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

5. Results

5.1. `Standard' model for NGC 6946

As a main result we achieve a basically axisymmetric magnetic field (ASS) with a clear maximum of the strength of the regular field within the interarm regions (Fig. 4, Fig. 5a). The field is strongest at  kpc radius (Fig. 6). The magnetic field mode m = 2 is given preference by the two-armed profile in and reaches almost 20% of the magnetic energy density (Fig. 8). The general structure of the simulated magnetic field is in agreement with the models based on small correlation times shown in Rohde & Elstner (1998): the field is strongly determined by the effect of differential rotation. Within our model, the relatively high mid-plane turbulence intensity ( = 12 km s^-1) requires a relatively small correlation time of interstellar turbulence, otherwise the turbulent dynamo would lead to magnetic pitch angles which are too large compared to the observations, or the dynamo would change to the -regime. For the model presented in this section we use values of 0.01 Gyr (interarm) to 0.02 Gyr (arm), comparable to the lifetime (merging time) of supernova remnants and of giant molecular clouds. Ferrière (1998) mentions merging times for superbubbles in the range 0.01 to 0.04 Gyr.

The magnetic pitch angle reaches values between ("magnetic arms") and ("magnetic interarm" = gaseous arms) (Fig. 5b) and thus varies by about around the pitch angle of the gaseous arms in our models (assumed to be constant). Fig. 7 shows the mean value of the magnetic pitch angle taken in the galactic mid-plane averaged over all azimuthal angles.

Fig. 4. Magnetic field geometry in the galactic mid-plane for an azimuthal variation of = 0.01 ... 0.02 Gyr. The optical spiral arms (enlarged ) are shown by light shades.

Fig. 5a and b. , where is the normalized regular magnetic field strength (solid) compared with the profile in correlation time (dotted, also normalized) a . The field shows a concentration between the gaseous spiral arms. We show to allow a comparison with the polarized intensity (Sect. 6.3). b  Magnetic pitch angle (solid) compared with the normalized magnetic field strength (dotted). The absolute value of the magnetic pitch angle is largest between the "magnetic arms", i.e. in the gaseous (optical) arms. Both plots are given for the galactic mid-plane at r = 6 kpc

Fig. 6. (normalized) for the `standard' model (solid) and a model based on the rotation curve given by Sofue (1996) (dotted) in the galactic mid-plane, averaged over all azimuthal angles

Fig. 7. Mean magnetic pitch angle in the galactic mid-plane averaged over all azimuthal angles for the `standard' model (solid), for the model with outward-decreasing turbulence intensity (dashed) (see Sect. 5.4) and for the model based on the rotation curve given by Sofue (1996) (dotted) (see Sect. 5.5)

Fig. 8. Normalized contribution of the magnetic field energy (`standard' model) of mode m = 0 (dashed), 1 (dotted) and 2 (solid) at several radii. The m = 2 magnetic field contribution reaches values up to 20% of the magnetic energy density

5.2. Role of the corotation radius

The corotation radius of the spiral pattern and the interstellar gas influences the geometry of the magnetic field. For radii smaller than the region with maximal magnetic field is shifted towards the spiral arm preceding in the sense of rotation (with enhanced correlation time); outside corotation radius the field is shifted towards the following arm (Fig. 9). Hence, a shift of 90^o occurs at the corotation radius.

Fig. 9a and b. Phase shift between region of maximal magnetic field and the preceding spiral arm for a model with a corotation radius of = 5 kpc a and 10 kpc b

This phenomenon is expected to be due to the fact that the interstellar gas rotates faster than the spiral pattern for radii smaller than . The partially frozen-in magnetic field lines are then transported by the gas from the interarm region towards the preceding arm. For radii larger than the process works in the opposite manner. This explains the observed similarity between the magnetic arms and their preceding optical arms (Frick et al. 1999) (see also Sect. 6.3).

According to Elmegreen et al. (1992) the corotation radius of NGC 6946 is located at about 2.6´ ( kpc at the distance assumed in this paper). The "magnetic arms" are observed between about 5 and 12 kpc radius (see Fig. 1) so that we would expect a phase shift with respect to the preceding arm of -. From the observational data, the mean shift is only between and for the four main arms (Frick et al. 1999, see also Sect. 6.3) which fits better to a large corotation radius (Fig. 9b). More detailed data of the galaxy's velocity field are required to determine the corotation radius with higher accuracy.

Note that a behaviour opposite to that in our models was found for the m = 1 mode excited by a spiral density wave (Mestel & Subramanian 1991, Subramanian & Mestel 1993), i.e. no phase shift between gas and magnetic field around the corotation radius. This is probably due to the neglection of and in their model (see Sect. 6.3). In our models, the m = 1 mode is unimportant unless the correlation time is very large, e.g. 0.1 Gyr (see Rohde & Elstner 1998).

5.3. Role of the spiral arm pitch angle

We further performed calculations with varied pitch angle of the gaseous arms and hence the turbulence profile (Eq. (12)).

The main and striking result is that the magnetic pitch angle is only weakly attached to the (optical arm) pitch angle of the underlying non-axisymmetric turbulence profile (Figs. 10, 11) whereas the strength of the regular field is always highest in the interarm regions (Fig. 12). That is, the magnetic pitch angle and the field strength are generally determined by the turbulent diffusivity, the differential rotation and the values of the -tensor, but not by the geometry of the optical spiral (see Sect. 4.2). This argumentation is of course only valid for an axially symmetric large-scale velocity field. Additional non-axisymmetric gas flows would strongly affect the magnetic field and its pitch angle (cf. Elstner et al. 1998).

