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Linear Isentropic Oscillations of Stars pp 1–8Cite as

Introduction

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Part of the book series: Astrophysics and Space Science Library ((ASSL,volume 371))

Abstract

The study of the stellar oscillations is the preeminent way for the investigation of the stability, and the interpretation of the variability of stars, as it has strikingly been stated by Ledoux (1963a):

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Notes

  1. 1.

    As we cannot make soundings in a star which would reveal us its internal constitution and chemical composition directly, we are restricted to construct stellar models whose macroscopic characteristics, mass M, luminosity L, radius R, or effective temperature T e, must be comparable to those of the real stars and satisfy various relations, relations M−R, L−R or L−T e, revealed by the observations …

    The diagram of Hertzsprung-Russell …is one of the most effective means to summarise a great number of properties of a stellar population …

    Not only the static models and the theory of stellar evolution must render the values of M, R, L observed for individual stars and the relations existing between these quantities, but also, for the majority of the stars, these models must be stable. So, one can consider the various criterions …as additional conditions imposed on the models and, consequently, as as many guides in their elaboration.

    On the other hand, …many …stars …are not stable. And first all intrinsic variable stars whose variations must have an origin in some form of instability capable of manifesting itself for a small perturbation that will tend to grow until nonlinear factors stabilise it at a finite amplitude …

    In that way the stability criteria also furnish a natural approach to the study of the origin of at least plus or minus important and plus or minus regular variations of which a large number of stars are the seat …

    Essentially two approaches exist for the discussion of the stability of a system. We can apply a small perturbation to it, so small that, in the equations that govern the evolution of this perturbation, we can content ourselves with keeping only the terms of the first degree in this perturbation. The equations so obtained are said to be linearised and their solution describes the behaviour of the perturbation.

    … This naturally lets escape cases of instability with respect to finite perturbations.

    The other method is related to the principle of Dirichlet according to which the equilibrium state of a conservative mechanical system is stable if it corresponds to a minimum of the potential energy …This last method has not much been used in the stellar problem.

  2. 2.

    Ritter devoted a particular attention to the pulsation problem. In Chap. …I have investigated the oscillations of gaseous spheres, which correspond to those of incompressible fluids …Ritter investigates another class of oscillations, those by which the compressibility of the gas appears to full advantage; the particles move along the radius, the density remains constant in concentric layers, and the sphere keeps its spherical form. The problem is solved for small oscillations on the assumptions that 1. the sphere is of constant density, 2. the sphere remains of constant density during the pulsation, 3. the particles follow Poisson’s equation p v γ=  const. during their motion.

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    Paul Smeyers

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Smeyers, P. (2010). Introduction. In: Linear Isentropic Oscillations of Stars. Astrophysics and Space Science Library, vol 371. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13030-4_1

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