New Analytic Solution with Continuous Thrust: Generalized Logarithmic Spirals

Abstract

A new fully analytic solution to the motion of a continuously accelerated spacecraft is presented. It is demonstrated that the same tangential control law that generates a logarithmic spiral yields an entire family of generalized spirals. The system admits two integrals of motion that follow from the equations of the energy and angular momentum. Three different subfamilies of spiral trajectories are obtained, depending on the sign of the constant of the generalized energy: elliptic, parabolic, and hyperbolic. Elliptic spirals are bounded, never escape to infinity, and the trajectory is symmetric. Parabolic spirals are equivalent to logarithmic spirals. There are two subfamilies of hyperbolic spirals, classified in terms of their generalized angular momentum. The first family has only one asymptote, whereas spirals of the second type are symmetric and exhibit two asymptotes. The new family of solutions is obtained when rigorously solving the equations of motion with no prior assumptions. Closed-form expressions for the trajectory, velocity, time of flight, and arc length are provided. The thrust magnitude decreases with the square of the radial distance and might be reproducible with solar sails or solar electric propulsion systems.