Slow pressure modes in thin accretion discs

Abstract

@ABSTRACT Thin accretion discs around massive compact objects can support slow pressure modesof oscillations in the linear regime that have azimuthal wavenumber m = 1. We con-sider finite, flat discs composed of barotropic fluid for various surface density profilesand demonstrate–through WKB analysis and numerical solution of the eigenvalueproblem–that these modes are stable and have spatial scales comparable to the size ofthe disc. We show that the eigenvalue equation can be mapped to a Schr¨odinger-likeequation. Analysisofthis equationshows that all eigenmodes havediscretespectra. Wefind that all the models we have considered support negative frequency eigenmodes;however, the positive eigenfrequency modes are only present in power law discs, albeitfor physically uninteresting values of the power law index β and barotropic index γ.Key words: accretion discs; hydrodynamics; waves; methods: analytical 1 INTRODUCTIONLow-mass discs orbiting massive compact bodies are a fea-ture of many astronomical systems. When the dynamics ofa disc is dominated by the Newtonian gravitational force ofthe central body, the disc may be considered nearly Kep-lerian. In a purely Keplerian potential eccentric orbits donot precess because the orbital frequency is equal to theepicyclic frequency. In a nearly Keplerian disc there is asmall difference between the orbital and epicyclic frequen-cies. This could be due to the self-gravity of the disc, ther-mal pressure in a gas disc, and random motions in a col-lisionless disc. This difference in frequencies manifests as aprecession of eccentric orbits at rates that are small com-pared to the orbital and epicyclic frequencies. Then the discmay be able to support large-scale, slow, lopsided modes(Kato 1983; Sridhar, Syer & Touma 1999; Lee & Goodman1999; Sridhar & Touma 1999). In the linear regime, thesemodes have azimuthal frequency, m = 1, whose first sys-tematic investigation is due to Tremaine (2001). He stud-ied slow modes in various types of discs (fluid, collision-less and softened gravity), with the focus largely on the ef-fect of the self-gravity of the disc. In particular, a WKBanalysis was used to show that the fluid disc can supportlarge-scale slow modes when the Mach number, M, is muchlarger than the Toomre Q parameter (both parameters aredefined in § 2). The assumption behind this analysis is thatthe self-gravity of the disc dominates fluid pressure. How-ever, such is not the case for thin accretion discs aroundwhite dwarfs and neutron stars. Indeed, for a disc around awhite dwarf (Frank, King & Raine 2002), we can estimateM ∼ 50 and Q ∼ 10