2002/07/11 by P. O. Vandervoort, Peter O. Vandervoort
Physics and Astronomy · #Astronomy and Astrophysical Research #Astrophysics #Classical mechanics #Continuous spectrum #Dispersion relation #Galaxies: Formation, Evolution, Phenomena #Galaxy #Normal mode #Oscillation (cell signaling) #Perturbation (astronomy) #Perturbation theory (quantum mechanics) #Physics #Quantum mechanics #Scientific Research and Discoveries #astro-ph
paper · pdf · doi:10.1046/j.1365-8711.2003.06196.x
published as Mon.Not.Roy.Astron.Soc. 339 (2003) 537 · 18 pages, 1 figure, submitted to the Monthly Notices of the Royal Astronomical Society
arxiv created 2002/07/11 · openalex publication_date 2003/02/21 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
This paper describes a general investigation of stationary oscillations of galaxies. It begins with a linear analysis of modes of oscillation with continuous spectra of real frequencies. Such modes are gravitational analogues of the van Kampen modes of oscillation in plasmas. The characteristic value problem governing modes of the van Kampen type in a galaxy is solved with the aid of a modified version of the matrix method of Kalnajs in which the perturbation of the distribution function is expressed in terms of generalized functions. In general, there is no characteristic equation governing the frequencies in the continuous spectrum. However, isolated frequencies in the continuous spectrum do satisfy a characteristic equation, which, for stellar systems, is a counterpart of the dispersion relation proposed by Vlasov for plasma oscillations. The linear analysis also provides a characteristic equation for modes with a discrete spectrum of real and/or complex frequencies. The second part of the paper describes a perturbation theory for a stationary oscillation of a galaxy with a small but finite amplitude. Integrals of the stellar motion are constructed with the aid of canonical perturbation theory and used in conjunction with Jeans' theorem to specify the density of stars in the six-dimensional phase space. These oscillations are slightly non-linear counterparts of the modes of the van Kampen type, and they are stellar-dynamical counterparts of the non-linear plasma waves described by Bernstein, Greene and Kruskal in the 1950s. Fully non-linear models of stationary oscillations of galaxies can be constructed with the aid of Schwarzschild's numerical method for the solution of the fundamental integral equation describing the self-consistency of a stellar system.