6.1. Summary of the main results
We have written the non-linear coupling coefficients between a spiral and two warps, taking into account the finite thickness of the galactic disk and the three dimensional motion in a warp. The efficiency of this mechanism is too weak in the stellar disk, except at its outer edge where the spiral slows down near its OLR and can be efficiently coupled to warps and extract them from the noise level, transferring to them a sizable fraction of its energy and angular momentum (with the rest transferred to the stars by Landau damping). As a net result we can consider that the spiral is transformed into two warps. One of them receives a third of the energy of the spiral and can propagate only outward. The other receives the remaining two thirds of the energy of the spiral, and splits into one wave traveling outward and one traveling inward, which we tentatively identify as the corrugation observed in many galaxies.
Order of magnitude estimates of the respective amplitudes of the outer warp (in the HI layer) and of the inner one (the corrugation), as well as of their wavelengths, lead to values in good agreement with observations. In particular we find from the observed amplitudes that the energy and angular momentum fluxes of the spiral and the warps are comparable, a result which receives a natural explanation in our model.
The expected long wavelength of the outer warp is in agreement with the observed trend to forming a straight line of nodes. Furthermore, our mechanism justifies the phase discontinuity of the warp at crossing the Holmberg radius (since the Holmberg radius is expected to be close to the Outer Lindblad Resonance of the spiral, where our mechanism takes place).
6.2. What about the halo ?
In the preceding sections we have not mentioned the role of a massive halo. We will not discuss a misalignment between the disk and the principal plane of the halo, since such a misalignment was introduced in an ad hoc model by Sparke and Casertano (1984) in order to explain the continuous excitation of the warp in isolated galaxies. Our mechanism does not require such an hypothesis.
A typical halo is about ten times more massive than the disk. If this halo has a spherical symmetry, it does not affect the vertical restoring force and thus leaves our results unchanged. On the other hand a flattened halo would give rise to a strong restoring force for a test particle (or a disk) displaced from the equatorial plane. The vertical oscillation frequency µ would be about three times the epicyclic frequency. Hence the forbidden band of Figs. 4-6 would invade more than the whole disk. Our analysis would then have to be altered, but we believe that its main result would remain valid.
Indeed, in order to find two bending waves that would propagate in the vicinity of the OLR of the spiral, we might invoke the "slow" branch of the warp dispersion relation, which uses intensively the compressional behavior of the gas (let us emphasize that we have already made use of this slow branch to find the wavelength of corrugation, which is in the same manner a bending wave of the gas in the deep potential well of the stars). This would not strongly change the expected efficiency of our mechanism, and would even rather increase it slightly, since we would be dealing with slower modes. Another solution would be to consider the pair of warps , which are both simultaneously propagating (with the former being retrograde). The efficiency of coupling with such a pair would be, in our framework (shearing sheet and steady state) exactly the same as the one we derived for the pair . Now the details of the scenario would change a little. We would have only one wave emitted outward and two inwards, but the order of magnitude of their amplitudes would not change. Note that observations of corrugation in external galaxies (Florido et al. 1991a) seem to reveal two distinct wavelength for corrugation waves.
Another matter concerns the response of the halo to the warp. A fully consistent treatment of this problem should be numerical. However, it is possible to get analytical trends by considering the halo as a thicker (and then warmer) disk than the stellar/gaseous disk. This can be done using the two-fluid dispersion relation of warps (see Masset and Tagger 1995). The two-fluid dispersion relation reveals three modes:
Fig. 8 shows that the participation of the halo to the global modes (i.e. the two waves corresponding to the one-fluid ones) is as strong as or stronger than that of the disk. Furthermore the wavelengths of these modes agree with the observations of a straight line of nodes. However, this figure has been obtained in the WKB assumption and moderately thick disk formalism (Masset and Tagger 1995), which are both far too rough in this case, so that a numerical solution taking into account the halo response would be necessary.
In the same manner, if we want to estimate the coupling efficiency taking into account the halo, a problem is a determination of for the spiral involving a participating halo. If we assume a behavior of matter independent of z, then should still be about 0.2, but once again only a numerical solution would allow us to give the real vertical profile of perturbed quantities in the spiral and in the warps. Furthermore, the problem is much more complex if we want to treat it fully correctly, since for distances to the galactic center less than the Holmberg radius we have to cope with three fluids: the halo, the stars and the gas, so the method which consists in treating the disk as a unique fluid will give only approximate results.
Nevertheless, despite the large number of allowed modes of propagation of bending waves in this realistic case, we see that global modes (i.e. involving a motion of the halo) will lead to the same efficiency as found in this paper. Numerical simulations would be needed to give an accurate evaluation of the e-folding length and to determine the fraction of incident energy that each wave carries away.
Finally, let us mention that the halo should be a collisionless fluid (i.e. star-like and not gas-like) if we still want the spiral to slow down at the OLR. Observations suggesting that the halo or outer bulge is composed of very low mass stars would confirm this hypothesis (see e.g. Lequeux et al. 1995).
6.3. Suggested observations for checking the spiral-warps coupling mechanism
In this paper we have shown that the spiral wave is converted into bending waves observed as an outer HI warp and a corrugation wave. We have already emphasized the observational fact that the fluxes of these three waves have the same order of magnitude for a typical galaxy. It would be interesting to test this coincidence with a better accuracy, and to search for a correlation between the fluxes of the spiral, the warp and the corrugation. The fluxes can be straightforwardly obtained from such observables as the disk surface density, the rotation curve (which gives values for and µ), the disk thickness, the perturbed surface density of the spiral (see e.g. Strom et al. 1976), and the deviation of the disk to the equatorial plane - although some of these quantities are better measured in face-on galaxies, and others in edge-on ones, making a complete set of observations in an individual galaxy a challenge. A correlation between the fluxes for a significant number of galaxies would be a strong argument for our mechanism.
© European Southern Observatory (ESO) 1997
Online publication: July 3, 1998