Triple encounters in the linear three-body problem with equal masses

Abstract

The paper considers triple encounters in the linear three-body problem for the case of equal masses. Triple encounters are described using two parameters: the virial coefficient k and the angle ϕ such that tan \(\varphi = \dot r/\dot \rho\), where \(\dot r\) and \(\dot \rho\) are the velocities of the “central” body relative to each of the “outer” bodies. The equations of motion are integrated numerically up to one of the following times: the time for a receding body to turn, the time for this body to reach some critical distance, the time for some escape criterion to be fulfilled, or to some critical time. Evolutionary scenarios for the triple system are determined as a function of the initial conditions. The dependences of the ejection length on k and \(\dot \varphi\) are derived. The initial conditions corresponding to escape form a continuous region with k>0.5. The regions into which the right and left bodies depart alternate and are symmetrical about the lines of triple close encounters (ϕ=45°,225°). Regions of stable motions in the vicinity of the central periodic orbit of Schubart (k⋍0.206; ϕ=135°,315°) are identified. Linear structures emanate from the peak of the region of stability, which divide the region for the initial conditions into alternating zones with identical evolutionary scenarios.