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Astron. Astrophys. 350, 423-433 (1999)

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6. Discussion

We confront our models with the radio polarization observations at [FORMULA]3.5 cm (Fig. 1) which are almost free from Faraday effects.

6.1. Radial variation of the polarized intensity

The observed polarized synchrotron intensity is concentrated in the central region and decreases strongly within a few kpc from the center (Fig. 17). If the density of cosmic-ray electrons is constant, the total synchrotron intensity scales roughly with [FORMULA] and the polarized intensity with [FORMULA] (see Beck et al. 1996 for details), where [FORMULA] and [FORMULA] are the strengths of the total and regular fields, respectively. In this case, none of our models can explain the strong central concentration of the polarized emission. If, on the other hand, the energy density of cosmic rays is in equipartition with the magnetic energy density, the scaling is [FORMULA] and our models predict a strong radial decrease of polarized intensity. However, the distribution in our `standard model' (Fig. 6) is still too broad. To achieve a sufficiently narrow concentration of [FORMULA], we need differential rotation of NGC 6946 from a few 100 pc radius until the outer galactic disk, as in the rotation curve by Sofue (1996) (see Fig. 2).

[FIGURE] Fig. 17. Radial variation of the mean expected polarized intensity (represented by [FORMULA] where [FORMULA] is the mean strength of the regular field) in the galactic mid-plane for our `best model', using Sofue's rotation curve (see Fig. 2), a radially constant turbulence intensity of 15 km s-1 and a correlation time of 0.02 Gyr (arm) and 0.01 Gyr (interarm). Observational data at [FORMULA] cm are given as stars; their errors are invisibly small

In the rotation curve by Carignan et al. (1990) differential rotation is so small within the inner kpc that the [FORMULA]-[FORMULA] dynamo cannot operate. For strong dynamo excitation an [FORMULA] dynamo is still possible, but no magnetic arms between the optical arms are expected in this regime (see Rohde & Elstner 1998). More velocity data, e.g. in CO lines, are required to obtain an accurate rotation curve.

At inner radii ([FORMULA] kpc) our models show a minimum of the magnetic field strength, inconsistent with the observations (Fig. 17). The reason is the insufficient approximation of Sofue's rotation curve in the very central region (Fig. 2). Dynamo models with increased spatial resolution in the inner region are necessary to achieve more realistic results at the very inner radii.

6.2. Radial variation of the magnetic pitch angle

Our `standard' model reveals a radial increase of the absolute value of the magnetic pitch angle, averaged over all azimuthal angles (Fig. 7), while it slightly decreases in the model with outward-decreasing turbulence intensity, consistent with radio observations. Note that the main optical arms also become more tightly wound up in the outer regions of the galaxy (Frick et al. 1999).

The absolute values of the pitch angles in the preferred model using Sofue's rotation curve and the correlation times of the `standard' model turn out to be too small. Increasing the correlation time leads to models with large-scale field reversals and therefore a ring-like structure. Such field configurations are especially a subject of discussion of the magnetic field in our Galaxy (cf. Indrani & Deshpande 1998), but they do not fit the situation observed in NGC 6946. To achieve reasonable agreement with the observations, we choose a model with enlarged turbulence intensity of 15 km s-1 in the galactic mid-plane. This value is larger than that given in Boulanger & Viallefond (1992), especially at larger radii (see Sect. 4.3). We do not have sufficient knowledge about the vertical stratification in NGC 6946. Thus we may expect that a more adequate profile in density will allow to avoid this inaccuracy in the mid-plane turbulence intensity, since the density stratification affects the turbulence intensity (cf. Eq. (13)).

The turbulence intensity is assumed to be constant along radius because the model with outward-decreasing turbulence is in conflict with Faraday rotation data (see Sect. 6.4). The correlation times are again 0.02 Gyr (arm) and 0.01 Gyr (interarm). This defines our `best model'.

However, our `best model' with constant turbulence intensity is not completely satisfactory because it is unable to explain the observed radial variation of pitch angles (Fig. 18). At inner radii the achieved absolute values of the pitch angles are too small, whereas they tend to be too large for radii [FORMULA] 6 kpc. With better observational data on the widths of spectral lines and cloud sizes, we hope to get refined values of turbulence intensity and/or correlation time and their radial variations as input parameters for improved models.

[FIGURE] Fig. 18. Radial variation of the mean magnetic pitch angle in the galactic mid-plane for our `best model', using Sofue's rotation curve (see Fig. 2), a radially constant turbulence intensity of 15 km s-1 and a correlation time of 0.02 Gyr (arm) and 0.01 Gyr (interarm). Observational data at [FORMULA] cm and their errors are given as stars

The magnetic pitch angle in our models is almost independent of the optical pitch angle (Sect. 5.3). The observed similarity between the magnetic and optical pitch angles may indicate some interplay between the spiral density wave and the dynamo wave. Preceding investigations (cf. Moss 1997 and references therein) dealt with the influence of a given non-axisymmetry onto the magnetic field, e.g. the influence of a density wave onto the dynamo wave (parametric resonance), but the behaviour of the magnetic pitch angle was not investigated.

