**3.1. Gravitational stability**

The conditions of stability of a dissipation-less component like the stellar one, is relatively well known. High enough velocity dispersions are required to suppress axisymmetric instabilities and even spirals, bars or z-instabilities. The latter provide heating to the medium which becomes un-responsive. The gas is dissipative and has a completely different behaviour; it is always unstable, and its velocity dispersion is fixed by regulation and feed-back.

The local stability criterion has been established by
Toomre (1964):
stabilisation is obtained through pressure gradients
(velocity dispersion *c*) at small scale, smaller than
the Jeans length:
= *ct*_{ff} =
*c*^{2} / *G*,
where is the
disk surface density.
At large scale, the rotation stabilises through centrifugal
forces, more precisely the scales larger than
L_{crit} = 4^{2}
*G* /
^{2}
(where is the epicyclic frequency).
The Safronov-Toomre criterion is obtained in
equalling and L_{crit}:

For one component, a radial mode
(, *k*) in a
linear analysis obeys the dispersion relation (where *k* is the
wave number):

which means that
self-gravity reduces the local frequency
.
If a stability criterium is easy to derive for one component, the same is
not true for a two-components fluid, since the coupling between the
two makes the ensemble more unstable than each one alone.
The one-component criterion can be applied separately
to the stellar and gas components, where the corresponding values of
and *c* are used, leading
to *Q*_{s} and *Q*_{g}.
Because
*c*_{g} << *c*_{s} however, only a
small percentage of mass
in gas can destabilize the whole disk, even when *Q*_{s}
> 1 and *Q*_{g} > 1.
For two fluids with gravitational coupling, the dispersion relation yields
a criterion of neutral stability
^{2} = 0:

which gives directly
an idea of the relative weight of gas and stars in the instabilities
(cf Jog 1996).
At low k (long waves), essentially the stars contribute
to the instability, while at
high k (short waves), the gas dominates.
The neutral equilibrium requires the simultaneous solution of
d^{2} (k) / dk = 0,
which leads to a
system of 2 polynomial equations 4th and 3rd order in k, and
no analytic criterion can be derived; there are only numerical solutions
in terms of a two-fluid
*Q*_{s - g} value, which is always lower than the
*Q*_{s} or *Q*_{g}
values. Q_{s - g} = 1 defines the neutral stability,
by analogy with the one-fluid treatment, at the fastest growing
F= 2/(1 + Q_{s - g}^{2}).
The stability depends strongly on the gas mass
fraction (between 5 and
25%), and there can be sharp transitions from high
to low values of , as the
mass fraction increases from
= 0.1 to 0.15 for
instance. When *Q*_{s} is high and *Q*_{g} is
low (a frequent situation, given the dissipation and cooling in the gas),
the main instability is at small-scales.
A gas-rich galaxy (with
> 0.25) is only stable
at very low
surface densities (this explains the Malin 1-type galaxies,
Impey & Bothun
1989);
the center of early-type galaxies, where *Q*_{s} is
high, and
*Q*_{g} is low (due to low gas velocity dispersion), could
be dominated
by the gas wavelength, even at very low gas mass fraction: this can
explain spiral arms in galaxies such as
NGC 2841
(Block et al 1996).
Also interaction of galaxies, by bringing in a high amount of gas, may
change abruptly the fastest growing wavelength from
_{s} to
_{g} , and trigger
star-formation.
Even beyond the neutral stability
criterium, when *Q*_{s - g} > 1, a galaxy disk can be
unstable with respect
to non-axisymmetric perturbations, such as bars or spirals; in this case
also, the two-fluid coupling increases the instability, i.e. the disk
will form a bar, even if the stars or the gas alone are stable with
respect to such perturbations
(Jog 1992).