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Astron. Astrophys. 339, 19-33 (1998)

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3. A model of NGC 6946

3.1. Representation of the interstellar medium

We have chosen to include two phases for the interstellar medium:
i) a dense molecular phase composed of spherical molecular clouds with masses ranging from 103 to 106 [FORMULA] and mean density 50 H2 cm-3. This mean density is used to determine cloud sizes from their masses. The half power width of the molecular layer, defined as [FORMULA], is equal to 65 pc. The global volume filling factor of this molecular phase in the disk, [FORMULA] (H2), amounts to 2.5%. The clumpy structure of molecular clouds is taken into account by using a larger density, [FORMULA] H2 cm-3, to compute the C+ emission. The C+ emission is not sensitive to the gas density when it is larger than [FORMULA] 103 cm-3 (see Tielens & Hollenbach 1985and Sect. 3.6). Indeed, C+ observations of nearby molecular clouds show that C+ emission is detected over the whole projected surface of molecular clouds (Jaffe et al. 1994). The total mass of dense molecular gas in the model is set to match the value deduced from CO(1-0) observations with a conversion factor of [FORMULA] H2 cm-2 (K km s-1)-1, [FORMULA] [FORMULA] (Table 1).
ii) a neutral diffuse phase, with a nearly constant HI density. We have adjusted the value of the mean HI density using the data from Boulanger & Viallefond (1992): it varies slowly from 0.8 H cm-3 at the center to 2 H cm-3 in the outer disk ([FORMULA] 6 kpc). We have assumed that the HI disk has a vertical thickness of 360 pc = [FORMULA], and that the density is uniform from [FORMULA] to [FORMULA]. The velocity dispersion of the diffuse gas is set to 10 km s-1. The total mass of diffuse neutral gas inside a radius of 6 arcmin is taken from the HI observations to be [FORMULA] [FORMULA].

In addition to these two phases, we take into account the formation of HII regions around OB associations (see below). For the numerical calculations, we represent the galaxy disk as a large 3D grid. The total grid length L and the cell size [FORMULA] are chosen to match the following constraints:

  • the grid size has to be at least as large as the optical disk, which has a diameter of 12.5 kpc, in order to be able to follow the path of photons in the external regions of the galaxy, and the heating of the neutral gas at large distance from the nucleus.

  • the spatial resolution, [FORMULA], should be good enough to describe the clumpy structure of the medium: in particular the smallest clouds in the ensemble should occupy at least one cell. With the adopted parameters for the clouds, the radius of the smallest clouds with a mass of 103 [FORMULA] is 4.5 pc. This number is only an estimate of the actual size of a given cloud. It is possible to use an alternative method based on the scaling relations for molecular clouds, the so-called Larson's laws. Using the mass-radius relation [FORMULA] (Falgarone et al. 1992), the radius of a 103 [FORMULA] cloud is then 3 pc. Another constraint on the spatial resolution is provided by the size of OB associations, which is generally larger than 30 pc (Garmany 1994) and their distance from molecular clouds. Leisawitz (1991) has estimated the mean distance between OB associations and their parent cloud to be about 50 parsecs.

We have chosen as the best compromise to represent the galaxy with a [FORMULA] cell structure. The galaxy size is then [FORMULA] = 12.5 kpc and the resolution [FORMULA] = 12.2 pc. The cells are filled with neutral atomic gas, or with molecular gas at the molecular cloud positions, an 106 [FORMULA] molecular cloud then occupies [FORMULA] cells. Finally, we include the ionized gas in the Strömgren spheres centered on each OB association, and replace the molecular and atomic gas with ionized gas whenever required. The radii of the Strömgren spheres are calculated assuming classical ionization in HI-bounded HII regions (Miller & Cox 1993). Because of the coarse spatial resolution of the dynamical code (200 pc), we do not attempt to reproduce the interstellar medium in the central region of NGC 946 ([FORMULA] pc).

