Discussion

Comparison with N-body Simulations

The predictions of W98 were checked by several groups using N-body simulations. Garcia-Ruiz et al. (2002) used a hybrid N-body particle-ring code and did not find the warp predicted in W98. They offer the failure of the linear theory as the culprit but did not investigate a variety of models. Similarly, Mastropietro et al. (2005) performed a simulation including both the gaseous, stellar and dark components of the Large Cloud and the Milky Way and remark that the effect on the Milky Way is negligible. Conversely, Tsuchiya (2002), using a hybrid code that includes both a potential expansion and a tree code, obtained amplitudes of m = 0,1,2 warps in the larger halo model that show very good agreement with the observed amplitudes in the Milky Way. He also shows that the amplitude of the warp depends strongly on the dark halo mass model.

It is difficult to reconcile these contradictory findings but three possibilities obviously occur: 1) the linear theory does not apply; 2) the simulations do not apply due to numerical difficulties; and 3) the two simulations with null warps have chosen unlucky sets of parameters. Both Garcia-Ruiz (2002) and Tsuchiya (2002) choose methods that explicitly treat the multiple scales. Garcia-Ruiz (2002) used tilted rings to represent the disk and a tree code to represent the halo. Although the use of rings limits the investigation to the $m = 1$ response, it should be sufficiently sensitive to the halo excitation without strong particle number issues. Tsuchiya (2002) used an expansion algorithm to represent the halo gravitational field and a tree code to represent the disk to better represent the multiple scales (see Tsuchiya 2002 for additional discussion of the dynamic range problems that motivate this choice). We feel that the discrepancy between the results of these two groups is most like Item (3): the Garcia-Ruiz et al. model is not particularly warp producing. The good qualitative correspondence between W98 and Tsuchiya (2002) suggests that the linear theory captures the underlying physics although is likely to differ in detail. For example, the linear theory over-predicts the height in the edge of the disk, and this motivates our truncation of the predictions in the outer disk. Mastropietro et al. (2005) primarily emphasize the affect of the Milky Way tides on the Large Cloud and do not tailor their approach to treat the multiple scale problem per se. We suggest that their report of no disk warp results from Items (2) and (3).

Other Explanations

Most warp theories depend on the bending response of the disk and this response is strongly affected by the existence of a self-gravitating dark matter halo. The theories may be roughly grouped as follows: 1) bending modes may be primordially excited and persistent (e.g. Sparke & Casertano 1988); 2) the response of the disk to the non-axisymmetric shape of the halo; 3) tidal excitation (as we have discussed here) and 4) the warp may be produced by the response of the disk to cosmic infall (e.g. Jiang & Binney 1999). The topic was nicely reviewed by Binney (1991). Subsequently, Nelson & Tremaine (1995) argued against long-lived modes. The triaxiality of the Milky Way halo remains uncertain. Helmi (2004) reports a prolate halo (q=1.25) while Johnston (2005) finds an oblate halo (q=0.8-0.9) using Sgr dwarf constraints. Such modest triaxiality seems unlikely to produce the observed vertical m=2 feature. In addition, cosmic infall more naturally produces tilted rings, an m=1 feature; the observed vertical m=2 may require a conspiracy of several inflow directions.

Relevance for Modified Gravity

Although the success of our model favors the existence of a dark-matter halo in Nature, many find it seductive to modify gravity to produce the observed rotation velocities in the Galaxy without dark matter (modified Newtonian dynamics or MOND, Milgrom 1983ab, Bekenstein & Milgrom 1984). Might the tidal theory also apply in MOND? It is beyond the scope of this letter to repeat our calculations using MOND, it seems plausible that direct forcing of the Galactic disk by the Clouds in MOND may provide warp amplitudes in excess of the original predictions without a dark halo (HT). Similarly, we would expect the disk modes and frequencies to be qualitively similar, the excess restoring force of the halo being produced by the MOND force. However, MOND would have to (1) admit bending modes with similar morphology to those in the Newtonian theory and (2) these modes would have to conspire to frequencies that couple them to the direct forcing by the Clouds in such a way that they assumed the same orientation as in the "clockwork" described in Section 5. Because this clockwork and the m=2 to m=1 amplitude ratio depends on the simultaneous halo and disk excitation, agreement with the observations described in Section 2 seems rather unlikely and thereby disfavors MOND. We encourage detailed predictions.

Relevance for other Galactic Systems

A large fraction of other warped galaxies also show warps in their HI layers and a significant fraction of these are asymmetric. Warps in these systems are generally analyzed with a program such at ROTCUR (Begeman 1989) or one of its derivatives, but invariably are forced to fit the m = 1 warp only. This paper, and the work of Levine et al. (2005) show that at least three harmonics ought to be fit to the warps of external galaxies, especially those with asymmetric warps, which can be caused by a superposition of these harmonics. The degree to which various harmonics are present in a warp can produce important constraints on whether the warp is due to a satellite, a triaxial halo, cold gas inflow, or some primordial excitation. All warps may not be alike.

Summary

We have demonstrated a plausible mechanism for the excitation of the warp that explains all of its general features. A number of warp producing mechanisms have been explored elsewhere in addition to tides. It is possible that different mechanisms may act in different galaxies and possibly in concert in a single galaxy. Nonetheless, our simple model nicely reproduces the observed features in the Milky Way HI gas layer. The existence of massive companions, the Magellanic Clouds, and the prediction from linear perturbation theory and at least one corroborating N-body simulation suggests that a tidal explanation is viable. Our model depends on the gravitational response of the halo and thereby suggests that the dark matter is not an artifact of modifying the laws of gravity. Conversely, given a dark halo, we argue that the tide from the Magellanic Clouds must be affecting the Milky Way disk, and, given the quality of the agreement, it seems to be the dominant mechanism. This model then promises an additional constraint on the distribution of dark matter. Although we have emphasized the gas layer response beyond the stellar disk, the effect of other satellite encounters such as the recent accretion of Sagittarius dwarf may be detectable in future high-resolution surveys and may help determine the properties of the inner halo. Warp observations are important because promise to reveal aspects of the dark matter distribution that are otherwise observationally inaccessible. Further analysis of warps may provide more precise constraints on the profile of dark matter in the Milky Way and nearby external galaxies.

Acknowledgements

L.B. would like to acknowledge partial support from NSF grant 02-28963. M.D.W. has been partially supported by NASA NAG5-12038 and NSF 02-05969. We thank Arend Sluis and the anonymous referee for comments on the manuscript


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Martin Weinberg <weinberg@astro.umass.edu>
Last modified: Sat Nov 5 23:33:09 EST 2005