We present idealized models of a razor–thin , axisymmetric , Keplerian stellar disc 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 .
The linear normal modes of waterbags are composed of sinusoidal disturbances of the edges of distribution function in phase space .
Waterbags which include circular orbits ( polarcaps ) have one stable linear normal 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 normal 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 eccentricities .