Generalized Sail Trajectory Approximation with Applications to MagSails

This paper investigates approximate solutions to the two-dimensional equations of motion of a generalized sail-based spacecraft. A generalized sail is a propellantless propulsion system that produces a continuous thrust the magnitude of which varies with the heliocentric distance as 1/r(eta), where eta is a positive real number that characterizes the propulsion system performance. Under the simplifying assumptions of low propulsive acceleration, constant thrust angle, and circular parking orbit, the spacecraft trajectory is approximated through closed-form relations. The general solution is then specialized to the case of a Magnetic Sail with a so-called "thick" operation mode, the latter being consistent with a kilometer-size loop radius. The effectiveness of the proposed mathematical model is verified by comparison with a numerical integration of the equations of motion, which confirms the accuracy of the analytical results even on very long timescales. Finally, the set of approximate equations is applied to the analysis of transfer trajectories towards a heliocentric target orbit with a given semimajor axis.