Though there have been suggestions that magnetic fields coupling to a slightly ionized ISM could explain the observed warps (Battaner et al. 1991) we will concentrate here on models invoking gravitational processes. The fact that regular disk galaxies show no major discrepancies between stellar and gaseous rotation curves (beyond the well-understood asymmetric drift) indicates that magnetic fields are in any case not a dominant contributor to large-scale galaxy dynamics. Moreover, in the Milky Way, recently Alard (2000) has shown that the stellar disk shares in the warp seen in the HI.
I will briefly discuss three scenarios for warps: the normal mode models, accretion of angular momentum, and interaction with satellites.
3.1. Normal Modes
The normal mode picture of warps (Toomre 1983; Dekel & Shlosman 1983; Sparke & Casertano 1988) envisions the disks as embedded in a flattened dark halo. If the disk is misaligned with the equator of the halo, the resulting torque combined with the disk's spin will cause it to precess about the halo minor axis. This precession rate is easily calculated (Kuijken 1991) as the ratio of the torque from the halo (determined from the circular and vertical frequencies H and H in the potential of the halo) to the spin of the disk (governed by the total circular frequency c):
where is the surface density of the disk at radius R.
Because different parts of the disk have a different free precession frequency in the potential of the halo, without disk selfgravity the warp would wind up and disappear, analogously to spiral perturbations in massless disks. However, the disk does have selfgravity, and Sparke and Casertano were able to show that solutions exist in which the disk precesses as a single, warped, unit in the halo. When the precession frequency difference across the disk is too large, or the disk is too lightweight, these solutions do not exist.
The problem with normal modes was already forseen by Toomre (1983), who pointed out that a crucial assumption in this picture is the neglect of backreaction from the halo to the precession of a massive disk in its center. This effect, analogous to dynamical friction, was investigated by Dubinski & Kuijken (1995), Nelson & Tremaine (1995), and Binney et al. (1998) after earlier investigation in terms of WKB density waves by Bertin and Mark (1980). It turns out to be very important: the precession frequencies implied by equation 1 are so low compared to the dynamical time of the inner halo that there is ample time for the inner halo to adjust itself to the reorientation of the disk. This aligned inner halo results in a diminished torque, which in turn further reduces the precession speed. The overall effect is an alignment wave which propagates outwards through the disk, as the precession grinds to a halt. The alignment between disk and halo is rather fast, on the order of three local orbital times.
Figure 1. N-body simulation of the evolution of an inclined disk in a live dark halo (Dubinski & Kuijken 1995). The azimuth from which the disk is viewed is indicated in each panel. Note that the precession very quickly comes to a halt, before even a quarter of a precession orbit has been completed.
Figure 2. The evolution of the inclination of the disk, the inner halo and the outer halo with time in the simulation shown in fig 1. The inner halo and the disk quickly align; over a longer time also the outer halo inclination approaches that of the disk.
Under some conditions, the close coupling between halo and disk can instead lead to a fast excitation. This can happen, for example, if the halo rotates retrograde with respect to the disk, or if it is prolate (Nelson & Tremaine 1995).