Abstract
This chapter is devoted to the relation between the convective stability in a star and the dynamic stability of the star with respect to non-radial isentropic normal modes. The condition that N 2(r) ≥ 0 at all points of a star is proven to be not only a sufficient but also a necessary condition for the eigenfrequencies of the star’s linear, non-radial, isentropic, spheroidal normal modes being real. The proof has been established by Lebovitz. In the supposition that the inner-product-space of the linear, isentropic normal modes of a star is complete, the condition that N 2(r) ≥ 0 everywhere in the star is then a sufficient and necessary condition for the dynamic stability of the star with respect to non-radial isentropic normal modes.
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Notes
- 1.
It seems …that in its most general form the question of the dynamic stability is related to the stability with respect to convective currents, with which the criterion of K. Schwarzschild is associated …
If this condition is not satisfied in a region of the star, convective currents must appear there. In the usual way to establish this criterion, one considers only a single small element of matter that is displaced from its equilibrium position and one completely neglects the effects of this perturbation on the rest of the star. When one tries to take account of these effects and the motion of the other elements in the unstable layer, the problem becomes very complex.
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Smeyers, P. (2010). N 2(r) Nowhere Negative as Condition for Non-Radial Modes with Real Eigenfrequencies. 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_14
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DOI: https://doi.org/10.1007/978-3-642-13030-4_14
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