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Astron. Astrophys. 318, 747-767 (1997)

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5. Comparison with observations

In this section we derive the expected characteristics of the warps produced by the coupling process. Let us first summarize the main results obtained in the previous section. The reader can also refer to Fig. 7 where the situation is depicted.

  • The spiral wave travels outward until it reaches its OLR. Before it reaches it, the coupling with warps is too weak to affect the spiral.
  • When it reaches the OLR (more precisely within a distance of the order of [FORMULA] from the OLR) the spiral slows down and is efficiently coupled with warps. This results in the "conversion" of the spiral into three warps which carry the incident flux of the spiral away from the coupling region: two warps at frequency [FORMULA], traveling respectively outward and inward, and the last warp at frequency µ, which is emitted outward.
  • The flux absorbed from the spiral is approximately equally shared by the three warps, for reasons of conservation of angular momentum and energy.
  • In the following we will neglect the absorption of the spiral wave by Landau damping. Thus our results on the warps amplitudes will be optimistic, but still give results of the correct order of magnitude since the length scale of Landau damping and non-linear coupling are similar.
[FIGURE] Fig. 7. This figure shows the spiral wave and the three warp waves excited near its OLR by non-linear coupling. See the text for more details.

5.1. Amplitude of the warp

We will now derive a rough prediction of the warp displacement resulting from our mechanism. For each of the three warps we can write:

[EQUATION]

where the factor [FORMULA] comes from the result (see Sect. 4.2.2) that the three warp waves share nearly equally the flux extracted from the spiral (the rest of the flux of the spiral being absorbed by Landau damping, as discussed in Sect. 4.3).

In order of magnitude, the flux of the spiral is given by:

[EQUATION]

Similarly, the fluxes of the warps emitted outward are:

[EQUATION]

In these expressions, [FORMULA] is the unperturbed surface density of the stellar disk, where the spiral propagates, and [FORMULA] is the unperturbed surface density of the external HI disk, where the two emitted warps propagate.

Now if we adopt a radial dependence of [FORMULA] such as given by Shostak and Van der Kruit (1984), that we will very roughly describe by :

[EQUATION]

where [FORMULA] is the radius of the optical limit of the disk, d the decaying length of surface density (a sizable fraction of [FORMULA]), we can deduce an approximate radial dependence of the deviation Z. We find :

[EQUATION]

[EQUATION]

and

[EQUATION]

(see e.g. Huchtmeier and Richter 1984 or Shostak and Van der Kruit 1984).

and we take as a value for the amplitude of the spiral [FORMULA] (but it could be larger than this widely adopted value, see Strom et al. 1976). This leads us to:

[EQUATION]

Of course [FORMULA] decreases progressively as the warps propagate outward, and Z accordingly grows to preserve the flux.

The dependency found above is typical of what is observed in isolated spiral galaxies. It is noteworthy - and we believe that this has not yet been noticed - that the flux of the spiral waves is of the same order of magnitude as the flux of the warps. We consider this coincidence as a strong point in favor of our mechanism - independently of its details -, and a challenge to any alternate model.

5.2. Expected characteristics of the corrugation

The computations concerning the reflected warp are slightly different. One should remember that it propagates in a region where both stars and gas are present, a fact which cannot be neglected when discussing its characteristics. In a previous paper (Masset and Tagger, 1995) we discussed the propagation of warps in a two-fluid disk. We found two types of waves: the first type corresponds to waves where both fluids move essentially together, while the second type corresponds to waves where the fluids move relative to each other; assuming that the stellar disk is much warmer and much more massive than the gaseous one, in this type of wave the stars stay essentially motionless while the gas moves in their potential well. We consider that this wave, which we call the corrugation mode, will be excited much more easily than the previous ones, since it moves much less mass and thus its amplitude for a given energy must be much higher. Its dispersion relation is:

