Janus and Epimetheus are two moons of Saturn with very peculiar motions. As they orbit around Saturn on quasi-coplanar and quasi-circular trajectories whose radii are only 50 km apart (less than their respective diameters), every four (terrestrial) years the bodies approach each other and their mutual gravitational influence lead to a swapping of the orbits: the outer moon becomes the inner one and vice-versa. This behavior generates horseshoe-shaped trajectories depicted in an appropriate rotating frame. In spite of analytical theories and numerical investigations developed to describe their long-term dynamics, so far very few rigorous long-time stability results on the "horseshoe motion" have been obtained even in the restricted three-body problem. Adapting the idea of Arnol'd (Russ Math Surv 18:85-191, 1963) to a resonant case (the co-orbital motion is associated with trajectories in 1:1 mean motion resonance), we provide a rigorous proof of existence of 2-dimensional elliptic invariant tori on which the trajectories are similar to those followed by Janus and Epimetheus. For this purpose, we apply KAM theory to the planar three-body problem.
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机译:Janus和Epimetheus是两个土星的卫星,具有非常奇特的动作。正如他们在氨基莲花和准圆形轨迹上的轨道上轨道,其半径分开只有50公里(小于它们各自的直径),那么每四个(地面)多年的身体彼此接近,它们的相互引力影响导致交换轨道:外部月亮成为内部,反之亦然。该行为产生在适当的旋转框架中描绘的马蹄形轨迹。尽管分析理论和制定的数字调查来描述其长期动态,但到目前为止,即使在受限制的三体问题中也已经获得了“马蹄运动”的严格长期稳定性。适应Arnol'd(Russ Math Surv 18:85-191,1963)到共振案例的想法(共轨运动与轨迹有关1:1平均运动共振),我们提供了严格的存在证明二维椭圆形不变扭矩,轨迹类似于Janus和Epimetheus的轨迹。为此目的,我们将KAM理论应用于平面的三体问题。
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