**13. SELF-GRAVITATING HYDRODYNAMICS**

The formation of rings by gas accumulation at resonances, through the action of bar gravitational torques, has been illustrated beautifully by non-self-gravitating simulations of sticky particles. We have seen that somewhat different results are obtained when considering the gas as a continuous fluid, submitted to pressure forces and shocks. The differences between the two gas models are amplified by taking into account self-gravity. In the fluid picture, the fast infall of gas towards the center produces large central mass concentrations that modify the whole dynamics. This can perturb the orbital structure of the stellar component, shift the resonances, and destroy the bar (e.g., Friedli & Benz 1993). At this stage, the consideration of star formation is necessary, as a process that can lock the gas into a non-dissipative phase, and stop the gas infall (Heller & Shlosman 1994).

**13.1. Stability**

The gas component, even though it represents only a few percent of the
total mass, has a considerable influence on the global stability of a
galaxy. The gravitational coupling between gas and stars makes the
ensemble unstable, even when each component would have been separately
stable (e.g.
Jog & Solomon 1984;
Bertin & Romeo 1988;
Elmegreen 1994b).
The stability of a one-component infinitely thin stellar disk
has been widely studied: the criterion introduced by
Toomre (1964)
indicates that stellar disks are stable against axisymmetric
perturbations if the ratio *Q* of their radial velocity dispersion
*c _{r}* to the critical dispersion

The stability of a two-fluid system with respect to axisymmetric perturbations has been studied by Jog & Solomon (1984), who showed that the star-gas system will be unstable if:

where *k* is the wave number, *µ*_{*} and
*µ*_{g} are the surface
densities, and *c*_{*} and *c _{g}* are the
velocity dispersions of stars
and gas, respectively. This formula emphasizes the relative
contribution of the two components, proportional to the surface
density, and weighted with the inverse of the velocity dispersion. This
shows why even a small fraction of gas, with a tiny velocity
dispersion, can de-stabilize the ensemble.
Bertin & Romeo (1988)
and Romeo (1992)
have shown that there exist two stability domains,
corresponding to the two bumps of the

Jog (1992) also studied the growth of non-axisymmetric instabilities in a two-fluid system, and concluded again that the ensemble can be unstable, even if each component is separately stable with respect to non-axisymmetric perturbations.

The crucial importance of gas comes from its dissipation. In galaxy evolution, gravitational instabilities heat the disk, but the gas component remains cool, radiating away the excess heating due to the waves. The spiral structure can then be continuously renewed (e.g. Miller et al. 1970; Sellwood & Carlberg 1984), even in the stellar disk. This is supported by observations, since spiral galaxies always possess gas, and early-type systems without gas are smooth and axisymmetric. Gas also has a damping effect on growing instabilities, because of its non-linear dissipation in shocks, and can then play a self-regulating role in spiral structure, as stressed by Bertin & Romeo (1988). While the gas is triggering spiral and bar instability when it represents only a few percent of the total mass, it can play the reverse role when its mass is above 10%. It is then unstable even to axisymmetric local perturbations. When the gas possesses too much self-gravity, it forms lumps through Jeans instability, and the lumpiness of the gas can scatter the stars, randomize their motions, and prevent any bar formation (Shlosman & Noguchi 1993).