Generalized logarithmic spirals for low-thrust trajectory design

Abstract

Shape-based approaches are practical for finding sub-optimal solutions during the preliminary design of low-thrust trajectories. Logarithmic spirals are the simplest, but of little practical interest due to having a constant flight-path angle. We prove that the same tangential thrust profile that generates a logarithmic spiral yields an entire family of generalized spirals. The system admits two integrals of motion, which are equivalent to the energy and the angular momentum equations. Three different subfamilies of spiral trajectories are obtained depending on the sign of the constant of the generalized energy: elliptic, parabolic, and hyperbolic. Parabolic spirals are equivalent to logarithmic spirals. Elliptic spirals are bounded; never escape to infinity and the trajectory is symmetric. Two types of hyperbolic spirals have been found: the first has only one asymptote; the second has two asymptotes, the trajectory is symmetric and never falls to the origin. The solution is obtained when solving rigorously the equations of motion with no prior assumptions. Closed-form expressions for both the trajectory and the time of flight are provided.