.
(3-2)
Early radio observations of the HI disk in our galaxy revealed that the gas layer rapidly thickens and deviates from the galactic plane at large radius (Kerr 1957; Burke 1957). HI observations of M31 found a similar warp, though more pronounced (Newton & Emerson 1977). Observations of M33 find an HI disk so distorted that our line of sight passes through the disk twice in many places (Baldwin 1978). Many other galaxies display warps in their gaseous disk (Sancisi 1976, 1983).
Morphologically, most warps take the antisymmetric form of an horizontal integral sign: on one side of the galaxy, the warp curves above the plane, and on the opposite side it curves below the plane (see Figure 1). Some warps appear bowl shaped (Sparke 1995). Others, like those in the MWG, NGC-5907, and NGC-4565 (Bosma 1983) are asymmetric.
While warps are primarily found in the HI disk, some observations suggest warping of the stellar disk (Innanen et al. 1982; van der Kruit & Searle 1981). Reed (1996) concludes that a stellar warp in the MWG follows the gaseous warp. Sanchez-Saavedra et al. (1990) report a remarkably high occurrence of optical warps (42 of 86 examined) in edge-on galaxies using Palomar Sky Survey plates, but it is not clear that the 23 "barely perceptible" warps are not the projection of nearly edge-on spiral arms or elliptical streamlines (e.g. Byrd 1978). Baldwin (1978) notes that the warps observed in the Local Group contain a lower HI density than those observed in distant galaxies. Many galaxies may contain warps too tenuous to observe. While the actual abundance of warped galaxies is unknown, observations amply demonstrate that warped galaxies are not uncommon.
Briggs (1990) studies 12 galaxies warped in HI and presents a list of the common features.
A leading spiral of nodes can be understood as the trailing edge of a pattern rotating counter to the disk's rotation. The straight line of nodes may be caused by a warp settling into a normal mode. Not surprisingly there exist exceptions to these rules (Bosma 1991), but the rules characterize the "generic" warp.
As a possible fourth rule, Sancisi (1976) indicates that warps appear in isolated galaxies. van Woerden (1979) observes that of five warped galaxies only two possess bright satellites or companions. More observational work and a larger sample size is needed to draw further conclusions.
Because we observe many warps in the present universe, warps must either be excited frequently, or once excited, endure for a great period of time. As Briggs (1990) notes, a warp contains a straight line of nodes inside RHo, implying that differential rotation has not yet wound up the warp. Warps occur at a large radius, therefore the rotation times are long. If the winding time is sufficiently long, then the warps we view today could be ancient warps preserved in situ. We may estimate the winding time by considering a ring of matter at radius r, tilted with respect to the principle plane. The tilt precesses with an angular frequency
wr = O - v (3-1)
where O is the azimuthal angular frequency, and v is the vertical angular frequency. In an oblate halo, v > O (observed in the Galaxy's stars), which means that the ring precesses counter to the disk's rotation. Now consider several tilted rings, extending from r1 to r2, containing a initially straight line of nodes. The line of nodes will wrap into a leading spiral spanning half of the disk in a time
.
(3-2)
We may obtain a naive estimate of the winding time in the limit where the inclination -> 0, and hence w ->O. The circular velocity of the disk, Vc, is 220 km/s and roughly constant. Therefore,
O(R) = Vc/R. (3-3)
Evaluating (3-2) at r1 = 20 kpc and r2 = 25 kpc, the winding time t = ~1.4 Gyr. This increases to ~3 Gyr at R = 30 kpc. Tubbs and Sanders (1979) estimate the winding time as ~1010 years within a halo three times as massive as the disk, but provide few details save that a near-spherical halo is required. Binney (1992) extensively calculates the winding time at RHo to be ~2 Gyr within a spherical halo. He notes that an oblate halo winds faster than a spherical one, quickly erasing any line of nodes. To maintain a straight line of nodes for longer than a few Gyr, some mechanism must be at work.
Like the vibrations of a drum head or plucked string, long-lived warps must correspond to a discrete bending mode to prevent winding or dissipation. The common integral-sign warp has the appearance of a m = 1 mode. Hunter and Toomre (1969; henceforth HT) use running waves to analyze the bending modes of isolated, thin, self-gravitating disks. While HT find that axisymmetric (m = 0) "bowl" modes and anti-symmetric (m = 1) integral-sign modes are stable, their results indicate that waves do not reflect in disks with realistically smooth edges. Instead, the waves continue radially outward into regions of low density and eventually dissipate. In their analysis, disks only contain a discrete mode when the disk's density falls swiftly to zero, forming a unrealistically sharp edge.
This analysis preceded the recognition of massive dark halos. The mass of the halo dominates on the outskirts of the disk, where the disk's self-gravitational response is weak. Toomre (1983) reviews HT's earlier work, lamenting that a halo does nothing to solve the problem of unreflected waves. Dekel & Shlosman (1983) find that a disk, misaligned to the fundamental plane of its halo, warps where the disk's self-gravity fails (Figure 4). Following up on this idea, Sparke and Casertano (1988) find that disks do support one mode insensitive to the details of the disk's edge: a trivial tilt of the entire disk. Within the potential of a oblate halo, a tilted disk precesses and warps. Sparke and Casertano use this "modified tilt mode" to model several warps. However, their results neglect the aligning friction between the disk and halo, a concern originally voiced by Toomre (1983).
Dubinski and Kuijken (1995) investigate the response of a misaligned disk within a massive halo. Contrary to the claims of Dekel & Shlosman (1983), they find that alignment occurs within a few orbital times. Their simulations show that the disk, due to its high angular momentum, realigns the halo in its vicinity, sending a wave of realignment into the outer halo. Only a few Gyr are required to align the halo to greater than R = 20 kpc. Could increasing ellipticity of the outer halo balance the disk realignment at some radius? Is a few Gyr alignment time plus a few Gyr winding time sufficient to preserve many warps?
