Normal modes of self-gravitating fluids in perturbed configurations. I - Perturbational-variational procedure. II - Perturbational-variational expansion of the g- and p-modes of a non-adiabatic fluid about the adiabatic limit

Astronomy and Astrophysics – Astrophysics

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Adiabatic Flow, Calculus Of Variations, Eigenvalues, Fluid Flow, Gravitational Effects, Perturbation Theory, Eigenvectors, Rayleigh-Ritz Method, Series Expansion, Stellar Structure, Variational Principles

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A perturbational-variational Rayleigh-Ritz (PV-RR) expansion scheme is developed for systematically obtaining the normal modes of small oscillations of self-gravitating fluids in perturbed configurations in powers of an external perturbing parameter that characterizes the eigenvalue equation. Some properties of the generalized perturbed eigenvalue equation are reviewed, a Rayleigh-Ritz ansatz for the eigendisplacements is introduced, and a concise matrix formulation is developed. Appropriate expressions are derived for the PV-RR expansions of the eigenvalues and the eigendisplacement vectors to arbitrarily high perturbation orders. The normal (g and p) modes of a nonadiabatic fluid are then expanded about those of an adiabatic structure. The procedure of Silverman and Sobouti (1977) is used to obtain a series expansion of the eigenfrequencies and eigendisplacements. This procedure is also extended to incorporate the bispectral (g and p) nature of the normal modes into the formalism.

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