3.2. Spatial distribution of molecular clouds

This section of the code uses the cloud-cloud collision code described by Combes & Gerin (1985), and used by Garcia-Burillo et al. (1993) and Gerin et al. (1991) to model the gas dynamics in nearby galaxies. The molecular clouds move in the gravitational potential of the galaxy, they grow through cloud-cloud collisions, which are sticky processes, and are disrupted by simulated SN events. The molecular gas is immediately recycled into small molecular clouds. The gravitational potential is deduced from an R band image of NGC 946 (Viallefond & Bonnarel, private communication, Bonnarel et al. 1988), with foreground stars removed. The image has been rotated to put the major axis vertical, and deprojected to face-on, using as projection parameters PA = [FORMULA] and i = [FORMULA]. The image has then been binned to [FORMULA] pixels. This R band image covers [FORMULA] on the sky, corresponding to a radius of 6.24 kpc at the assumed distance of NGC 946, 5 Mpc. The actual spatial resolution amounts to a few times the cell size, about 200 pc. The following step is to build a good gravitational potential from this image. As in Garcia-Burillo et al. (1993), the calculation is done in two steps. The axisymmetric part of the potential is obtained by assuming a constant mass to light ratio, and adjusting this constant to reproduce the observed CO and HI rotation curves. The perturbations due to the bar and spiral arms are included in the non axisymmetric part of the potential. The last parameter to adjust is [FORMULA], the pattern speed of the density wave (bar + spiral arms). The gas distribution and velocity field are very sensitive to [FORMULA]. We find that [FORMULA] = 42 km s-1 kpc-1 gives the best results. The corotation is found at a radius of 3.5 kpc, close to the end of the bar, and the OLR lies outside of the disk. Fig. 3a presents the deprojected R-band image. Figs. 3b and 3c present an example of the obtained molecular cloud distribution. The gas clouds are concentrated in the spiral arms, with few molecular clouds at radii larger than 4 kpc. The model rotation curve is shown in Fig. 4, together with the angular frequency [FORMULA]. We have not attempted to model the compression of diffuse gas in the spiral structure, and have kept an axisymmetric distribution of diffuse atomic gas.

[FIGURE] Fig. 3a. R band image of NGC 946 used as input for the calculation of the gravitational potential. Field stars have been removed, the image has then been rotated and deprojected to get a face-on view of NGC 6946.

[FIGURE] Fig. 3b. A face-on view of the model galaxy, with molecular clouds drawn as circles and OB associations drawn as stars. Only 20% of the OB associations and molecular clouds are drawn.

[FIGURE] Fig. 3c. A close-up view of Fig. 3b. The clouds are drawn at their exact size in the model.

[FIGURE] Fig. 4. Adopted rotation curve (dot-dashed line), angular frequency [FORMULA] (full line), and [FORMULA] curves. The corotation for the adopted pattern speed is located at [FORMULA] kpc.

3.3. Stellar population

Once the gas is distributed over the galactic disk, we create OB associations, with star masses ranging from 10 to 60 [FORMULA] (stellar types from B2 to O5). We have chosen not to include stars less massive than 10 [FORMULA] because their lifetime becomes a significant fraction of the rotation period. Shortward of 2000 Å, the UV radiation is mostly produced by OB stars and the contribution by late B type stars is at most 20% (Walterbos & Greenawalt 1996, Mathis et al. 1983). The choice of the upper mass cutoff at 60 [FORMULA] is motivated by the work of Heydari-Malayeri & Beuzit (1994) who have shown that suspected very massive stars ([FORMULA] 100 [FORMULA]) are actually clusters of less massive stars. Also, a very short time step is required to sample adequately the lifetime of these very massive stars.

