火星重力场中准圆形冻结轨道的分析研究
Analytical investigations of quasi-circular frozen orbits in the Martian gravity field
摘要：
冻结轨道由于其偏心率和近心辐角平均保持不变的特性，一直是轨道设计的重要研究对象。本研究探讨准圆形冻结轨道，并探讨其基本性质分析使用两种不同的方法。首先，基于拉格朗日公式的分析方法，适用于获得火星冻结轨道的约束条件。其次，利用李变换精确定位这些轨道，并绘制出哈密顿量的等值线，以显示平衡点的演化。这两种方法在80 \times 80火星重力场的数值积分验证。仿真结果表明，这两种分析方法可以提供足够准确的结果。通过比较发现，两种方法具有很好的一致性，都发现了四类火星冻结轨道：三类小偏心率轨道和一类接近临界倾角的轨道。结果还表明：对于大多数火星冻结轨道，由于J3与J2同号，近心角保持在270 °，而对于少数低海拔、低倾角的冻结轨道，由于高次奇环带的影响，近心角有可能保持在90 °;对于临界倾角的情况，近心角也可以保持在90度。值得注意的是，存在一些特殊的冻结轨道，它们具有极小的偏心率，这将为侦察提供很大的便利。最后，基于单值矩阵的迹估计了火星冻结轨道的稳定性。分析研究可以为更复杂模式的数值修正方法提供良好的初始条件。
英文摘要：
Frozen orbits are always important foci of orbit design because of their valuable characteristics that their eccentricity and argument of pericentre remain constant on average. This study investigates quasi-circular frozen orbits and examines their basic nature analytically using two different methods. First, an analytical method based on Lagrangian formulations is applied to obtain constraint conditions for Martian frozen orbits. Second, Lie transforms are employed to locate these orbits accurately, and draw the contours of the Hamiltonian to show evolutions of the equilibria. Both methods are verified by numerical integrations in an 80\times80 Mars gravity field. The simulations demonstrate that these two analytical methods can provide accurate enough results. By comparison, the two methods are found well consistent with each other, and both discover four families of Martian frozen orbits: three families with small eccentricities and one family near the critical inclination. The results also show some valuable conclusions: for the majority of Martian frozen orbits, argument of pericentre are kept at 270 degrees because J3 has the same sign with J2; while for a minority of ones with low altitude and low inclination, argument of pericentre are possible to be kept at 90 degrees because of the effect of the higher degree odd zonals; for the critical inclinations cases, argument of pericentre can also be kept at 90 degrees. It is worthwhile to note that there exist some special frozen orbits with extremely small eccentricity, which could provide much convenience for reconnaissance. Finally, the stability of Martian frozen orbits is estimated based on the trace of the monodromy matrix. The analytical investigations can provide good initial conditions for numerical correction methods in the more complex models.
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