Balmforth, N. J.; Howard, L. N.; Spiegel, E. A. Instability of rapidly rotating polytropes. (English) Zbl 0821.35111 SIAM J. Appl. Math. 55, No. 2, 298-331 (1995). Summary: The linear stability theory of rapidly rotating, self-gravitating polytropes is developed by an asymptotic, shallow-layer method. This reduces the general three-dimensional stability problem to an integro- differential eigenvalue problem (a Fredholm integral equation of the second kind) for normal modes. At leading order, the asymptotic analysis produces familiar, zero-thickness disk equations. In subsequent orders, stabilizing effects due to compressibility enter. We solve the stability equations numerically and construct approximate solutions using short- wavelength arguments. The eigenspectrum of a disk can have various forms; instabilities of a pressure-less configuration form a continuous piece of the spectrum, but polytropic disks can have discrete, unstable eigenvalues. A further example is provided by the rapidly rotating disk, which, at leading order, can be solved exactly. MSC: 35Q35 PDEs in connection with fluid mechanics 85A30 Hydrodynamic and hydromagnetic problems in astronomy and astrophysics 35C20 Asymptotic expansions of solutions to PDEs Keywords:astrophysical fluid dynamics; matched asymptotic expansions PDFBibTeX XMLCite \textit{N. J. Balmforth} et al., SIAM J. Appl. Math. 55, No. 2, 298--331 (1995; Zbl 0821.35111) Full Text: DOI