Continuum eigenmodes in some linear stellar models

Abstract

Abstract. We apply parallel approaches in the study of continuous spectra to adiabatic stellar models. We seek continuum eigenmodes for the LAWE formulated as both finite difference and linear differential equations. In particular, we apply methods of Jacobi matrices and methods of subordinancy theory in these respective formulations. We find certain pressure-density conditions which admit positive-measured sets of continuous oscillation spectra under plausible conditions on density and pressure. We arrive at results of unbounded oscillations and computational or, perhaps, dynamic instability. IntroductionThe problem of radial, adiabatic (isentropic) stellar oscillations is well studied in cases where discrete eigenmodes are calculated in the frameworks of linearized differential equations and in parallel applications of operator theory (see, for instance, [2,3,9,22]).1 However, continuous eigenvalues, although previously suggested in the general literature, are typically absent, either excluded from particular physical models or explicitly avoided to assure dynamic or numerical stability [4,14,17,23] (see also the Appendix of [6]). In this article we investigate some cases where continuous spectra are present which have ramifications on the associated motion -and perhaps on the perturbation method itself. Our study involves linearized perturbations of the differential systemHere, m = M(r) is total mass measured from the origin to distance r from the center of a spherically symmetric star, and P and ρ are pressure and density, respectively, depending on r. Moreover, the differential mass dm is interpreted as that of a spherical mass shell (or mass element) with mean radius r about 1991 Mathematics Subject Classification. 47N50, 85-08, 85-A30. 1 In the general literature (cf. [9]), the problem is often referred to as the LAWE -the linear adiabatic wave equation.

An MHD spectral theory approach to Jeans’ magnetized gravitational instability

“…The spectral theory of linear operators has a wide field of applications in physics. It is at the foundation of quantum mechanics (von Neumann 1955) as well as of MHD (Lifschitz 1989;Goedbloed et al 2019), it is useful in astrophysics (Adam 1986a,b;Winfield 2016) and in Earth seismology (Margerin 2009;Margerin et al 2009), to name but a few examples.…”