution manifold for a fairly well knowlow-thrust problem,the constant outward radial acceleration.The trajectories are always planarbecause the acceleration is permanently in the plane of the orbit. Therefore, we treat it asatwo-degree-of-freedomproblem.Theproblemisknown tohave a closed-form analytical solution, and, therefore, a good number of its properties are already evident. 1i4 The analytical solution can be expressed in terms of the standard elliptic integrals that actually do not give too much insight into their basic properties. To gain more insight into these properties, our present investigation will have recourse to some methods that are usually employed in classical mechanics and celestial mechanics, such as the boundaries imposed by the potential function, as well as the two integrals of the problem, the energy and the angular momentum. These conceptsalready allowusto distinguish between the escape trajectories and the bounded trajectories. The use of the effective potential, as well as the value of the energy constant, allows us to dee ne forbidden, as well as allowable, regions of motion. Note that the concept of an effective potential energy has been well employed earlier by Prussing and Coverstone-Carroll 3 in a similar context, but their initial conditions are restricted to that of a circular orbit.