Analytical Approximation Method for Continuous Trajectories of Electric Sails

In response to the specific requirements of continuous thrust angle for the low attitude maneuverability of electric sails，and recognizing the current focus on near-circular parking orbits and fixed thrust angles，a two-dimensional trajectory analytic approximation method for electric sails considering a specific continuous variation of thrust angle is proposed. Firstly，the equation of motion of an electric sail in polar coordinates considering a standard elliptical parking orbit is derived by introducing a perturbation parameter. Subsequently，the state variables are represented as combinations of different powers of perturbation coefficient and solved separately to obtain the analytical approximation trajectory of the electric sail. Finally，rectification step is added to the analytical solution to further improve its accuracy. Numerical simulation analyzed the relative error between analytical solutions and numerical integral solutions，and compared with that of splicing analytical solutions obtained by fixed thrust angle. The impact of initial parameter was also analyzed. The results show that rectification step greatly reduces approximation errors while eliminating integration time in the analytical solution，and compared to splicing analytical solutions obtained by fixed thrust angle，the proposed analytical solution greatly improves approximation accuracy. The proposed analytical solution offers an efficient approach for the continuous transfer trajectory design of future complex interstellar missions involving electric sails，as well as initial value estimation for large-scale search missions.