Abstract We discuss the linear theory of non-axisymmetric normal modes in self-gravitating gaseous discs. These instabilities occur when the disc is stable to axisymmetric modes. They can have co-rotation situated either inside or outside the disc. The profile of the ratio of vorticity to surface density is found to be important in determining the properties of the normal modes. These modes may be important for redistributing the angular momentum in the disc.

Introduction

Discs and rings in which the internal self-gravity plays an important role are important in astronomy. Examples are the rings around Saturn and Uranus, (Goldreich & Tremaine 1982), and the Milky Way and other spiral galaxies (Toomre 1977, 1981). They may also exist around active galactic nuclei and T. Tauri stars. In both of these cases, instabilities may be important for driving mass accretion and angular momentum transport (Paczynski 1977, Lin & Pringle 1987). An understanding of non-axisymmetric instabilities is clearly important because they may play a significant role in determining the structure and evolution of all of these objects. In this paper we discuss the linear theory of stability as applied to self-gravitating gaseous discs. We find various kinds of instabilities, some of which are generalizations of those found in the non-self-gravitating case (Papaloizou & Pringle, 1984, 1985, 1987). These are essentially due to the unstable interaction of waves on either side of co-rotation. However, when self-gravity is included, there are other modes which have co-rotation outside the system.