TIME EVOLUTION OF GALACTIC WARPS

We have followed the time evolution of a warped galactic disk that is initially tilted with respect to an oblate halo potential. If the disk has a discrete normal mode of bending, the disk settles toward the mode shape, from the center outwards: the settled inner warp has a straight line of nodes, while the outer warp eventually twists into a leading spiral. Even if the disk as a whole has no discrete bending mode, the inner part still settles into a coherent warp without spirality. The timescale for settling may be estimated from the group velocity of bending waves; propagation times in the very outer disk exceed a Hubble time. We have run models with parameters corresponding to the observed systems NGC 4013, NGC 4565, NGC 2903, NGC 3198, and NGC 2841; just as with the discrete normal modes, comparing our time-evolving models to observed warps allows us to place constraints on the core radius and on the oblateness of the dark halo.