Statistical approach to triple systems in three-dimensional motion

This paper considers disruption of triple close approaches with low initial velocities and equal masses in the framework of statistical escape theory in a three-dimensional space. The statistical escape theory is based on the assumption that the phase trajectory of a triple system is quasi-ergodic. This system is described by allowing for both energy and angular momentum conservation in the phase space. In this paper, "possibility of escape" is derived with the formation of a binary on the basis of relative distances of the participating bodies. The complete statistical solutions (i.e. the semi-major axis $a$, the distributions of eccentricity $e$ of the binary, binary energy $E_b$, escape energy $E_s$ of escaper, and its escape velocity $v_s$) of the system are derived from the allowable phase space volumes and are in good agreement with the numerical results in the range of perturbing velocities $v_i$(${10}^{-1}\le v_i\le {10}^{-10}$) and directions of $v_i(0\le {\alpha }_i,{\beta }_i,{\gamma }_i\le \pi )$, $i=1,2,3$. In this paper, the double limit process has been applied to approximate the escape probability. Through this process, it is observed that the perturbing velocity $v_i\to 0^{+}$, as the product of the semi-major axis $a$ of the final binary and the square of the escape velocity $v_s$ approach 2/3, i.e. $a{v}_s^2\to 2/3$, whatever direction of $v_i$ may be.