Analytical approximation of the propulsion dynamics of magnetic sail with fixed solar incidence angle

Magnetic sails are a highly promising propellantless propulsion method for interplanetary spaceflight. This study investigated the approximate analytical solution for the two-dimensional equations of motion of a magnetic sail, employing the latest single-loop magnetic sail thrust model with a fixed solar incidence angle. Based on linear perturbation theory (neglecting higher-order perturbations), an analytical expression for the heliocentric transfer orbit of a magnetic sail was derived. A trajectory obtained via numerical integration was used as the reference for assessing the proposed method’s level of approximation precision. A simulation probed the effects of varying the magnetic sail coil radius and the parameters of the initial parking orbit, and the results indicated that the proposed analytical method demonstrates excellent approximation accuracy for small-scale magnetic sails and general elliptical orbits. During the early stages of the mission, the approximate relative error was as low as 10⁻⁴, and the computation time was only 1 % of that required by the numerical integration method. Furthermore, to address long-duration missions and large-scale magnetic sails, the analytical solution was corrected by updating the orbital parameters at each step of the approximation process. This method achieved an approximation accuracy that was 100 times more precise than that of the uncorrected analytical solution. Even for a long-duration mission spanning 30 orbits, the corrected analytical solution maintained an accuracy on the order of 10⁻⁵. The approximation method proposed in this study is a valuable tool for the preliminary design and optimization of trajectories in deep space missions employing magnetic sails, particularly for long-duration missions and large-scale transfer scenarios.