We allow only clouds more massive than [FORMULA] [FORMULA] to form massive stars. In the cells satisfying this criterion, the OB associations are born at the outer edge of the molecular clouds. We have tested different star formation laws, so as to match as well as possible the H[FORMULA] radial profiles. The best fits are obtained with a star formation rate (SFR) depending on the angular frequency and the local gas surface density as proposed by Wyse & Silk (1989):


with [FORMULA] the star formation efficiency, [FORMULA] the angular frequency at radius r and [FORMULA] the HI+H2 surface density at position [FORMULA] in the disk. We find that [FORMULA] provides the best match to the H[FORMULA] radial profile by Kennicutt (1989).

The stellar mass distribution inside an OB association is drawn from the Initial Mass Function (IMF). We have chosen an index close to the Salpeter IMF, as suggested by measurements in Galactic and extragalactic OB associations (Massey et al. 1995): [FORMULA].

The stellar associations are born with a mean velocity relative to their parent cloud chosen from a Gaussian of mean value 10 km s-1 and FWHM [FORMULA] 10 km s-1. These figures are in good agreement with observed data by Leisawitz et al. (1989) in their study of the relation of star clusters with molecular clouds. With this value, the average motion of OB stars relative to their parent cloud is 10 pc in 106 years. Finally, stars die after a time equal to their Main Sequence lifetimes from Güsten & Mezger (1983). Effective temperatures, stellar luminosities and radii, averaged over the main-sequence lifetime, are taken from Cox et al. (1986). Because we are only interested in broad band fluxes and colors, we have calculated all stellar fluxes, from UV to B bands, in the black-body approximation. Lyman continuum radiation production rates, averaged on the Main Sequence lifetime, are also taken from Cox et al. (1986) and Güsten & Mezger (1983) to have a coherent set of parameters. We use the parametrization as a function of the stellar mass, M:


to compute the thermal radio continuum flux, H[FORMULA] luminosity and Strömgren sphere diameters.

We also include the effects of the radiation of massive stars on neutral gas, and allow molecular clouds to be partially eroded and ionized by the radiation of nearby OB associations, if they overlap with the Strömgren sphere of an OB association. We compute the radius of this sphere, [FORMULA], assuming that all stars are located at the same position and have a global production rate of Lyman continuum photons [FORMULA], equal to the sum of the contribution of the individual stars, and for case B recombination in a diffuse medium of density [FORMULA]: [FORMULA]. In this formula, [FORMULA] is the hydrogen recombination coefficient (Osterbrock 1989). The mean radius for RHII is 35 pc.

To constrain the population of massive stars, we calculate different stellar outputs, namely we perform a detailed calculation of the UV radiation field at 912-2000 Å, and also compute the global U flux and U radial profile, as well as the thermal radio continuum and H[FORMULA] emission.

[FORMULA] 6 cm radio-continuum.
According to Mezger (1972) and Turner & Ho (1994), the radio continuum emission of HII regions for case B recombination, at an electronic temperature [FORMULA] of [FORMULA] K and with 45% of the ionizing photons being converted into H[FORMULA] photons, is directly related to NLyc by:


This formula does not include any correction for dust absorption of the ionizing radiation within the HII region. Current estimations are that nearly 50% of the Lyman continuum may be absorbed by dust.

The relationship between the H[FORMULA] luminosity and the production rate of Lyman continuum photons, using the same assumptions as above, can be deduced from Mezger (1972) and Peimbert et al. (1975):


The correction for the extinction may amount to about 1 magnitude at H[FORMULA] but it is highly uncertain (Mc Kee & Williams 1997, van der Hulst et al. 1988). Because of the uncertainties involved in this correction, we have preferred not to do it. This allows us to check that the energy is conserved with a good accuracy in the simulation.

