  The standard Q criterion (with Q > 1) describes the local stability of a disc
supported by rotation and random motion. Most astrophysical discs, however, are
under the influence of an external gravitational field which can affect their
stability. A typical example is a galactic disc embedded in a dark matter halo.
Here we do a linear perturbation analysis for a disc in an external field, and
obtain a generalized dispersion relation and a modified stability criterion. An
external field has two effects on the disc dynamics: first, it contributes to
the unperturbed rotational field, and second, it adds a tidal field term in the
stability parameter. A typical disruptive tidal field results in a higher
modified Q value and hence leads to a more stable disc. We apply these results
to the Milky Way, and to a low surface brightness galaxy UGC 7321. We find that
in each case the stellar disc by itself is barely stable and it is the dark
matter halo that stabilizes the disc against local, axisymmetric gravitational
instabilities. This result has been largely missed so far because in practice
the value for Q for a galactic disc is obtained in a hybrid fashion using the
observed rotational field that is set by both the disc and the halo, and hence
is higher than for a pure disc.
