Annu. Rev. Astron. Astrophys. 1992. 30: 51-74
Copyright © 1992 by . All rights reserved

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The literature contains two attempts to elevate the halo from the role of a passive byestander or a glum damping force to an active excitor of bending waves. Binney (1978, 1981) showed that orbits in a principal plane of a triaxial halo are in certain circumstances unstable to vertical excitations. This is interesting from the point of view of warps, since the principal axes of the halo that lie in the disk plane provide directions along which the warp's line of nodes may naturally align, thus holding it straight in the face of differential precession.

If the halo has negligible pattern speed, closed orbits are vertically unstable only if the disk's spin axis coincides with the halo's middle axis, and then orbits of every radius are unstable. If the halo has non-zero pattern speed, there is an annulus in which orbits circulating around the halo's shortest axis are vertically unstable. This annulus only lies inside the corotation circle for counter-rotating orbits.

In view of the general instability of orbits around the middle axis of a triaxial halo, it seems unlikely that disks ever form around such axes. As regards the possibilities of unstable annuli, one that lies beyond corotation is not very interesting from the point of view of halos, since their corotation resonances must lie hundreds of kiloparsecs from their centers. So vertical instabilities can be important for warps only if disks generally conter-rotate with respect to the patterns of their embedding halos.

Sparke (1984b) has studied the propagation of bending waves away from disk/halo resonances and concludes that a weakly barred halo of constant shape, flat rotation curve and resonable mass can amplify a small disturbance to an observable warp over a Hubble time provided it is strongly flattened towards the disk and its figure rotates in the opposite sense to that of disk rotation. Warps generated in this way are not unlike observed warps in that the lines of nodes is straight over the first few scale lengths, and then becomes a leading spiral.

Bertin & Mark (1980) have argued that halos can excite bending waves through a more subtle mechanism than Binney's vertical resonance. They reason that forward-running bending waves generate a shadow in the more slowly rotating halo, and that the gravitational field of this shadow drags the waves backwards. Consequently, the bending waves surrender angular momentum to the halo, and, being themselves negative-angular momentum perturbations of the disk, their amplitude grows in response to this loss. Thus Bertin & Mark argued that the torque associated with frictional drag between bending waves and a responsive halo acts in the sense required to further tip over already inclined rings.

Bertin & Mark (1980) analyze this process in the context of tightly-wound WKB bending waves and suggest that warps might be made up of bending waves trapped between corrotation and inner Lindblad resonances, much as spiral structure can be decomposed into similarly trapped spiral waves (Toomre 1981). In this analogy, amplification of bending waves by the Bertin & Mark process takes the place of swing amplification in making good losses suffered by the waves as a result of their excitation of resonant particles, and the leakage of wave energy into the outermost disk, where it is thermalized. Unfortunately this elegant analogy is flawed because bending waves have no corotation resonance; Bertin and Mark were led to believe that they did because they approximated the true dispersion relation Omegap (k) of tightly-wound bending waves (Hunter 1969)

Equation 12a (12a)


Equation 12b (12b)

and later supposed that the right-hand side can become small compared with Omega 2 (Toomre 1983). Since kappa2 appeq 2Omega 2 for the systems of interest, this supposition is clearly false.

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