# Statistical approach to triple systems in three-dimensional motion

Original Article

## Abstract

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 $$\mathbf{v}_{{i}}$$ may be.

## Keywords

Astrophysics Three-body problem Triple close approaches Statistical theory

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## Authors and Affiliations

• Ranjeet Kumar
• 1
• Navin Chandra
• 2
• Surekha Tomar
• 1
1. 1.Department of PhysicsR.B.S. College (B.R. Ambedkar University)AgraIndia
2. 2.Department of MathematicsDeshbandhu College (University of Delhi)KalkajiIndia