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

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2. NAIVE THEORY

In the simplest approximation HI clouds move as test particles in a fixed, axisymmetric galactic potential Phi(R, z). Since they have random velocities ltapprox 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 nu = sqrt[ð2Phi / ðz2]. Meanwhile its azimuthal coordinate phi circulates with angular frequency Omega = R-1/2Phi / ðR)1/2. Since for an oblate potential nu > Omega, the angle at which the cloud crosses the equatorial plane regresses with angular frequency omegar = nu - Omega.

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

Equation 1 (1)

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

Equation 2 (2)

it is easy to show that

Equation 3 (3)

where kappa2 = R-3R4 Omega 2 / ðR) is the square of the radial epicycle frequency. Equation 3 shows that omegar is intimately connected with the density rho far out where warps are observed; warps are a potentially powerful probe of dark halos.

For a flat rotation curve, Omega propto 1 / R, we have kappa2 = 2 Omega 2 and thus that

Equation 4 (4)

Here sqrt[4 pi Grho (R)] is the value the circular frequency Omega would take if the galaxy were perfectly spherical and the density at general spherical radius r were (R / r)2rho (R, 0). We expect this to differ from the actual circular frequency Omega (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 Omega (R) is independent of q, while rho (R, 0) is inversely proportional to q. Thus we have omegar (R) = Omega (R)(1 / sqrtq - 1).
(b) At the radii of interest, an exponential disk contributes significantly to Omega 2 (R) but negligibly to rho (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)]

Equation 5 (5)

while the circular speed vh of a spherical halo satisfies

Equation 6 (6)

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

Equation 7 (7)

One frequently identifies vc with vd's peak value

Equation 8 (8)

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

Equation 9 (9)

This gives, for example, omegar (4Rd) appeq 0.16 Omega and omegar (6Rd) = 0.05 Omega.

In real life omegar 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 gtapprox 4Rd unless the halo is remarkably spherical.

Typically R25 ltapprox 4Rd. So in the region R25 ltapprox R ltapprox RH0 in which Briggs finds the line of nodes straight, we have twrap appeq pi / omegar (4Rd) appeq 6pi / Omega ltapprox 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 omegar 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)

    Equation 10 (10)

    where Sigma 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 gtapprox 40 kpc, where Sigma 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.

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