We consider the planar Hill’s lunar problem with a homogeneous gravitational potential. The investigation of the system is twofold. First, the starting conditions of the trajectories are classified into three classes, that is, bounded, escaping, and collisional. Second, we study the no-return property of the Lagrange point \(L_2\) and we observe that the escaping trajectories are scattered exponentially. Moreover, it is seen that in the supercritical case, with the potential exponent \(\alpha \ge 2\), the basin boundaries are smooth. On the other hand, in the subcritical case, with \(\alpha <1\) the boundaries between the different types of basins exhibit fractal properties.

我们考虑具有均质引力势的平面希尔月球问题。对系统的研究是双重的。首先，轨迹的起始条件分为三类，即有界、逃逸和碰撞。其次，我们研究了拉格朗日点\(L_2\)的不返回性质，我们观察到逃逸轨迹呈指数分布。此外，可以看出，在超临界情况下，当势指数为\(\alpha \ge 2\) 时，盆地边界是平滑的。另一方面，在亚临界情况下，随着\(\alpha <1\)，不同类型盆地之间的边界表现出分形特性。