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

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4. The model for NGC 6946

We consider galaxies to be differentially rotating turbulent disks embedded in a plasma of given conductivity (Elstner et al. 1990). In the simplest case the "plasma" is vacuum and the conductivity therefore vanishes. Thickness and height of the galactic disk are determined by the density scales, see Sect. 4.3. The calculated volume is restricted to the radius R = 15 kpc and the half-thickness H = 1.5 kpc. The simulations are performed with a 3D time-stepping code using cylindrical polar coordinates with a resolution of 41 gridpoints in each direction. The code is described in more detail in Elstner et al. (1990) and Rohde et al. (1998a). All calculations shown here are performed with an initial magnetic field composed of modes m=0,1 and a mixed parity with respect to the equatorial plane, but our tests have shown that the resulting magnetic field is independent of the initial field.

4.1. Differential rotation

We describe the differentially rotating gas by a Brandt-type law

[EQUATION]

By selecting the values [FORMULA] = 72 Gyr-1, [FORMULA] = 2.5 kpc and n = 1.3 we approximate the rotation curve given in Carignan et al. (1990) on the basis of neutral hydrogen (HI) data. The asymptotic velocity reaches a value of [FORMULA] = 180 km s-1. Our modeled rotation curve is shown in Fig. 2 (solid line).

[FIGURE] Fig. 2. Rotation velocities given by Carignan et al. (1990) (stars, with error bars) and by Sofue (1996) (dashed). Approximations used for our simulations are also shown (solid and dotted, respectively)

The shape of rotation curves is still a topic of investigation. Sofue (1996) claimed that HI data compared to observations of the molecular gas (CO) yield too small velocity values, especially at small radii. An approximation to his (CO+HI) rotation curve for NGC 6946 is also shown in Fig. 2 (dashed line). For this rotation curve we choose the parameters [FORMULA] = 220 Gyr-1, [FORMULA] = 1 kpc and n = 2. In Sect. 5.5 we discuss the influence of the different rotation curves onto the magnetic field generation.

4.2. Non-axisymmetry

Since investigations of the H2 and HI surface densities in the galaxy NGC 6946 have revealed that there are rather weak azimuthal variations of gas density (Tacconi & Young 1990; Boulanger & Viallefond 1992), we assume an axisymmetric density distribution for our galaxy model. Note that the influence of a non-axisymmetric density distribution onto the dynamo-induced magnetic field is very small and is thus neglected (Rohde & Elstner 1998).

The turbulence intensity [FORMULA] can be measured in principle by the velocity dispersion of spectral lines, e.g. HI (Boulanger & Viallefond 1992). However, other effects like unresolved velocity gradients or vertical gas motions will also increase the observed velocity dispersion. On the other hand, a strong arm/interarm contrast in turbulence intensity, as assumed in several previous papers (Shukurov 1998; Rohde & Elstner 1998; Moss 1998; Schreiber & Schmitt 1999) should be detectable through a significant difference in line widths. No such effect has been found in any galaxy so far. In particular, HI data for NGC 6946 do not show any difference in linewidth between arm and interarm regions near the minor axis (Kamphuis 1993 and Kamphuis, priv.comm.). Consequently, [FORMULA] is also assumed to be axisymmetric in our models.

We introduce a spiral profile attached to only the correlation time [FORMULA] of interstellar turbulence, whose value we take as a free parameter since it is indeed unknown. We assume that the correlation time is enlarged within the optical spiral arms of the galaxy. This configuration seems justified by the fact that small molecular clouds collide to form giant clouds in the arms and thus are larger than in the interarm regions (Casoli 1991). Assuming constant turbulence intensity [FORMULA], larger molecular clouds are then correlated with enhanced correlation length and correlation time [FORMULA].

The arm/interarm contrast is described via the shape of a logarithmic spiral

[EQUATION]

varying between 1 and q. The pitch angle [FORMULA] of the optical spiral arms was set to [FORMULA] following Kennicutt (1981). The real spiral shape of NGC 6946 is of course more complicated than in our model (Elmegreen et al. 1992). On a red-light image Frick et al. (1999) identified four main spiral arms with average pitch angles between [FORMULA] and [FORMULA]. In Sect. 5.3 we discuss the influence of different (optical arm) pitch angles [FORMULA] onto the magnetic field in order to justify our simplification.

4.3. Vertical stratification and radial scale lengths

Lacking knowledge about the vertical stratification of the gas density in NGC 6946 we adopt the empirical HI distribution of Dickey & Lockmann (1990) for our Galaxy: a combination of two Gaussians with central densities 0.395 and 0.107 cm-3 and scale heights of 212 and 530 pc, respectively, and an exponential with central density 0.064 cm-3 and a scale height of 403 pc. We add a further Gaussian with mid-plane density 0.3 [FORMULA] cm-3 and half-width 70 pc to include the molecular gas layer (cf. Bloemen 1987), and an exponential with a scale height of 1.5 kpc and a mid-plane density of 0.025 cm-3 representing the extended ionized gas (Reynolds 1989). The total gas mass of NGC 6946 is approximately twice that of our own Galaxy (Carignan et al. 1990). Note that the total gas mass does not influence the vertical stratification (Eq. (13)). It only determines the equipartition field strength [FORMULA] and therefore the strength of the magnetic field that saturates the dynamo by [FORMULA]-quenching, but has no influence onto the field structure.

We further introduce exponential scale lengths for the different components of the gas disk of NGC 6946 (Tacconi & Young 1986): 10 kpc (HI), 3.5 kpc (HII) and 3 kpc (H2).

Based on the given density stratification we adopt the common simplification of the vertical momentum equation as a possibility to calculate the turbulence intensity:

[EQUATION]

The used potential [FORMULA] is essentially due to a self-gravitating isothermal sheet of stars with constant thickness [FORMULA] and we assume

[EQUATION]

Lacking knowledge on the exact density distribution in NGC 6946 and for the sake of simplicity we neglect radial dependencies within the potential [FORMULA]. We set an average value [FORMULA] = 21.5 km s-1 for the vertical velocity dispersion of the old disk stars and [FORMULA] = 0.6 kpc. The value for [FORMULA] should be larger than 1 (Elstner et al. 1996); here we assume [FORMULA] = 3 (see Fröhlich & Schultz (1996) and Elstner et al. (1996) for details).

The velocity dispersion in HI shows a radial dependency decreasing outwards from approximately 16 km s-1 to 8 km s-1 (Boulanger & Viallefond, 1992). This may be interpreted as a radial decrease of the turbulent gas velocity, but there may be other reasons (see Sect. 4.2). In our `standard' model we assume a constant mid-plane turbulence intensity with an average value of [FORMULA] = 12 km s-1, in Sect. 5.4 we also investigate a model with radially decreasing mid-plane turbulence intensity. Note that below and above the galactic mid-plane the turbulence intensity slightly depends on r because the galactic density decreases outwards (cf. Fig. 3 and Eq. (13)).

[FIGURE] Fig. 3. The maximal turbulence intensity (rms turbulence velocity) calculated from Eq. (13) at radius r = 3.75 kpc (solid) and r = 11.25 kpc (dashed)

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© European Southern Observatory (ESO) 1999

Online publication: October 4, 1999
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