Like the wake generated by the motion of a satellite through the dark halo, a warp experiences dynamical friction from its precession through the halo. Nelson and Tremaine (1995) indicate that dynamical friction between a warp and the halo generally damps the warp in much less than a Hubble time, perhaps in under a Gyr. Warps may possess a straight line of nodes purely because they do not persist sufficiently long for the nodes to precess.
While disks possess a trivial mode, the mode is unstable due to dynamical friction and the relentless alignment of the disk and halo. Therefore, it is unlikely that primordial warps have endured to the present. To reconcile this transience with the ubiquity of warps, the warps viewed today must have been recently excited.
In the absence of a stable mode, warps require periodic excitation or forcing. Battaner et al. (1990) argue that extra-galactic magnetic torque could twist disks, but Binney (1992) claims that this argument is problematic, citing the required strength of the magnetic field as much larger than observed or easily justified. Nelson and Tremaine (1996) suggest that gravitational noise from halo black holes or dark matter can excite warps. Infalling dark matter may slew the axes of the dark halo (Binney 1992). Likewise, taxation of diffuse luminous matter may slew the disk (E. Ostriker 1991).
Astronomers implicate satellite galaxies in a wide variety of galactic features including starbursts (Hernquist & Mihos 1995), spiral arm formation (Toomre & Toomre 1972), and disk misalignments (Quinn 1986; Hernquist 1991; Walker et al. 1996). Because a misaligned disk will warp where its self-gravity fails (see §3.2.1), the merger of a large satellite on an inclined orbit should generate a warp. However, Sancisi (1976) and van Woerden (1979) indicate that warps appear in isolated galaxies, suggesting that the tidal force of a large satellite is not a necessary condition.
Warped galaxies may only appear isolated because progenitor satellites: a) are too dim; or b) have merged with the disk. We are unable to observe low luminosity objects like dSph outside of the local group, and the lower limit of infalling mass needed to induce a warp is unknown. Phookun et al. (1992) present observations of a spiral stripping gas from a small satellite into an inclined ring; it is instructive to note that this satellite is only 5% as luminous as its primary and would likely have been missed in van Woerden's survey.
Many open questions regarding warps await observations to resolve. Do warped galaxies possess signs of recent infall: multiple nuclei, distorted spiral arms? Do warped galaxies with visible satellites, or signs of recent infall, generally contain asymmetric warps? Do other peculiar features appear in Sparke's (1995) bowl-shaped warps?
The earliest observed HI warp is that of the Milky Way (Kerr 1957; Burke 1957). Recent observations indicate that the warp also exists in young stars (Reed 1996), old stars (Carney & Seitzer 1991), and in infrared (Freudenreich et al. 1994). Seen in Figure 5, the warp is weakly antisymmetric, the greatest height of the HI warp occurs at longitude 90°, where the warp reaches 1.5 kpc above the plane at a distance of 20 kpc. On the other side of the Galaxy, the warp occurs near longitude 270°, at 15 kpc with a height of roughly -1 kpc settling back to zero by 25 kpc (Freudenreich et al. 1994). This corresponds to an inclination of ~4° in the North and ~3° in the South. (The Andromeda warp has an antisymmetric inclination of ~5°.)
Avner and King (1967) conclude that the Large Magellanic Cloud must "create distortions of the galactic plane which in order of magnitude approach the observed distortions." However these results are eclipsed by HT's exhaustive investigation of the LMC and the Galaxy's warp. Limited by the present near-perigalacticon distance of ~50 kpc and a mass of ~1010 Mu, HT indicate that the direct tidal force from the LMC is insufficient, by an order of magnitude, to produce the observed galactic warp. Even possessing a mass of 2 x 1010 Mu, direct force calculations indicate the LMC requires a perigalacticon of ~20 kpc to sufficiently warp the disk.
However, these early analyses neglect the effect of a massive halo. As noted by Mulder (1983), the passage of a satellite though a halo induces a trailing wake of deflected matter (Figure 6). The gravitational field of the wake retards the satellite's motion, producing the deceleration known as dynamical friction. In addition to eroding the satellite's orbit, the wake enhances the satellite's apparent mass to a distant observer. Therefore satellites within a dynamic halo have a greater influence than a direct approximation predicts.
Weinberg (1995) explores this line of inquiry, examining the wake's effect with a perturbation expansion of the collisionless Boltzmann equation. He finds the dominant response of the halo is a barycentric shift; that is, the center of mass of the halo is displaced from the center of mass of the combined system (see Figure 11). The displacement may be considered a region of overdensity, opposite the barycenter from the satellite but orbiting in the same sense. While the distance to the Clouds produces a significant displacement (~8 kpc in Weinberg's LMC/MWG system), the ratio of the orbital times of the disk at ~8 kpc and the displacement is between 1:7 and ~1:13 (assuming a semi-major axis of ~75 kpc for the Clouds). This low ratio and the displacement's polar orbit are not conducive to strong coupling. He estimates that the halo response increases the effective mass of the satellite by a factor of two.
In a system similar to the Milky Way, employing a satellite with a mass 30% that of the disk, Weinberg finds that the satellite induces a slight symmetric warp with a maximum amplitude of 0.45 kpc at ~25 kpc. The warp possesses an orientation and sign equivalent to that observed in the Milky Way. While Weinberg concludes that a pure tidal excitation of the Galaxy's warp requires a higher LMC mass or closer perigalacticon than can be justified by observations, he expresses optimism that resonance in a self-consistent model might provide the additional force.