独立开普勒轨道中两物体碰撞概率的简化推导
Simplified derivation of the collision probability of two objects in independent Keplerian orbits
摘要：
行星研究中的许多课题都需要估计两个在近似开普勒轨道上运动的物体碰撞的概率。在Öpik（1951）和Wetherill（1967）的经典著作中，碰撞概率是通过对碰撞点附近的运动进行线性化来推导的，现在有大量文献使用他们的方法。我们在这里提出了一个更简单，更物理动机的推导非切向碰撞开普勒轨道，以及切向碰撞，以前没有考虑。我们的公式有一个额外的优点，即两个碰撞体的参数明显对称。与Öpik-Wetherill的处理方法一样，我们将轨道交叉点附近（或两个轨道之间最小距离点附近）的物体运动线性化，并假定碰撞半径内碰撞参量均匀分布。我们指出，线性近似导致奇异结果的情况下切向遭遇。我们正则化这个奇异性的抛物线近似的运动在附近的切向遇到。
英文摘要：
Many topics in planetary studies demand an estimate of the collision probability of two objects moving on nearly Keplerian orbits. In the classic works of Öpik (1951) and Wetherill (1967), the collision probability was derived by linearizing the motion near the collision points, and there is now a vast literature using their method. We present here a simpler and more physically motivated derivation for non-tangential collisions in Keplerian orbits, as well as for tangential collisions that were not previously considered. Our formulas have the added advantage of being manifestly symmetric in the parameters of the two colliding bodies. In common with the Öpik-Wetherill treatments, we linearize the motion of the bodies in the vicinity of the point of orbit intersection (or near the points of minimum distance between the two orbits) and assume a uniform distribution of impact parameter within the collision radius. We point out that the linear approximation leads to singular results for the case of tangential encounters. We regularize this singularity by use of a parabolic approximation of the motion in the vicinity of a tangential encounter.
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