Secular Instabilities of Keplerian Stellar Discs around a Massive Black Hole.

We present idealized models of razor--thin, axisymmetric, Keplerian stellar discs around a massive black hole, and study non-axisymmetric secular instabilities in the absence of either counter-rotation or loss cones. These discs are prograde mono-energetic waterbags, whose phase space distribution functions are constant for orbits within a range of eccentricities (e) and zero outside this range. Waterbags which include circular orbits (polarcaps) have one stable linear edge-mode for each azimuthal wavenumber m. The m=1 mode always has positive pattern speed and, for polarcaps consisting of orbits with e < 0.9428, only the m=1 mode has positive pattern speed. Waterbags excluding circular orbits (bands) have two linear edge-modes for each m, which can be stable or unstable. We derive analytical expressions for the instability condition, pattern speeds, growth rates and normal mode structure. Narrow bands are unstable to modes with a wide range in m. Numerical simulations confirm linear theory and follow the non-linear evolution of instabilities. Long-time integration suggests that instabilities of different m grow, interact non-linearly and relax collisionlessly to a coarse-grained equilibrium with a wide range of e.