Possibility of Exchange of a Rectilinear Three-Body System with Zero Energy

Abstract

Possibility of exchange is discussed for a rectilinear three-body system with zero energy. By introducing some reguralized coordinates which relate to McGehee s ones and using a separability of the reguralized coordinates in the equations of motion, we can get a boundary manifold T² which describes the motion of configuration of the system. The fixed points of the flow on T² correspond to parabolic escape, hyperbolic-elliptic escape and triple collision. By using the correspondence between an orbit and an evolution of the system and introducing a concept of “Critical System”, we can obtain a result that all of HE^--HE⁺ orbit are exchange type in critical system whose orbits of parabolic-parabolic escape type experience odd times of binary collision and no exchange occurs in critical systems whose orbits of parabolic-parabolic escape type experience even times of binary collision. Moreover, in non-critical system, we can find the both orbits of exchange type and non-exchange type.