Unstable Bar and Spiral Modes of Disk Galaxies
  We study bisymmetric modes of angular wavenumber 2 for flat stellar disks in potentials with smooth cores. Stars either all circulate in the same direction or a small fraction may counter-rotate. The bisymmetric modes are unstable unless there is a sufficiently large external halo or bulge. We find two modes for each disk: a more central fundamental mode and a more extensive and more spiral (trailing) secondary mode. The fundamental mode is particularly sensitive to the orbital population in the central part of the disk. Depending on that population, it varies from a small compact bar to a trailing spiral that is almost as wound as the secondary mode. All modes rotate too rapidly for there to be an inner Lindblad resonance. They transfer angular momentum from the central to the outer regions of the disk. Most of them release gravitational energy and convert it to kinetic energy, which also flows outwards through the disk. 
M.A. Jalali and C. Hunter