Fig. 10a and b. Magnetic field geometry for different turbulence profiles in the galactic mid-plane. The optical spiral arms (enlarged ) with = 40o a and = 15o b are shown by light shades. The magnetic pitch angle does not depend on the galactic spiral profile, see also Fig. 11

Fig. 11a and b. Magnetic pitch angle in the galactic mid-plane at for (optical arm) pitch angles = 40o a and = 15o b of the -profile. The pitch angle of the -profile shows almost no influence on the values of the magnetic pitch angle

Fig. 12a and b. (normalized) (solid) compared with the turbulence profile (dotted, also normalized) in the galactic mid-plane at for (optical arm) pitch angles = 40o a and = 15o b of the -profile. The regular field is always concentrated between the spiral arms

We expect that a more complicated and more realistic spiral arm profile would not lead to qualitative different results.

5.4. Radial decrease of mid-plane turbulence intensity?

The model we discussed in Sect. 5.1 is based on a mid-plane turbulence intensity that does not vary in radial (and azimuthal) direction. Since there are observations suggesting an outward-decreasing turbulence intensity profile (Boulanger & Viallefond 1992, see Sect. 4.3) we try to understand the consequences of such a profile by an appropriate simulation. We chose a slightly reduced decrease where the turbulence intensity in the galactic mid-plane varies linearly from 14 km s^-1 in the center to 10 km s^-1 at the outer radius (15 kpc).

Two main consequences appear being different to the `standard' model presented in Sect. 5.1:

• The absolute value of the magnetic pitch angle now decreases in the outward direction, but it has also larger values at inner radii compared with the `standard' model (Fig. 7).

• The magnetic field shows a considerable asymmetry with respect to the galactic mid-plane: The magnetic field modes A0 and A2 are excited besides the S0 and S2 modes. The asymmetry is well seen in a meridional plot of the azimuthal magnetic field () in Fig. 14b. This asymmetry leads to a strong vertical magnetic field () passing through the inner galactic disk without changing its sign which should be observable via strong Faraday rotation (Fig. 15) (see Sect. 6.4). The parity of the magnetic field is shown in Fig. 13. The asymmetry mentioned above leads to a value of 0.5 (dashed curve).

Fig. 13. Parity of the magnetic field for the `standard' model (solid), for the model with outward-decreasing turbulence intensity (dashed) (see Sect. 5.4) and for the model based on the rotation curve given by Sofue (1996) (dotted) (see Sect. 5.5)

Fig. 14. a Meridional plot of the toroidal magnetic field () taken from the `standard' model discussed in Sect. 5.1. The field is clearly symmetric with respect to the galactic mid-plane and shows a strong concentration there. b  Meridional plot of the toroidal magnetic field () taken from the model with outward-decreasing turbulence intensity in the mid-plane (see Sect. 5.4). The field also contains antisymmetric components with respect to the mid-plane (A0, A2)

Fig. 15a and b. Meridional plot of vertical magnetic field component taken from the `standard' model discussed in Sect. 5.1 a and from the model with outward-decreasing turbulence intensity b . In the latter case the vertical magnetic field passes through the inner galactic disk without changing sign due to the contribution of A-modes

The behaviour of the magnetic pitch angle can be estimated by

(Beck et al. 1996). In the linear case, the approximations and with the characteristic scale height h, inserted into the expressions of the induction coefficients

give the estimation In the nonlinear case (adopted here) a quenching function must be added in order to represent the influence of -quenching. This gives

The quenching function, which in a simple estimation is (Beck et al. 1996), forces the absolute value of the magnetic pitch angle to increase outwards since the magnetic field is large around and decreases outwards. This behaviour is well seen in the calculations shown in Fig. 7 (standard model) and Fig. 18. This effect is blurred in case of outward-decreasing turbulence intensity where the absolute value of the magnetic pitch angle decreases outwards (Fig. 7, dashed).

Using the estimation (17) we can also explain the enlarged magnetic pitch angles within the the gaseous arms where the correlation time is assumed to be larger than in the magnetic arms.

Note that, according to Eq. (17), the same radial variation of the magnetic pitch angle occurs if one assumes a radial decrease of the correlation time e.g. because the mean size and/or lifetime of molecular clouds decreases outwards.

5.5. Role of the rotation curve

We discuss here how the field generation is influenced by the shape of the rotation curve. To avoid numerical problems due to the limited resolution of our simulations at the sharp increase of the velocity for 1 kpc, we approximate the rotation curve by Sofue (1996) using a Brandt-type law with = 220 Gyr^-1, = 1 kpc and n = 2 (Fig. 2). The shape of the spiral arms and the distributions of density, turbulence intensity and correlation time are the same as for our `standard' model discussed in Sect. 5.1.

The investigation of Sofue's rotation curve leads to a magnetic field with rather small absolute values of the magnetic pitch angle due to the stronger differential rotation (Fig. 7, dotted). The magnetic field is more concentrated at inner radii (Fig. 6). The contribution of higher magnetic field modes (2) and thus the influence of the spiral arms is smaller than in the `standard' model. These field properties can be seen by comparing the magnetic fields of different models in the galactic mid-plane (Fig. 16).

Note that for very inner radii the approximation of the rotation curve given in Sofue (1996) is not very accurate. Therefore we expect that, for a better approximation of the sharp velocity increase, the above mentioned differences to the `standard' model would be even larger.

Fig. 16a-c. Magnetic field geometry taken in the galactic mid-plane. a  For the `standard' model discussed in Sect. 5.1 (cf. Fig. 4). b  For a model with outwards decreasing turbulence intensity as discussed in Sect. 5.4. c  For a model with a rotation curve based on Sofue (1996) as discussed in Sect. 5.5

© European Southern Observatory (ESO) 1999

Online publication: October 4, 1999
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