[EQUATION]

where the subscripts g and [FORMULA] apply to gaseous and stellar values, and where [FORMULA] is the stellar density at midplane. Because of the moderate thickness of the stellar disk, the square frequency [FORMULA] is much larger than the other characteristic frequencies of the problem, and in particular larger than the frequency [FORMULA] of the wave which can be non-linearly excited. A solution can nevertheless be found because inside the OLR of the warp the last term in the dispersion relation is negative, and can for large enough values of q balance the dominant term [FORMULA]. Neglecting the self-gravity of the gas, and writing [FORMULA] (a realistic assumption since the ratio of these quantities is about one tenth), we find:

[EQUATION]

Assuming a [FORMULA] vertical stellar density profile (i.e. the disk is dominated by the local stellar gravity, which is a reasonable assumption), this can be rewritten as:

[EQUATION]

Taking [FORMULA], and [FORMULA] of the order of one kiloparsec, we find an order of magnitude of one kiloparsec for the wavelength of the corrugation. This is consistent with the observed wavelength for the corrugation, in the Milky Way (Quiroga and Schlosser 1977, Spicker and Feitzinger 1986) or in other galaxies (Florido et al.  1991a). Despite the observations by Spicker and Feitzinger of a one kiloparsec corrugation in our galaxy, it seems that the typical wavelength of corrugation is rather about 2 kiloparsecs, or even larger (Florido et al.  1991b). The discrepancy of [FORMULA] between observations and our estimate could be due either to the magnetic field which affects the gas motions and thus should imply the use of magnetosonic speeds instead of the acoustic speed, or to a lack of resolution of some observations. From equation (6), we find that the wavelength of corrugations increases as we approach the edge of the optical disk, in good agreement with observations.

Now let us give an order of magnitude of the expected amplitude of the corrugation. As already explained, we can write that its flux is about one third of the spiral flux. Now the flux of the corrugation is roughly given by (the index c is for corrugation):

[EQUATION]

where:

[EQUATION]

since [FORMULA], and:

[EQUATION]

Hence:

[EQUATION]

which gives, taking into account [FORMULA]:

[EQUATION]

Taking [FORMULA] and [FORMULA], we obtain an expected amplitude for the corrugation of about one hundred parsecs, which is typically the observed value. Note that, as discussed above, energy can also be given to the other modes, involving both the gas and the stars, for the reflected warp. Their expected wavelength is larger than the disk radius, so that their line of nodes should be almost straight, corresponding to the observations of Briggs (1990), and so that the corrugation can easily be detected when superimposed on such large scale bending waves.

5.3. Straight line of nodes

We have found thus far that our mechanism leads to results that fit well the observations for the amplitudes of the "outer" warps (the two waves emitted beyond the OLR of the spiral) and of the corrugation, and for the wavelength of the corrugation. We now turn to the wavelength of the outer warps. Actually the line of nodes of the observed warps is always nearly straight, leading to radial wavelengths larger than the galactic radius. This is known as the problem of the straight line of nodes, and is one of the most serious challenges to models of warps. In his work on the "Rules of Behavior of Galactic Warps", Briggs (1990) has emphasized this remark: the line of nodes of the outside warps appears nearly straight or with a slight spirality, always almost leading. Later observations have shown that outside warps could also be slightly trailing, but always nearly straight. This can be interpreted from our mechanism, which predicts the emission of two warps outward from the OLR. In fact, the frequency of the first warp is µ, leading from the dispersion relation to a wavevector [FORMULA], hence a straight line of node. The second one is emitted at [FORMULA], i.e. near its OLR. From the dispersion relation of warps taking into account the compressibility effects near the OLR (see e.g. Masset and Tagger 1995), we see that [FORMULA] must tend to zero in order to balance the behavior of the resonant denominator, thus also leading to a straight line of nodes.

Of course this is a qualitative and simple interpretation, since we use results obtained in the framework of the WKB assumption (tightly wound waves) where [FORMULA] is precisely not allowed to become small. Hence this discussion about the straight line of nodes is just the very beginning of a more complete work which should take into account non-WKB effects and the cylindrical geometry. This work is in progress, using the numerical code introduced in Masset and Tagger 1995. Finally, we should mention that the combination of two warps outside OLR can easily justify the large and arbitrary jump in the line of nodes across the Holmberg radius observed by Briggs, since the relative phase between warps 1 and 2 is arbitrary.

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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998
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