受Kozai效应驱动的抛射体的Öpik型碰撞频率：处于倾斜圆轨道的目标天体
Öpik-type collision frequency for Kozai-driven projectiles: Target bodies on inclined circular orbits
AI总结：
本文将Vokrouhlický等人的半解析碰撞频率框架扩展至倾斜圆轨道目标，引入相对交点经度修正，其结果与直接动力学模拟匹配，可用于预测抛射体剩余比例的衰减曲线。
AI中文摘要：
现有的Öpik型碰撞频率方法已纳入高倾角抛射体轨道要素的Kozai驱动长期演化，但这些方法通常假设目标天体的轨道位于摄动天体轨道定义的参考平面内。本文将Vokrouhlický等人（2012）开发的、适用于目标天体在参考平面内圆轨道的半解析框架，扩展到目标天体相对于该平面具有非零倾角的圆轨道情形。目标天体的交点进动率被设定为常数，可为零。在此几何构型中，两天体轨道是否相交不仅取决于抛射体轨道的长期状态，还取决于两者的相对交点经度。为考虑这种相关性，将相对交点经度作为额外的几何变量引入，并对框架进行相应扩展。当目标天体的圆轨道位于参考平面内时，本文的公式可解析退化为Vokrouhlický等人（2012）描述的零倾角情形，数值结果也证实了该退化关系。对于目标天体轨道倾斜的两种情形，利用本文框架计算的碰撞频率被用于预测剩余抛射体比例的半解析衰减曲线，这些预测曲线与直接动力学模拟得到的曲线高度吻合。
英文摘要：
Existing Öpik-type collision-frequency methods already incorporate the Kozai-driven secular evolution of the orbital elements of high-inclination projectiles. However, these methods generally assume that the target body's orbit lies in the reference plane defined by the orbital plane of the perturbing body. The present paper extends the semi-analytical framework developed by Vokrouhlický et al. (2012) for a target body on a circular orbit in the reference plane to the case of a target body on a circular orbit with a non-zero inclination relative to that plane. The target body's nodal precession rate is prescribed to be constant and may be zero. In this geometry, whether the two orbits intersect depends not only on the secular state of the projectile's orbit but also on the relative nodal longitude between the two orbits. To account for this dependence, the relative nodal longitude is introduced as an additional geometrical variable, and the framework is extended accordingly. When the target body's circular orbit lies in the reference plane, the present formulation analytically reduces to the zero-inclination case described by Vokrouhlický et al. (2012). The numerical results also confirm this reduction. For the two cases with inclined target-body orbits, the collision frequencies computed with the present framework are used to predict semi-analytical decay curves for the fraction of projectiles remaining. These predicted curves closely match those obtained from direct dynamical simulations.
