Skip to main content
Log in

Numerical Treatment of Small Stellar Systems with Binaries

  • Published:
Celestial Mechanics and Dynamical Astronomy Aims and scope Submit manuscript

Abstract

The use of regularization methods based on the Kustaanheimo-Stiefel transformation (KS) is reviewed. The history of the development of such methods is summarized and anecdotal information about the related events is told. Details of the multi-particle regularization methods are given, including the most recent developments.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  • Aarseth, S. J.: 1971, “Binary Evolution in Stellar Systems”, Astrophys. Space Sci. 13, 324.

    Article  ADS  Google Scholar 

  • Aarseth, S. J.: 1972, “Direct Integration Methods of the N-body Problem”, in Gravitational N-body Problem, proceedings of IAU colloquium No. 10, ed. M. Leear, Reidel, Dordrecht, pp.373–387.

    Google Scholar 

  • Aarseth, S. J.: 1974, “Dynamical Evolution of Simulated Star Clusterrs. I. Isolated Models”. Astron. Astrophys., 35, 237.

    ADS  Google Scholar 

  • Aarseth, S. J.: 1977, Rev. Mex. Astron. Astrofis., 3, 199.

    ADS  Google Scholar 

  • Aarseth, S. J.: 1988, “Integration Methods for Small N-Body Systems”, in The Few Body Problem, (Valtonen M. J., ed.), 287–306, Kluwer, Dordrecht, Holland.

    Google Scholar 

  • Aarseth, S. J. 1996, in Dynamical Evolution of Star Clusters, ed. Hut, P. and Makino, J. Kluwer.

  • Aarseth, S. J. and Zare, K.: 1974, “A Regularization of the Three-Body Problem”, Cel. Mech., 10, 185–205.

    Article  MATH  ADS  Google Scholar 

  • Aarseth, S. J. and Heggie, D. C.: 1976, “The Probability of Binary Formation by Three-Body Encounters”, Astron. Astrophys., 53, 259–265.

    ADS  Google Scholar 

  • van Albada, T. S.: 1968, “The evolution of small stellar systems and its implications for the formation of double stars”, Bull. astr. Inst. Neth., 20, 57.

    MathSciNet  ADS  Google Scholar 

  • Alexander, M. E.: 1986, “Simulations of binary-single star and binary-binary scattering”, J. Comp. Phys., 64, 195–219.

    Article  MATH  ADS  Google Scholar 

  • Bulirsch, R. and Stoer, J.: 1966, “Numerical Treatment of Differential Equations by Extrapolation Methods”, Num. Math., 8, 1–13.

    Article  MATH  MathSciNet  Google Scholar 

  • Heggie, D. C.: 1974, “A Global Regularisation of the Gravitational N-Body Problem”, Cel. Mech., 10, 217–241.

    Article  MATH  ADS  Google Scholar 

  • Heggie, D. C.: 1988, “The N-Body Problem in Stellar Dynamics”, in Long-Term Dynamical Behaviour of Natural and Artificial N-Body Systems, (Roy A. E., ed.), 329–347, Kluwer, Dordrecht, Holland.

    Google Scholar 

  • von Hoerner, S.: 1960, Z. Astrophys. 50, 184.

    MATH  MathSciNet  ADS  Google Scholar 

  • von Hoerner, S.: 1963, Z. Astrophys. 57, 47.

    MATH  ADS  Google Scholar 

  • Hopf, H. 1931, “Uber die Abbildung der dreidimensionalen Sphäre auf die Kugelfläche”, Math. Ann. 104. Reprinted in: Selecta Heinz Hopf, 38–63, Springer 1964.

  • Kustaanheimo, P.: 1964, “Die Spinordarstellung der energetischen Identitäten der Kepleibewegung”, p. 333–340. In: E. Stiefel (ed.): Mathematische Methoden der Himmelsmechanik und Astronautik, Mathematisches Forschungsinstitut Oberwolfach, Berichte 1. Bibliographisches Institut Mannheim, 1966, 350 pp.

  • Kustaanheimo, P. and Stiefel, E.: 1965, “Perturbation theory of Kepler motion based on spinor regularization”, J. Reine Angew. Math., 218, 204–219.

    MATH  MathSciNet  Google Scholar 

  • Lemaitre, G.: 1955, “Regularization of the Three-Body Problem”, in Vistas in Astronomy, (ed. A. Beer), Vol 1, Pergamon Press.

  • Levi-Civita, T.: 1920, “Sur la régularisation du problème des trois corps”, Acta Math. 42, 99–144.

    Article  Google Scholar 

  • Mikkola, S.: 1983, “Encounters of Binaries-I. Equal Energies”, Mon. Not. R. Astr. Soc, 203, 1107–1121.

    ADS  Google Scholar 

  • Mikkola, S.: 1985, “A Practical and Regular Formulation of the N-Body Equations”, Mon. Not. R. Astr. Soc, 215, 171–177.

    MATH  MathSciNet  ADS  Google Scholar 

  • Mikkola, S. and Aarseth, S. J.: 1990, “A Chain Regularization Method for the Few-Body Problem”, Cel. Mech., 47, 375–390.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  • Mikkola, S. and Aarseth, S. J.: 1993, “An Implementation of N-body Chain Regularization”, Celest. Mech. Dyn. Astron. 57, 439

    Article  MATH  MathSciNet  ADS  Google Scholar 

  • Mikkola, S. and Aarseth, S. J.: 1996, “A Slow-Down Treatment of Close Binaries”, Celest. Mech. Dyn. Astron. (to appear).

  • Stiefel, E. L. and Scheifele, G.: 1971, Linear and Regular Celestial Mechanics, Springer, Berlin.

    MATH  Google Scholar 

  • Szebehely, V.: 1967, Theory of Orbits, Academic Press, New York.

    MATH  Google Scholar 

  • Szebehely, V. and Bettis, D. G. 1972, in Gravitational N-body Problem, proceedings of IAU colloquium No. 10, ed. M. Lecar, Reidel, Dordrecht, pp. 136–147.

    Google Scholar 

  • Szebehely, V. and Peters, C. F.: 1967, “Complete solution of a general problem of three bodies”, Astron. J., 72, 876.

    Article  ADS  Google Scholar 

  • Szebehely, V. and Zare, K.: 1975, “Time transformations in the extended phase-space”, Cel. Mech., 11, 469.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  • Waldvogel, J.: 1972, “A new regularization of the planar problem of three bodies”, Cel. Mech., 6, 221.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  • Waldvogel, J., Kirchgraber, U. Schwarz, H. R, Henrici, P.: 1979, “In Memoriam E. Stiefel.” ZAMP 30, 133–142.

    Article  MathSciNet  Google Scholar 

  • Zare, K.: 1974, “A Regularization of Multiple Encounters in Gravitational N-body Problem” Cel. Mech., 10, 207–215.

    Article  MATH  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mikkola, S. Numerical Treatment of Small Stellar Systems with Binaries. Celestial Mechanics and Dynamical Astronomy 68, 87–104 (1997). https://doi.org/10.1023/A:1008291715719

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1008291715719

Navigation