In the general three-body systems, one encounters motions of different types under certain initial conditions. In fact, the main problem is the partition of the phase space of the initial conditions. According to Henon (1974), the region of phase space with bounded motion is mixed with escape.
The conjecture of Birkhoff (1922, 1927) and later reformulated by Szebehely & Peters (1967), Szebehely (1973) states that sufficiently triple close simultaneously asymmetric approach results with the formation of a binary and escape of the third body. The above statement is numerically confirmed by Agekian & Anosova (1967), Szebehely & Peters (1967), Szebehely (1974b), Chandra & Bhatnagar (1998a) and others. Closely related to this problem has also been investigated by Waldvogel (1976) and Marchal & Losco (1980).
According to Sundman (1912) for a triple collision the total angular momentum C must be equal to zero and for triple close approach, C should be sufficiently small.
Here, the subject is the general problem of three bodies and the equilateral Lagrangian solution in a symmetric rotating configuration. The masses of the participating bodies are equal. If the mean motion is such that the virial coefficient is unity, the distances between the bodies are constant. In Lagrangian solution, the symmetric configuration is never destroyed, therefore, escape does not occur and all motions are periodic, even when angular momentum C is small. It is equally important for unstable Lagrangian solutions.
In the equilateral configuration, if C is small and asymmetric changes of the initial conditions are introduced, it leads to escape instead of periodic orbits. Furthermore, if the total energy of the system is positive, instability always occurs and the system either explodes or a binary is formed and the third body escapes, but in the astronomically more important case of negative total energy, one or several triple close approaches proceed and an eventual escape, if it occurs at all. Therefore, the detailed behaviour during triple close approaches might be considered relevant to stellar dynamics.
In the present paper, our aim is to study which body is likely to escape and the other two forming a binary with the help of relative distances of the participating bodies when perturbation is used, and compare the results with the Agekian's escape probability.
According to the Szebehely (1971) and Agekian & Martinova's (1973) classification of the states of motion in the general problems of three bodies, the family presented in this paper belongs to class `0'.
© European Southern Observatory (ESO) 1999
Online publication: May 21, 1999