Abstract
We study the various approximations used to investigate the eigenmode spectrum for systems with highly elongated stellar orbits. The approximation in which the elongated orbits are represented by thin rotating spokes, with the rotation imitating the precession of real orbits, is the simplest and most natural one. However, we show that using this pictorial approximation does not allow the picture of stability to be properly presented. We show that for stellar systems with a plane disk geometry, this approach does not allow unstable spectral modes to be obtained even in the leading order in small parameter, which characterizes the spread of nearly radial orbits in angular momentum. For spherical systems, where the situation is more favorable, the spectrum can be determined but only in the leading order in this parameter. A rigorous approach based on the solution of more complex integral equations given here should be used to properly investigate the stability of stellar systems.
Similar content being viewed by others
References
V. A. Antonov, Astron. Zh. 37, 918 (1960) [Sov. Astron. 4, 859 (1960)].
V. A. Antonov, Vestn. Len. Gos. Univ. No. 19, 96 (1962).
J. Barnes, Dynamics of Star Clusters, IAU Symp. No. 114 Ed. by J. Goodman and P. Hut (Reidel, Dordrecht 1985), p. 297.
A. M. Fridman and V. L. Polyachenko, Physics of Gravitating Systems (Springer, New York, 1984).
D. Lynden-Bell, Mon. Not. R. Astron. Soc. 187, 101 (1979).
D. Merritt, in Proc. of IAU Symp. on Structure and Dynamics of Elliptical Galaxies, Ed. by T. de Zeeuw (Princeton, 1987), p. 315.
A. B. Mikhailovskii, Theory of Plasma Instabilities (Atomizdat, Moscow, 1970), Vol. 1 [in Russian].
V. L. Polyachenko, Pis’ma Astron. Zh. 7, 142 (1981) [Sov. Astron. Lett. 7, 79 (1981)].
V. L. Polyachenko, Pis’ma Astron. Zh. 15, 890 (1989) [Sov. Astron. Lett. 15, 385 (1989)].
V. L. Polyachenko, Pis’ma Astron. Zh. 17, 691 (1991a) [Sov. Astron. Lett. 17, 292 (1991)].
V. L. Polyachenko, Pis’ma Astron. Zh. 17, 877 (1991b) [Sov. Astron. Lett. 17, 371 (1991)].
E. V. Polyachenko, Mon. Not. R. Astron. Soc. 348, 345 (2004).
E. V. Polyachenko, Mon. Not. R. Astron. Soc. 357, 559 (2005).
V. L. Polyachenko and A. M. Fridman, Physics of Gravitating Systems (Nauka, Moscow, 1976; Springer, New York, 1984).
V. L. Polyachenko and E. V. Polyachenko, Astron. Zh. 81, 963 (2004) [Astron. Rep. 48, 877 (2004)].
V. L. Polyachenko and I. G. Shukhman, Preprint SibIZMIR SO RAN SSSR No. 1-72 (Sib. IZMIR SO Ross. Akad. Nauk, 1972).
V. L. Polyachenko, E. V. Polyachenko, and I. G. Shukhman, Pis’ma Astron. Zh. 33, 261 (2007a) [Astron. Lett. 33, 227 (2007)].
V. L. Polyachenko, E. V. Polyachenko, and I. G. Shukhman, Zh. Éksp. Teor. Fiz. 131, 443 (2007b) [JETP 104, 396 (2007)].
E. L. Polyachenko, V. L. Polyachenko, and I. G. Shukhman, Mon. Not. R. Astron. Soc. 379, 573 (2007c).
V. L. Polyachenko, E. V. Polyachenko, and I. G. Shukhman, Pis’ma Astron. Zh. 34, 185 (2008a) [Astron. Lett. 34, 163 (2008a)].
E. L. Polyachenko, V. L. Polyachenko, and I. G. Shukhman, Mon. Not. R. Astron. Soc. 386, 1966 (2008b).
J. Touma and S. Tremaine, Mon. Not. R. Astron. Soc. 292, 905 (1997).
S. Tremaine, Astron. J. 121, 1776 (2001).
S. Tremaine, Astrophys. J. 625, 143 (2005).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © V.L. Polyachenko, E.V. Polyachenko, I.G. Shukhman, 2009, published in Pis’ma v Astronomicheskiĭ Zhurnal, 2009, Vol. 35, No. 2, pp. 100–113.
Rights and permissions
About this article
Cite this article
Polyachenko, V.L., Polyachenko, E.V. & Shukhman, I.G. On various approaches to investigating the instability of stellar systems with highly elongated stellar orbits. Astron. Lett. 35, 87–99 (2009). https://doi.org/10.1134/S1063773709020030
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1063773709020030