A bowl-shaped mode of galactic disks

A thin self-gravitating galactic disk in an axisymmetric halo potential can have a discrete bowl-shaped mode of vertical oscillation which is not sensitive to details of the disk edge. The mode arises as a modification of the trivial mode of an isolated disk consisting of a displacement perpendicular to its plane; it is neutrally stable, neither growing nor decaying. The discrete mode has no radial nodes, and it becomes increasingly curved at the outer edge as the halo core radius shrinks. A disk that is given an initial axisymmetric vertical perturbation settles toward the discrete mode in the inner regions, while developing corrugations in the outer parts. Axisymmetric warping modes may explain the "banana" shapes observed in a number of polar ring systems, and systematic effects seen in the analysis of the velocity fields of spiral galaxies.