Axisymmetric disks of eccentric Kepler orbits are vulnerable to an instability that causes orbits to exponentially grow in inclination, decrease in eccentricity, and cluster in their angle of pericenter. Geometrically, the disk expands to a cone shape that is asymmetric about the mid-plane. In this paper, we describe how secular gravitational torques between individual orbits drive this "inclination instability". We derive growth timescales for a simple two-orbit model using a Gauss N-ring code, and generalize our result to larger N systems with N-body simulations. We find that two-body relaxation slows the growth of the instability at low N and that angular phase coverage of orbits in the disk is important at higher N. As N → ∞, the e-folding timescale converges to that expected from secular theory.

中文翻译：

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关于倾角不稳定性的动力学

偏心开普勒轨道的轴对称圆盘容易受到不稳定性的影响，这种不稳定性会导致轨道的倾斜度呈指数增长，偏心度减小，并在其周心角上聚集。在几何上，圆盘扩展为关于中平面不对称的圆锥形状。在本文中，我们描述了各个轨道之间的长期引力如何驱动这种“倾角不稳定性”。我们使用高斯 N 环代码为简单的两轨道模型推导出增长时间尺度，并将我们的结果推广到具有 N 体模拟的更大 N 系统。我们发现二体弛豫减缓了低 N 下不稳定性的增长，并且盘中轨道的角相位覆盖在较高 N 时很重要。当 N → ∞ 时，e 折叠时间尺度收敛到世俗理论预期的时间尺度。