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Treatment of Close Approaches in the Numerical Integration of the Gravitational Problem of N Bodies

Published online by Cambridge University Press:  12 April 2016

D. G. Bettis  and
V. Szebehely
D. G. Bettis
Affiliation:
The University of Texas at Austin
V. Szebehely
Affiliation:
The University of Texas at Austin

Abstract

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One of the main difficulties encountered in the numerical integration of the gravitational n-body problem is associated with close approaches. The singularities of the differential equations of motion result in losses of accuracy and in considerable increase in computer time when any of the distances between the participating bodies decreases below a certain value. This value is larger than the distance when tidal effects become important, consequently, numerical problems are encountered before the physical picture is changed. Elimination of these singularities by transformations is known as the process of regularization. This paper discusses such transformations and describes in considerable detail the numerical approaches to more accurate and faster integration. The basic ideas of smoothing and regularization are explained and applications are given.

Type
Research Article
Information
International Astronomical Union Colloquium , Volume 10: Gravitational N-Body Problem , November 1971 , pp. 133 - 150
Copyright
Copyright © Reidel 1971

References

Aarseth, S.: 1970, this issue, p. 20.Google Scholar
Bettis, D. G. : 1970, Numer. Math. 14, 421–434.CrossRefGoogle Scholar
Fehlberg, E.: 1968, NASA TR R-287, also 1969, Computing 4, 93.Google Scholar
Heggie, D.: 1970, this issue, p. 35.Google Scholar
Henrici, P. : 1962, Discrete Variable Methods in Ordinary Differential Equations, John Wiley and Sons.Google Scholar
Hurwitz, A.: 1933, Math. Werke, Vol. 2, Birkhauser, Basel.Google Scholar
Kustaanheimo, P. and Stiefel, E.: 1965, J. Math. 218, 204.Google Scholar
Levi-Civita, T.: 1903, Ann. Math. 9, 1.Google Scholar
Peters, C. F.: 1968, Bull. Astron. 3, 167.Google Scholar
Schubart, J. and Stumpff, P.: 1966, Veröff. Astron. Rechen-Inst. Heidelberg, No. 18, Karlsruhe.Google Scholar
Stiefel, E. and Bettis, D. G.: 1969, Numer. Math. 13, 154–75.Google Scholar
Stiefel, E. and Scheifele, G.: 1970, Linear and Regular Celestial Mechanics, Springer-Verlag.Google Scholar
Sundman, K. F.: 1912, Acta Math. 35, 105.Google Scholar
Szebehely, V.: 1967, Theory of Orbits, Academic Press, New York.Google Scholar
Szebehely, V.: 1968, Bull. Astron. 3, 91.Google Scholar
Szebehely, V. and Bettis, D. G.: 1970, Astrophys. Space Sci. 13, 365.Google Scholar