Unstable $m=1$ modes of counter-rotating Keplerian discs
  We study the linear $m=1$ counter-rotating instability in a two-component,
nearly Keplerian disc. Our goal is to understand these \emph{slow} modes in
discs orbiting massive black holes in galactic nuclei. They are of interest not
only because they are of large spatial scale--and can hence dominate
observations--but also because they can be growing modes that are readily
excited by accretion events. Self-gravity being nonlocal, the eigenvalue
problem results in a pair of coupled integral equations, which we derive for a
two-component softened gravity disc. We solve this integral eigenvalue problem
numerically for various values of mass fraction in the counter-rotating
component. The eigenvalues are in general complex, being real only in the
absence of the counter-rotating component, or imaginary when both components
have identical surface density profiles. Our main results are as follows: (i)
the pattern speed appears to be non negative, with the growth (or damping) rate
being larger for larger values of the pattern speed; (ii) for a given value of
the pattern speed, the growth (or damping) rate increases as the mass in the
counter-rotating component increases; (iii) the number of nodes of the
eigenfunctions decreases with increasing pattern speed and growth rate.
Observations of lopsided brightness distributions would then be dominated by
modes with the least number of nodes, which also possess the largest pattern
speeds and growth rates.
