Normal modes of Bardeen discs - II. A sequence of n = 2 polytropes

Abstract

We study in detail the normal modes of a sequence of differentially rotating, inviscid polytropic fluid discs in the Bardeen approximation. The equilibrium isentropic self-gravitating discs have comparable pressure and rotational support. At the low-angular-momentum end of the sequence, discs are secularly and dynamically stable; at the high end they possess two dynamically and two extra secularly unstable modes. We identify two types of modes, called p- and r-modes, for which we introduce a classification scheme based upon a number of criteria, including winding numbers in phase diagrams. Most of the r-modes have corotation points and seem to be finite-dimensional approximations to a continuous spectrum, caused by differential rotation, but we also observe p-modes with corotation points which seem to remain discrete modes even inside the continuous spectrum. In fact, dynamical instability sets in through such modes, in a disc in which the ratio τ of kinetic to potential energy is 0.27. Secular instability to gravitational radiation sets in via a zero-frequency p-mode when τ = 0.1221. The r-modes are never unstable. Apart from obvious effects of differential rotation, the qualitative behaviour of eigenfrequencies along this sequence bears a striking resemblance to that along the Maclaurin sequence of incompressible fluids. In an appendix we describe an inexpensive test of the numerical accuracy of eigenfrequencies.