Bending instabilities in stellar systems

We discuss the physics of bending instabilities in inhomogeneous stellar systems and propose a simple criterion for stability. We derive the bending modes of a family of thin disk models and show that long- wavelength modes in thin systems with realistic density profiles are not always stabilized by gravity, in contrast to an infinite sheet. We also present the results of N-body experiments with finite-thickness disks that confirm instability to bending even when the velocity anisotropy is much less extreme than the critical value for instability in an infinite slab. We suggest that the most important mechanism that stabilizes bending in realistic models is the out-of-phase response of stars that encounter the bend with a frequency greater than their free vertical oscillation frequency Kᵦ_. This mechanism successfully accounts for our N-body results, and for the behavior of the infinite slab at short wavelengths, where gravity can be neglected. It further predicts stability at all wavelengths to longitudinal bending modes for pressure- supported systems in which the ratio of orbital oscillation frequencies is less extreme than about 2: 1, that is, for which the isodensity contours are rounder than about 1: 3. (Oblate systems can be flatter if the velocities are azimuthally biased. ) The latter number is in agreement with the behavior of bending modes in N-body models and in uniform spheroids, as well as with the absence of elliptical galaxies flatter than about E7.