Slow pressure modes in thin accretion discs

TL;DR

This paper investigates slow pressure oscillation modes in thin, flat, barotropic accretion discs around compact objects, demonstrating their stability, discrete spectra, and conditions for positive or negative frequency modes.

Contribution

It provides a detailed analysis of the stability and spectral properties of pressure modes in finite accretion discs using WKB and numerical methods, including a Schrödinger-like mapping.

Findings

Modes are stable with spatial scales comparable to the disc size.

Eigenvalue spectra are discrete for all considered models.

Negative frequency eigenmodes exist in all models; positive modes are limited to certain power-law discs.

Abstract

Thin accretion discs around massive compact objects can support slow pressure modes of oscillations in the linear regime that have azimuthal wavenumber $m=1$. We consider finite, flat discs composed of barotropic fluid for various surface density profiles and demonstrate--through WKB analysis and numerical solution of the eigenvalue problem--that these modes are stable and have spatial scales comparable to the size of the disc. We show that the eigenvalue equation can be mapped to a Schr\"odinger-like equation. Analysis of this equation shows that all eigenmodes have discrete spectra. We find that all the models we have considered support negative frequency eigenmodes; however, the positive eigenfrequency modes are only present in power law discs, albeit for physically uninteresting values of the power law index $\beta$ and barotropic index $\gamma$.