On the other hand, the discussion of a back-reaction of the magnetic field onto the large-scale gas motion (e.g. the density wave) is still lacking because this back-reaction is neglected in the frame of present-day kinematic dynamo models. This simplification is generally justified by the weakness of galactic magnetic fields, i.e. the energy density of the magnetic field is small compared to the kinetic energy of the large-scale motion of the interstellar gas. This assumption is possibly not true in the magnetic arms since the strength of the regular field is enhanced there Both energy densities are comparable for [FORMULA] 10 µG and [FORMULA] 0.02 cm-3 (assuming [FORMULA] 150 km s-1) (Beck & Hoernes 1996). The gas density there is still unknown because deep CO observations are not available. A further nonlinear effect that should be taken into account in future models is the back-reaction of the magnetic field onto the turbulent diffusivity ([FORMULA]-quenching) (cf. Elstner et al. 1996; Rohde et al. 1998a; Moss 1998).

6.3. Azimuthal variations of the field

Fig. 19 shows the azimuthal variations of [FORMULA] and the magnetic pitch angle of our `best model' which are both in reasonable agreement with the observational data. The strongly deviating pitch angles at [FORMULA] azimuth correspond to the kink in the magnetic arm in the south ([FORMULA], [FORMULA], see Fig. 1). The optical arm in the southeast ([FORMULA] ahead in azimuth) shows a similar kink.

[FIGURE] Fig. 19a and b. Azimuthal variation of the expected polarized intensity ([FORMULA]) a and of the magnetic pitch angle b of our `best model', both at a radius of 6 kpc. The turbulence intensity is 15 km s-1 (mid-plane value), the correlation time varied between 0.02 Gyr (arm) and 0.01 Gyr (interarm). Observational data (averaged between 5 and 7 kpc radius) are given as stars. The azimuthal angle runs counterclockwise in the plane of the galaxy ([FORMULA], [FORMULA]), starting from the northeastern major axis

Our models predict that the absolute values of the pitch angle should be larger in the gaseous arms than in the magnetic arms (Fig. 5). The gaseous arms coincide with the arms of total radio emission where the magnetic field is strongest, but mostly irregular (Beck & Hoernes 1996). In the ring between 5 and 7 kpc radius two broad arms can be identified, located at [FORMULA]-[FORMULA] and [FORMULA]-[FORMULA] azimuthal angle. These regions are those with the largest absolute values of the pitch angle (Fig. 19b). However, the errors are large due to the low polarized intensity in the gaseous arms. Future polarization observations with higher resolution may partly resolve the field structure so that the detectable polarized intensity might become higher.

The phase shift between the magnetic and the preceding gaseous arms is predicted to increase with radius (Fig. 9b), from [FORMULA] at 5 kpc to [FORMULA] at 10 kpc radius (corotation). Frick et al. (1999) found that the phase shifts as determined from the radio polarization and red-light images are roughly constant. Taking into account the observational errors, this result agrees with our model.

It is remarkable that the concentration of regular magnetic field between the gaseous spiral arms in our models is only caused by the non-axisymmetric distribution of correlation time [FORMULA], whereas in all previous papers this effect was achieved via modulation of the turbulence intensity (Rohde & Elstner 1998; Shukurov 1998; Moss 1998; Schreiber & Schmitt 1999). The pronounced "magnetic arms" in the interarm regions are due to the interplay of [FORMULA] and [FORMULA]-[FORMULA] induction processes and are only weakly influenced by the nonlinear back-reaction on the turbulence. This is why the process is also working for uniform turbulence intensity, contrary to earlier concepts. In Rohde et al. (1998b) we estimated a local dynamo number

[EQUATION]

reflecting the interplay between the two interfering induction processes described by [FORMULA] and [FORMULA] (cf. Eq. (16)). (The quenching function [FORMULA] is introduced via [FORMULA]-quenching, see Sect. 5.4.) Due to the rather small values of the correlation time our models for NGC 6946 work in the [FORMULA]-[FORMULA] regime. In this case the induction is dominated by the [FORMULA] induction coefficient within the dynamo number (Eq. (18)). Increasing [FORMULA] in the optical arms then decreases the dynamo number in this region: The field generation works preferably in the interarm region. (Remember that the turbulence intensity [FORMULA] has no arm/interarm contrast in our models.) The back-reaction of the magnetic field onto the turbulence ([FORMULA]-quenching) cannot explain the existence of magnetic arms: As the equipartition field [FORMULA] (Eq. (9)) is axisymmetric in our models but the regular field [FORMULA] is stronger in the magnetic arms, [FORMULA]-quenching suppresses the formation of such arms. Our models indicate that the effect of higher local dynamo numbers in the interarm regions is more important. For a strong [FORMULA]-effect the second term in Eq. (18) could be neglected and an enhanced [FORMULA] (higher local dynamo number) in the arms leads to stronger magnetic fields in the optical arms with a bisymmetric structure. This was first noticed by Mestel & Subramanian 1991 and verified with a full 3D simulation by Rohde & Elstner (1998).

6.4. Faraday rotation

The large-scale distribution of Faraday rotation in NGC 6946 indicates that the field points towards us in the northeastern magnetic arm and away from us in the southwestern arm (Krause & Beck 1998). This is the typical signature of a mixture of [FORMULA] and [FORMULA] modes, in agreement with the models discussed in this paper.

No significant increase of Faraday rotation has been observed within the inner 2 kpc, as predicted by our model with outward-decreasing turbulence intensity (Fig. 15b). Consequently, our `best model' assumes a constant turbulence intensity. This is not necessarily in conflict with the observed decrease of velocity dispersion (see Sect. 4.3) because other phenomena like unresolved velocity gradients or vertical gas motions affect the observed dispersion. Radio data in the HI and CO lines with better angular resolution are required.

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© European Southern Observatory (ESO) 1999

Online publication: October 4, 1999
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