Though their inner regions are generally flat, many disk galaxies exhibit significant warping at larger radii. To account for such warps, disk galaxies may be embedded in massive, oblate, misaligned dark halos.
Warps are very common. For example, the Milky Way, M31, and M33 are all warped (e.g. Binney 1992); the warp of M33 is so strong that some lines of sight intersect the disk more than once (MB81, Fig. 8-29).
In edge-on galaxies such as NGC 5907 such warps are seen as so-called `integral sign' contours bending symmetrically away from the plane of the disk (e.g. MB81, Fig. 8-30). In disks seen more nearly face-on warps are revealed by characteristic distortions of the iso-velocity contours (MB81, Fig. 8-27) such that the kinematic principal axes remain roughly perpendicular at all radii. Such distorted velocity maps can be modeled by treating the disk as a collection of concentric circular rings, each tilted by a small amount with respect to its neighbors (MB81, Fig. 8-28).
Unlike the spirals of disk galaxies, which occur in a profusion of forms, galactic warps obey some fairly simple rules (Briggs 1990):
There are at least two reasons why the kinematics of extended gas in disk galaxies is especially useful in studying warps. First, atomic hydrogen may be detected far beyond the optical radii of galactic disks, so the gas provides the best constraints on the gravitational field at large radii. Second, the gas is usually moving on closed orbits, which greatly simplifies the analysis of the observations.
To explain the warps observed in galactic disks, start by considering a ring which has been warped by displacing particles in the z direction by an amount proportional to exp(i m phi), where phi is the angular coordinate in the disk plane and m is the azimuthal wavenumber. If the vertical displacements are small the particles will execute harmonic motions with vertical frequency kappa_z (see BT87, Ch. 3.2.3). The net result of the combined rotation and vertical oscillations is that the ring pattern rotates about the z axis with a pattern speed
(7) Omega_p = Omega - kappa_z/m ,where Omega is as usual the angular velocity of a circular orbit (Toomre 1983).
In a spherical potential kappa_z = Omega and so any warp pattern constructed out of non-coplanar rings will persist unchanged and unchanging. But in an oblate potential, kappa_z > Omega, so an m = 1 disturbance precesses backwards with pattern speed Omega_p = Omega - kappa_z. Unless this Omega_p is the same at all radii, the warp pattern will wind up with time, much like the winding-up of kinematic spiral density waves. The good alignment of neighboring rings within R_26.5 (rule #2 above) can't be explained if individual rings freely precess.
If warps are long-lived structures, affairs must be arranged so that Omega_p does not change with radius. Ways to do do this include
The longevity of warps can be plausibly explained if disk galaxies have misaligned halos (Toomre 1983, Dekel & Shlosman 1983). Misalignments can arise if dark halos formed by collisionless gravitational clustering provide the potential wells in which visible galaxies accumulate dissipatively (White & Rees 1978), because dark and luminous components have very different dynamical histories. In a high-density universe, galaxies would be accreting matter right up to the present, and the late-arriving material is likely to prefer a rather different rotation axis (e.g. Binney 1990).
At large radii the halo presumably dominates the gravitational field and the tenuous outer disk should settle into the principal plane of the halo, while at small radii the gravitational attraction of adjacent rings of disk material tends to enforce a coplanar geometry. Thus a disk which is misaligned with respect to its halo should slowly precess as a unit out to a few times the disk scale length r_0, and then go smoothly but rather quickly over to the plane of the halo (e.g. Toomre 1983, Fig. 2). A numerical model which treats the disk as warped surface and the halo as a rigid potential finds a discrete warped mode which does not depend on the details of the disk edge (Sparke & Casertano 1988). This model does an impressive job of reproducing the warp of NGC 4013.
For this explanation to work several requirements must be met (Sparke & Casertano 1988):
The picture of warps as products of misaligned halos might be refined in several ways:
Last modified: April 8, 1997