A test particle in a non-coplanar orbit about a member of a binary system can undergo Kozai–Lidov oscillations in which tilt and eccentricity are exchanged. An initially circular highly inclined particle orbit can reach high eccentricity. We consider the non-linear secular evolution equations previously obtained in the quadrupole approximation. For the important case that the initial eccentricity of the particle orbit is zero, we derive an analytic solution for the particle orbital elements as a function of time that is exact within the quadrupole approximation. The solution involves only simple trigonometric and hyperbolic functions. It simplifies in the case that the initial particle orbit is close to being perpendicular to the binary orbital plane. The solution also provides an accurate description of particle orbits with non-zero but sufficiently small initial eccentricity. It is accurate over a range of initial eccentricity that broadens at higher initial inclinations. In the case of an initial inclination of π/3, an error of 1 per cent at maximum eccentricity occurs for initial eccentricities of about 0.1.

中文翻译：

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Kozai-Lidov 演化方程的解析解

围绕双星系统成员的非共面轨道上的测试粒子可以经历 Kozai-Lidov 振荡，其中倾斜和偏心交换。初始圆形高度倾斜的粒子轨道可以达到高偏心率。我们考虑先前在四极近似中获得的非线性长期演化方程。对于粒子轨道的初始偏心率为零的重要情况，我们推导出粒子轨道元素作为时间函数的解析解，该解析解在四极近似内是精确的。该解决方案仅涉及简单的三角函数和双曲函数。它简化了初始粒子轨道接近垂直于二元轨道平面的情况。该解决方案还提供了对具有非零但足够小的初始偏心率的粒子轨道的准确描述。它在一定范围的初始偏心率上是准确的，该偏心率在较高的初始倾角处变宽。在初始倾角为 π/3 的情况下，对于大约 0.1 的初始偏心率，在最大偏心率处会出现 1% 的误差。