2. NAIVE THEORY

In the simplest approximation HI clouds move as test particles in a fixed, axisymmetric galactic potential (R, z). Since they have random velocities 10 km s-1 (van der Kruit & Shostak 1982), they may be assumed to move on nearly circular orbits and their motion may be followed in the epicycle approximation. If a cloud's orbit is inclined to its galaxy's equatorial plane, its height z above this plane then oscillates harmonically at angular frequency = sqrt[ð2 / ðz2]. Meanwhile its azimuthal coordinate circulates with angular frequency = R-1/2 / ðR)1/2. Since for an oblate potential > , the angle at which the cloud crosses the equatorial plane regresses with angular frequency r = - .

Suppose a warp extends from R1 to R2 and initially has a straight line of nodes. Then the line of nodes will wrap into a leading spiral which wraps half way round the galaxy in a time

(1)

How big is r? Writing Poisson's equation for an axisymmetric system in the form

(2)

it is easy to show that

(3)

where 2 = R-3R4 2 / ðR) is the square of the radial epicycle frequency. Equation 3 shows that r is intimately connected with the density far out where warps are observed; warps are a potentially powerful probe of dark halos.

For a flat rotation curve, 1 / R, we have 2 = 2 2 and thus that

(4)

Here sqrt[4 G (R)] is the value the circular frequency would take if the galaxy were perfectly spherical and the density at general spherical radius r were (R / r)2 (R, 0). We expect this to differ from the actual circular frequency (R) for two reasons:

(a) The halo may be flattened to axis ratio q < 1. Then the mass M (R) within cylindrical radius R required to generate (R) is independent of q, while (R, 0) is inversely proportional to q. Thus we have r (R) = (R)(1 / q - 1).
(b) At the radii of interest, an exponential disk contributes significantly to 2 (R) but negligibly to (R, 0). In fact, for R > 4 Rd, where Rd is the disk's scale length, the circular speed vd of an exponential disk approximately satisfies [e.g., equation (2P-5) of Binney & Tremaine (1987)]

(5)

while the circular speed vh of a spherical halo satisfies

(6)

Equating to zero the derivative w.r.t. R of vc2 = vh2 + vd2 and then making the approximation vd2 = GMdisk / R yields

(7)

One frequently identifies vc with vd's peak value

(8)

so that vd2 (R) (vc2 / 0.642) (Rd / R). Making this identification one has

(9)

This gives, for example, r (4Rd) 0.16 and r (6Rd) = 0.05 .

In real life r will receive contributions from both halo flattening and the presence of an embedded disk. Given the strong dependence on R of the second contribution, halo flattening will dominate at R 4Rd unless the halo is remarkably spherical.

Typically R25 4Rd. So in the region R25 R RH0 in which Briggs finds the line of nodes straight, we have twrap / r (4Rd) 6 / 2Gyr even in a spherical halo. How do lines of nodes avoid winding up on this or a shorter time-scale? Evidently some simplifying assumption has invalidated the calculation. Some possible culprits are:

1. The assumption that the halo's ellipticity q is radius independent. Petrou (1980) pointed out that r would not decline outwards, and the warp would not wind up, if q were to decrease outwards in the right way. Unfortunately, there seems to be no reason why in every warped galaxy q should vary with radius in the approved fashion.

2. Neglect of non-gravitational forces. Perhaps gaseous disks warp in response to pressures applied by an intergalactic medium. The necessary pressure may be hydrodynamical in origin (Kahn & Woltjer, 1959), or due to a magnetic field (Battaner et al., 1990, 1991). With simple hydrodynamical pressure it is difficult to generate an integral-sign warp; the dominant response to the IGM's ram pressure of a disk moving through the IGM will be axisymmetric, taking the form of a rim rather than a warp. The anisotropy of magnetic pressure in relatively stationary, magnetized IGM can in principle generate an integral-sign warp, but the magnitude of the field required to distort a disk near R25 is implausibly large: a simple calculation leads to the estimate (Binney 1991)

(10)

where is the warped disk's surface density, z0 the vertical scale of the warp, and Trot is the rotation period at the radius of the warp. Given that a field of 2.8 µG has the same energy density as the cosmic microwave background, it is clear that magnetic forces will be unimportant except possibly at R 40 kpc, where is extremely small.

3. The halo has been assumed to be axisymmetric, whereas it is probably triaxial. Is the warp's line of nodes somehow aligned with the halo's principal axes? This might arise because the orbits of clouds are forced in a systematic way by the triaxiality of the halo's potential.

4. The disk's potential has been assumed to be fixed. Actually the warp modifies the potential, and one must take this into account by modelling the disk as a self-gravitating collisionless system. This is most conveniently done by asking what running bending waves the disk supports. Alternatively (but not equivalently) one can look for normal modes of oscillation.

5. The halo potential has been assumed to be an inanimate, unresponsive thing. The orbits of the halo's constituent particles must respond to the variations in the disk's gravitational potential to which the warp gives rise. This greatly complicates the already difficult mathematical problem posed by study of bending waves and normal modes in self-gravitating disks.

6. Galaxies have been assumed to be closed, isolated things. In reality they inhabit groups and clusters and sometimes experience close encounters with companions. Moreover, unless we live in an open Universe, cosmic infall onto galaxies must be an important process. Could warps arise from the response of galaxies to changing external circumstances?

Suggestions (3)-(6) are closely connected and raise problems of formidable difficulty which are far from being adequately understood. Many of them require an understanding of the dynamical excitations of self-gravitating disks.