UNSTABLE BAR AND SPIRAL MODES OF DISK GALAXIES

submitted to The Astrophysical Journal We present new analytical and numerical results for modes of at stellar disks which lie in potentials with soft centers. Stars primarily circulate in one direction. We identify two modes of angular wavenumber 2: a more central fundamental mode and a more extensive and more spiral (trailing) secondary mode. The fundamental mode is particularly sensitive to the population of stars of low angular momentum. Depending on that population, a small fraction of the whole, the fundamental mode varies from a small compact bar to a trailing spiral that is almost as wound as the secondary mode. Modes 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 ows outwards through the disk. Few Fourier components contribute signicantly to this transfer. All modes rotate too rapidly to have an inner Lindblad resonance, and are unstable unless there is a suÆciently large external halo or bulge