外共振中的倾斜非对称振动
Inclined asymmetric librations in exterior resonances
摘要：
天体力学中的平动运动通常与稳定共振组态的存在相关联，并通过稳定周期解的存在和临界（共振）角的振荡来表示。当这种振荡发生在一个不同于0或$π$的值附近时，这种振动称为不对称振动。在平面圆形限制性三体问题（CRTBP）的背景下，对于外部平均运动共振（MMR）1：2，1：3等以及共轨运动（1：1），已经确定了非对称振动。在外部MMR中，无质量物体是外部物体。本文研究三维空间中的非对称振动。我们采用Markellos（1978）的计算方法计算非对称周期轨道族及其稳定性。稳定的，不对称的周期轨道所包围的相空间域的初始条件，对应于稳定的演变和振动的谐振角。我们的计算集中在太阳-海王星-TNO系统（TNO= trans-Neptunian object）的空间圆形限制三体模型上。我们比较我们的结果与观察到的TNO的数值积分，这表明，他们中的一些进行1：2共振，倾斜的非对称振动。对于稳定的1：2 TNOs库，我们发现它们的天平动似乎与我们模型的垂直稳定的平面非对称轨道有关，而不是在本研究中发现的三维轨道。
英文摘要：
Librational motion in celestial mechanics is generally associated with the existence of stable resonant configurations and signified by the existence of stable periodic solutions and oscillation of critical (resonant) angles. When such an oscillation takes place around a value different than 0 or $π$, the libration is called asymmetric. In the context of the planar circular restricted three-body problem (CRTBP), asymmetric librations have been identified for the exterior mean-motion resonances (MMRs) 1:2, 1:3 etc. as well as for co-orbital motion (1:1). In exterior MMRs the massless body is the outer one. In this paper, we study asymmetric librations in the 3-dimensional space. We employ the computational approach of Markellos (1978) and compute families of asymmetric periodic orbits and their stability. Stable, asymmetric periodic orbits are surrounded in phase space by domains of initial conditions which correspond to stable evolution and librating resonant angles. Our computations were focused on the spatial circular restricted three-body model of the Sun-Neptune-TNO system (TNO= trans-Neptunian object). We compare our results with numerical integrations of observed TNOs, which reveal that some of them perform 1:2-resonant, inclined asymmetric librations. For the stable 1:2 TNOs librators, we find that their libration seems to be related with the vertically stable planar asymmetric orbits of our model, rather than the 3-dimensional ones found in the present study.
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