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 η 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.

本文研究了广义帆基航天器二维运动方程的近似解。广义帆是一种无推进剂的推进系统，它产生一个连续的推力，其大小随日心距离变化为$1/r^{\eta }$, 其中η是表征推进系统性能的正实数。在低推进加速度、恒定推力角和圆形停放轨道的简化假设下，通过闭式关系来近似航天器轨迹。然后，通用解决方案专门用于具有所谓“厚”操作模式的磁帆的情况，后者与公里大小的环路半径一致。通过与运动方程的数值积分进行比较，验证了所提出的数学模型的有效性，即使在很长的时间尺度上也证实了分析结果的准确性。最后，这组近似方程应用于分析具有给定半长轴的日心目标轨道的转移轨迹。