Slow m=1 instabilities of softened gravity Keplerian discs
  We present the simplest model that permits a largely analytical exploration
of the m=1 counter-rotating instability in a "hot" nearly Keplerian disc of
collisionless self-gravitating matter. The model consists of a two-component
softened gravity disc, whose linear modes are analysed using WKB. The modes are
slow in the sense that their (complex) frequency is smaller than the Keplerian
orbital frequency by a factor which is of order the ratio of the disc mass to
the mass of the central object. Very simple analytical expressions are derived
for the precession frequencies and growth rates of local modes; it is shown
that a nearly Keplerian disc must be unrealistically hot to avoid an
overstability. Global modes are constructed for the case of zero net rotation.
