ARlogo Annu. Rev. Astron. Astrophys. 1992. 30: 51-74
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Within the optical disk the halo rarely contributes as much as half the mass and radial force (e.g., Kent, 1988). So treating the halo as a rigid, unreponsive body as Sparke & Casertano (1988) do, is a very doubtful approximation. What one really should do is treat the disk and the halo together as a single dynamical system and seek normal modes of the whole in which the disk is warped. Such a calculation would be very taxing, and may still be beyond the reach of even officianados of normal-mode calculations. So can we guess its likely outcome? If there is a mode of the specified type, it would surely amount to a systematic precession of the inner part of the system with respect the surrounding envelope. Indeed, the inner halo is dynamically more tightly coupled to the disk embedded in it than it is to the outer halo. So disk and inner halo may be expected to precess together. This situation differs significantly from that envisaged by Sparke & Casertano, in which inner and outer halo are at all times aligned. Moreover, as the work of Mathur (1990), Weinberg (1991) and Louis (1992) have shown, the modes of hot stellar systems are no more likely to be discrete than those of an isolated disk. In particular, systems like real disk/halo galaxies in which the characteristic frequencies of stars span a great range from the center to the periphery, are not expected to exhibit any discrete modes at all. So it would be very surprising if any realistic combined disk-halo system had a discrete mode in which the inner portion precesses as a rigid body.

If the warping modes of disk/halo systems all lie in the contiunuum, any warp will inevitably Landau damp away. The questions are then: (a) ``How fast?'' and (b) ``What configurations will damp most slowly?''

Dekel and Shlosman (1983) argue from a study of the orbits of certain halo objects in a precessing disk potential that the damping time is longer than the Hubble time. However, no reliable calculation of the damping time seems to be available, and its evaluation via semi-analytical tools such as dynamical friction is bound to be a hazardous enterprise. To see this, consider the effect of a slow warp on the halo in the frame that rotates with the warp's pattern. In this frame the halo will stream in the same sense as the disk rotates unless in an inertial frame it displays significant counter-rotation, which is unlikely. Hence the enhancement in the halo's density to which the warp gives rise - let's call it the warp's ``shadow'' - will be displaced from the warp itself in the direction of stellar rotation. Consequently, the torque applied by the halo will have a small component along the line that is perpendicular to both the halo's symmetry axis and the line of nodes (see Figure 1). The sense of this perpendicular component of torque is not self-evident. But suppose it acts in the opposite sense to that shown in the figure. Then it will tend to tip the disk's angular momentum vector more over towards the negative y-axis than it already is, thus enhancing the warp. Conversely, if the perpendicular component of torque acts in the sense shown, it will tend to align disk and halo and thus diminish the warp.

Figure 1

Figure 1. Schematic representation of a disk tilted with respect to an axisymmetric halo. The z-axis is the halo's symmetry axis and the x-axis the line of nodes. The shaded region marks the disk's ``shadow'' of halo particles. In the frame in which the line of nodes is stationary, the halo rotates in the same sense as the disk and the centre of the shadow is displaced from the y-axis in the sense of rotation. Conseqnently there is a component of torque about the y-axis. If this component is in the sense shown, it tends move the disk's spin axis towards the z-axis.

Which way does the perpendicular component of torque act? If the shadow is global, as Weinberg's (1989) work on satellites suggests, the raised portion of the disk at positive y will produce an enhancement in the halo's density at positive z and negative x, and this will torque the disk in the sense marked in Figure 1, causing the warp to damp. But the validity of this picture is by no means clear, and, indeed, a calculation of Bertin and Mark (1980 - see below) suggests that the shadow may excite rather than damp the warp.

If the damping rate is uncertain, can we say anything about the likely form of the longest lived configurations? Surely the fast modes of systems with rigid halos will have no long-lived counterparts in the real world of responsive halos. Indeed, relative motion of the central halo and the disk will damp quickly in consequence of the short dynamical time-scale near the center. Consequently, the only long-lived warped configurations will be those in which the disk and inner halo precess as a single rigid body with respect to the outer halo. Such configurations will resemble slow warps in that they involve the outer disk dragging a reluctant inner disk around with it at something less than the outer disk's already low natural precession frequency.

This is precisely the situation Toomre (1983) and Dekel & Shlosman (1983) advocated at the 1982 Besançon meeting. These authors pointed out that in this picture there is a useful analogy between the configuration of the disk and the Laplace plane of planetary theory: Moons orbiting near an oblate planet tend to decay into orbits lying in the planet's equatorial plane. Far from the planet the quadrupole moment to which a moon is exposed is dominated by the Sun and planets, and lunar orbits decay into the ecliptic plane. The transition between these regimes is rather abrupt because it is associated with competition between rival quadrupole moments. So a disk of massless moons would consist of two nearly planar annuli joined by a sharp warp. In the Toomre-Dekel-Shlosman picture, warps are associated with the abrupt transition from the orientation of the inner galaxy (the ``planet'') to that of the outer halo (the ``ecliptic'').

The key question at this point would seem to be whether a visible galaxy can precess for a Hubble time as a reasonably rigid unit at the center of a misaligned halo. It should be possible to answer this question with security from suitably designed n-body experiments.

Fascinating as the problem of stellar warps is, we should not forget that warps are fundamentally a gaseous phenomenon. The velocity fields of warped galaxies are generally remarkably regular, indicating that the gas is moving on circular, if tilted orbits. The gas has undoubtedly been driven to these circular orbits by dissipation. Is dissipation expected to drive the gas into a fundamental plane of the potential at the same time as it confines the gas to circular orbits? Most hydrodynamical simulations (Tohline et al. 1982; Habe & Ikeuchi, 1988; Pearce & Thomas, 1991) answer this question in the affirmative: gaseous material settles to rings aligned with a principal axis of the confining potential rather than tipped rings. On the other hand, Quinn (1991) finds that if the gas is concentrated in clouds of such compactness that the cloud-cloud collision time is comparable with or greater than the orbital time, inclined rings can form. Also Katz & Rix (1992) find that sufficient cooling and spatial resolution will allow a warped, tilted disk to form in a hydrodynamical simulation provided the initial angular momentum vector of the gas lies almost within the symmetry plane of the potential - that is, providing the gas starts from nearly polar orbits. So the verdict here is mixed; if the gas behaves like gas, it will usually be quickly driven by dissipation to a fundamental plane. But if the gas becomes extremely lumpy or there is no suitable fundamental plane handy, it may settle into inclined orbits, which will then evolve as for a stellar system.

In the absence of an embedding hot atmosphere, one would not expect gas pressure to modify significantly the linear propagation of bending waves that have wavelengths long compared with the thickness of the gas layer; in such waves gravity is the only significant source of vertical acceleration. This expectation is confirmed for gas layers of negligible mass by the analytic study of Nelson (1981) and the numerical simulations of Johns & Nelson (1987). Bertin & Casertano (1982) have considered the potential of a hot embedding medium to excite bending waves in a gaseous sheet. However, their results, which were intended to serve as a guide to stellar dynamical processes and not to model real gas layers, are not directly applicable to real gaseous disks since the disk and halo are assumed to move freely through one another.

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