The formed OB associations have typical production rates of Lyman continuum photons, [FORMULA], in the range 1045-1051 s-1. The cumulative Lyman continuum luminosity function of the population is shown in Fig. 5. Its shape is similar to the distribution for Galactic HII regions (McKee & Williams 1997). There is a small excess around [FORMULA] [FORMULA] 1049.5 s-1, dominated by the most massive stars in the younger associations, and a deficiency of associations with [FORMULA] [FORMULA] 1050.5 s-1. As a whole, the match of the two distributions is very good. The distribution of the intrinsic UV luminosity of OB associations extends over four orders of magnitude from 104 to 107 [FORMULA], i.e. from small associations with about 20 stars and a total stellar mass of 600 [FORMULA], up to large associations gathering 200 OB stars, including a few 50-60 [FORMULA] stars, and having a total stellar mass of 6000 [FORMULA].

[FIGURE] Fig. 5. Distribution of the production rate of Lyman continuum photons [FORMULA], for the modelled population of OB associations (full line) in NGC 6946, and for the Galaxy (dot-dashed curve) (McKee & Williams (1997)). The plot can be read for instance as 500 OB associations out of 12000 have NLyc [FORMULA] 1050 s-1.

3.4. Dust properties

We use the average Galactic extinction curve from Fitzpatrick & Massa (1988) at UV wavelengths. For visible and near infrared wavelengths, we use the work by Seaton (1979). This curve is probably valid on a large scale in the diffuse medium of NGC 6946 which has a similar metallicity to the Milky Way. We assume a constant metallicity and gas to dust ratio in the disk, and use the average value for the Milky Way: [FORMULA] H cm-2 mag-1 and AV = 3.1 [FORMULA] (Bohlin et al. 1978). The extinction through a cell filled with molecular gas, with the adopted spatial resolution, is 2.5 magnitudes. The dust properties have been summarized by Bruzual et al. (1988) and Witt & Gordon (1996). Apart from the enhanced absorption in the 2175 Å bump, the dust albedo is fairly constant at UV and visible wavelengths at [FORMULA]. We include the coherent scattering of UV light by dust grains. The anisotropy is described using the Henyey-Greenstein (1941) function, where the anisotropy parameter is defined as [FORMULA] and [FORMULA] is the scattering angle.

3.5. Radiative transfer

The observed properties of external galaxies depend on the propagation of the stellar radiation in the interstellar medium. To determine the local radiation field, we follow the propagation of UV photons (912-2000 Å) emitted from the OB stars in the two phases medium. In each cell, we compute a local radiation field. We define the local radiation field in the UV, [FORMULA], relative to the mean radiation field in the UV at the Solar radius, [FORMULA], established by Mathis et al. (1983), so that [FORMULA]. In the Galaxy and at the solar radius, the InterStellar Radiation Field, ISRF in the UV, has been defined as [FORMULA] = [FORMULA] erg cm-2 s- 1 and through visible/IR bands as G0 = [FORMULA] = [FORMULA] erg cm- 2 s-1, where [FORMULA] is the specific intensity of the radiation field averaged over 4[FORMULA] sr (Mathis et al. 1983).

On their path, photons can be scattered and/or absorbed by dust in both neutral phases. We have chosen not to follow individual photons which would have been time consuming, but to use instead pseudo-photons representing a collection of N photons. We are then able to probe a larger area with a lower number of photons. We sample the 912-2000 Å interval with 20 bins of constant wavelength width [FORMULA], and launch [FORMULA] pseudo-photons per association per wavelength bin. The longward limit has been set to 2000 Å, to avoid having to take into acount the contribution from star types later than B to the ISRF (Walterbos & Greenawalt, 1996).

The pseudo-photons emitted by the OB association number i carry a fraction [FORMULA] [FORMULA] of the luminosity [FORMULA] radiated by this OB association. They travel from the center of a cell (size [FORMULA]) to an adjacent one, and the energy absorbed by the interstellar medium, when non-zero, is left on the common face of these 2 cells. The absorbed energy is reprocessed in the far-infrared. The pseudo-photon energy along its path from association number i can be written as:


In this formula, [FORMULA] is the product of the probabilities for non absorption in each cell along the travel from the OB association number i to the cell position [FORMULA], or with [FORMULA] the dust albedo and [FORMULA] the total cell opacity including scattering:

in the diffuse phase, and

in the H2 phase.

