Modal analysis of gravitational instabilities in nearly Keplerian,
  counter-rotating collisionless discs
  We present a modal analysis of instabilities of counter-rotating,
self-gravitating collisionless stellar discs, using the recently introduced
modified WKB formulation of spiral density waves for collisionless systems
(Gulati \& Saini). The discs are assumed to be axisymmetric and in coplanar
orbits around a massive object at the common center of the discs. The mass in
both discs is assumed to be much smaller than the mass of the central object.
For each disc, the disc particles are assumed to be in near circular orbits.
The two discs are coupled to each other gravitationally. The perturbed dynamics
of the discs evolves on the order of the precession time scale of the discs,
which is much longer than the Keplerian time scale. We present results for the
azimuthal wave number $m=1$ and $m=2$, for the full range of disc mass ratio
between the prograde and retrograde discs. The eigenspectra are in general
complex, therefore all eigenmodes are unstable. Eigenfunctions are radially
more compact for $m = 1$ as compared to $m = 2$. Pattern speed of eigenmodes is
always prograde with respect to the more massive disc. The growth rate of
unstable modes increases with increasing mass fraction in the retrograde disc,
and decreases with $m$; therefore $m=1$ instability is likely to play the
dominant role in the dynamics of such systems.
