Slow pressure modes in thin accretion discs

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 $\gamma$.