When leaving an OB association, the direction for each pseudo-photon is uniformly chosen on the unit sphere. These pseudo-photons travel in the two-phase interstellar medium. There are different possibilities when reaching a new cell:
i) The gas in the cell is diffuse and atomic. Then the pseudo-photon can either:

  • be absorbed partially in the cell. A fraction of the luminosity is left, and the pseudo-photon continues in the same direction with a lower luminosity.

  • be scattered.

ii) The cell is filled with molecular gas. The pseudo-photon is partly absorbed, and partly backward scattered, the ratio between the energy left in the cell and the total energy of the pseudo-photon depends on the albedo as [FORMULA]. We have chosen backward scattering in that case to prevent the pseudo-photon from interacting with the same molecular cloud several times. After a scattering event, a scattering angle [FORMULA] is chosen according to the anisotropy function, and the azimuthal angle [FORMULA] is uniformly chosen in the interval [FORMULA]. Then the new direction is easily deduced from the previous one (Witt 1977).

Since we cannot store all the pseudo-photons' incident directions, and in order to define an isotropic radiation field in each cell, we assume that the equivalent surface of the cell is [FORMULA]. Then, the UV radiation field in the cell [FORMULA] due to the contributions of all pseudo-photons travelling through this cell, can be expressed relative to the Galactic ISRF [FORMULA] as:


Because of the coarse angular resolution and the low number of pseudo-photons leaving each OB association, the resultant map of the radiation field presents strong fluctuations. We have chosen to smooth the map of the radiation field by averaging the data in nearby cells, typically 3x3 cells. Furthermore, some cells are never visited by UV photons, for example in the interarm region or in the outer disk ([FORMULA] kpc). In that case, we use as incident radiation field, the Galactic ISRF (longward of 2000 Å), scaled by the local surface brightness in the R band image to take into account the radial variation of the radiation field from the old stellar population between the central regions ([FORMULA] kpc) and the outer disk ([FORMULA] kpc). The reference value is given in units of [FORMULA] = G0 - [FORMULA] = [FORMULA] erg cm-2 s- 1. The position for the reference value has been chosen at the edge of a spiral arm, at a distance [FORMULA] kpc from the center, where the UV radiation field [FORMULA] is close to 1.

3.6. Emergent emission

Once the local UV energy density has been calculated, models are used to determine the C+ and FIR emissions from each cell, assuming that they arise from the same area as the one used for the calculation of the UV radiation field. For the dust emission, we use the model by Désert et al. (1990), which has 3 different components: PAHs which are fully ionized when [FORMULA] [FORMULA] 1, very small grains and big grains. We calculate the dust emission in the four IRAS bands at 12, 25, 60 and 100 [FORMULA] plus an additional band at 200 [FORMULA], as the reprocessing of the combination of the UV radiation field (912 - 2000 Å) described by [FORMULA] and of the ISRF for the 2000 Å -2 µm part of the spectrum. We decrease the PAH abundance in large radiation field environments, as suggested by Ryter et al. (1987), to one fifth of the standard value when [FORMULA].

Very large molecular clouds, with masses larger than 106 [FORMULA] occupy more than 4 cells in the grid. The inner cell is not directly exposed to the UV radiation and for this cell we assume that the ISRF is attenuated by 2.5 magnitudes of visual extinction.

We assume low optical depth in the mid and far infrared. From the analysis of the COBE maps of the Galaxy, Boulanger et al. (1996) deduce [FORMULA]/NH= 1 [FORMULA] 10-25 cm2 H-1 ([FORMULA] /250 µm)-2, which combined with the mean column density of individual clouds, NH = 1.5 [FORMULA] 10 22 cm-2, gives an opacity of 2 [FORMULA] 10-3 at 200 [FORMULA].

