Single close encounters in the planetary problem

We consider a large massM and two small massesm ₁ andm ₂ (m ₁ m ₂;m ₁,m ₂M). The orbit ofm ₁ is initially circular and the motion ofm ₂ hyperbolic with respect toM. The orbital elements of the small masses are strongly modified after a close, single encounter betweenm ₁ andm ₂.An approximative method, similar to the theory of stellar encounters, is used to determine the probabilities of collisions, hyperbolas, direct and retrograde ellipses, as well as the mean values of the semimajor axes and their root mean square deviation after the encounter.The results are close to those which are obtained if the massm ₂ is negligibly small, (Mm ₁m ₂;m ₂ 0), as should be also expected on general grounds.