# 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

## Notes

### Acknowledgements

We are thankful to Prof. M. Valtonen, Tuorla Observatory, University of Turku, Finland for his suggestion and for providing related research papers. We wish to put on record our gratitude to Prof. K.B. Bhatnagar who had given us valuable suggestions on our work when he was alive.

## References

1. Anosova, J.P., Bertov, D.L., Orlov, V.V.: Astrofizika 20, 327 (1984)
2. Chandra, N., Bhatnagar, K.B.: Astron. Astrophys. 346, 652–662 (1999)
3. Heggie, D.C.: Mon. Not. R. Astron. Soc. 173, 729 (1975)
4. Henon, M.: Celest. Mech. 10, 375 (1974)
5. Kumar, R., Chandra, N., Tomar, S.: Astrophys. Space Sci. 361, 79 (2016)
6. Marchal, C., Yoshida, J., Sun, Y.S.: Celest. Mech. 34, 65–93 (1984)
7. Mikkola, S.: Mon. Not. R. Astron. Soc. 269, 127 (1994)
8. Mikkola, S., Valtonen, M.: Mon. Not. R. Astron. Soc. 223, 269 (1986)
9. Monaghan, J.J.: Mon. Not. R. Astron. Soc. 176, 63 (1976a)
10. Monaghan, J.J.: Mon. Not. R. Astron. Soc. 177, 583 (1976b)
11. Nash, P.D., Monaghan, J.J.: Mon. Not. R. Astron. Soc. 184, 119 (1978)
12. Saslaw, W.C., Valtonen, M., Aarseth, S.: Astrophys. J. 190, 253 (1974)
13. Standish, M.: Celest. Mech. 4, 44 (1971)
14. Sundman, K.F.: Acta Math. 36, 105 (1912)
15. Szebehely, V.: Celest. Mech. 4, 116 (1971)
16. Szebehely, V.: In: Tapley, B., Szebehely, V. (eds.) Recent advances in dynamical astronomy, p. 75. Reidel, Dordrecht (1973)
17. Valtonen, M.J.: In: Kozai, Y. (ed.) Stability of the solar system and of small stellar systems. Proc. IAU Symp., vol. 62, p. 211. Reidel, Dordrecht (1974)
18. Valtonen, M.J.: Astrophys. Space Sci. 42, 331–347 (1976)
19. Valtonen, M., Millari, A., Orlov, V., Rubinov, A.: ASP Conference Series 316 (2004) Google Scholar
20. Valtonen, M., Karttunen, H.: The Three Body Problem. Cambridge University Press, Cambridge (2006)