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P. O. Vandervoort (Univ. Chicago)
This paper describes a theory of the modes of oscillation of a galaxy with a continuous spectrum of real frequencies. Such modes in an inhomogeneous system are generalizations of the van Kampen modes in the theory of plasma oscillations. In the present case, the characteristic value problem governing the modes is solved with the aid of a generalized version of the matrix method of Kalnajs. The Eulerian perturbations of the densities and the gravitational potentials are represented as superpositions of the elements of a biorthonormal set of basis densities and potentials. The linearized collisionless Boltzmann equation is solved in terms of action-angle variables. As in the case of the van Kampen modes in a homogeneous plasma, the solutions for the Eulerian perturbations of the distribution function are expressed in terms of generalized functions. The condition for the self-consistency of a mode requires that the density perturbation of the response of the system to the perturbation of the gravitational potential is equal to the density perturbation that is the source of the perturbation of the potential. The condition of self-consistency reduces to an inhomogeneous system of linear, algebraic equations governing the constant coefficients in the superpostions of basis functions representing the perturbation. 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 inhomogeneous systems, is a counterpart of the dispersion relation derived by Vlasov for plasma oscillations.
The author(s) of this abstract have provided an email address for comments about the abstract: voort@oddjob.uchicago.edu