According to the dominant phase in a given cell of the model, the emergent infrared emission can be:

  • reprocessing of the whole incident stellar radiation for molecular clouds which are totally optically thick in the UV. All impinging radiation is completely reprocessed in infrared emission from the outer cells of molecular clouds,

  • proportional to the gas column density for the diffuse medium, which is optically thin in the UV. We assume no UV extinction at the 12 pc scale.

We use the PhotoDissociation Region (PDR) model by Le Bourlot et al. (1993) to estimate the [FORMULA] C+ emission at the surfaces of molecular clouds. We use a constant molecular hydrogen density of 5 [FORMULA] 103 H2 cm-3 and the incident UV field [FORMULA]. The density is not a critical parameter as long as it is higher than the critical density for collisional excitation of C+ (1000 H cm-3) (see Tielens & Hollenbach 1985). Furthermore, we assume that the whole surface of clouds contributes to the C+ emission. As for the C+ emission from the diffuse neutral phase, we use the model by Wolfire et al. (1995) for a two phase neutral atomic medium. Finally, the contribution from the ionized gas in the HII regions around the OB associations is also included. We assume that the gas has the same density as the diffuse medium, [FORMULA] and an electronic temperature of 104 K. The total C+ luminosity from an HII region of radius [FORMULA] is then proportional to the volume of the HII region with a correction factor to take into account the other ionization stages of carbon.

Because we deal with a line, the opacity may not be small depending on the gas distribution and viewing geometry. In fact, opacity effects are important at large inclination angles. To obtain an edge-on view of the model galaxy in the [FORMULA] C+ line, we have made an accurate calculation of the radiative transfer in this line. For each line of sight through the disk, we sample the line profile with bins of 1 km s-1 width, and calculate the emergent intensity in each velocity bin, including saturation effects. We assume that the intrinsic velocity dispersion of a PDR is 1 km s-1. For the diffuse medium, the velocity dispersion is 10 km s-1. We do not account for absorption in the far infrared continuum. We obtain [FORMULA] = 0.40 for an edge-on view, and [FORMULA]= 0.10 for a face-on view.

3.7. Implementation of the model

To avoid transient stages of the simulation, the code is evolved during a few time steps. We stop the simulation when stable results are obtained on a time scale of 20 Myrs. This time scale corresponds to about half the lifetime of a giant molecular cloud before disruption by photoevaporation. This is the reason why we can not integrate further in time without treating gas recycling. The time step has been fixed at 106 years, shorter than the lifetime of the most massive stars. We have checked the reliability of the calculations by different tests:

  • We have verified that the total UV luminosity from the stellar population emerges from the galaxy either at the same wavelength, or at far infrared wavelengths for the light reprocessed by dust grains. The total luminosity is conserved with an accuracy of 1%.

  • When the number of pseudo-photons leaving each OB association [FORMULA] is too small, the map of UV radiation field is noisy with a few extremely bright spots and large voids. This is due to undersampling of the galaxy volume. The number of pseudo-photons should be as large as possible, but we have verified that we obtain a good map of the UV radiation field with 100 pseudo-photons per OB association. The map is smooth in the vicinity of the OB associations, hence the ratio of FIR emissions from the diffuse and dense gas stays constant with increasing [FORMULA].

  • The cell size is also a critical parameter: since [FORMULA] is proportional to [FORMULA], it might be underestimated for small clouds very close to OB associations. This has severe consequences for the C+ emission, which scales roughly as [FORMULA], but little or none for the FIR emission which varies as [FORMULA] since in that case there is no resultant scaling with [FORMULA]. To test the validity of the adopted resolution, we have performed a run restricted to one quadrant only, with a cell size of 6.1 pc. We observed no large variation in the C+ emission and thus conclude that the adopted resolution of 12 pc is correct for our purpose. Note that the volume filling factor decreases to 1.3% in the high resolution run, because we fill the space in a more accurate way using a higher spatial resolution.

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© European Southern Observatory (ESO) 1998

Online publication: September 30, 1998