The purpose of this work is to study the phenomenon of tidal locking in a pedagogical framework by analyzing the effective gravitational potential of a two-body system with two spinning objects. It is shown that the effective potential of such a system is an example of a fold catastrophe. In fact, the existence of a local minimum and saddle point, corresponding to tidally locked circular orbits, is regulated by a single dimensionless control parameter that depends on the properties of the two bodies and on the total angular momentum of the system. The method described in this work results in compact expressions for the radius of the circular orbit and the tidally locked spin/orbital frequency. The limiting case in which one of the two orbiting objects is point-like is studied in detail. An analysis of the effective potential, which in this limit depends on only two parameters, allows one to clearly visualize the properties of the system. The notorious case of the Mars' moon Phobos is presented as an example of a satellite that is past the no return point and, therefore, will not reach a stable or unstable tidally locked orbit.

中文翻译：

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潮汐锁定和引力褶皱灾难

这项工作的目的是通过分析具有两个旋转物体的双体系统的有效引力势，在教学框架中研究潮汐锁定现象。结果表明，这种系统的有效潜力是折叠灾难的一个例子。事实上，对应于潮汐锁定圆形轨道的局部最小值和鞍点的存在是由一个无量纲控制参数调节的，该参数取决于两个物体的特性和系统的总角动量。这项工作中描述的方法导致圆形轨道半径和潮汐锁定自旋/轨道频率的紧凑表达式。详细研究了两个轨道物体之一是点状的极限情况。有效潜力分析，在这个限制中，它只取决于两个参数，允许一个人清楚地形象化系统的属性。臭名昭著的火星卫星火卫一的例子是卫星通过不返回点，因此不会到达稳定或不稳定的潮汐锁定轨道。