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

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4. Results

Table 2 summarizes the input parameters for the Standard Model, and Table 3 presents the results. The star formation rate from 2 to 60 [FORMULA] is fixed at 4.0 [FORMULA]yr-1, with a star formation efficiency [FORMULA] of 5%, as defined in Sect. 3.4. With these values, the modelled H[FORMULA] luminosity and H[FORMULA] radial profile are in quite good agreement with the observed data (Kennicutt 1989). This is also true for the UV luminosity at 2000 Å and the 6 cm luminosity. We are thus confident that the massive star population is well constrained by the observed data. With a low mass cutoff at 10 [FORMULA] stars, we overestimate the UV luminosity at 2000 Å of the modelled stellar population by 20%, because we miss the contribution to the UV continuum of lower mass stars, between 2 and 10 [FORMULA].


Table 2. Input parameters for the standard model.


Table 3. Output values from the simulation for the standard model. Notes: All luminosities are emergent luminosities. The total UV luminosity generated by OB associations in the disk between 912 and 2000 Å is [FORMULA] [FORMULA].

4.1. Disk opacity in the UV

We have computed an average opacity over the galaxy in the UV and for the H[FORMULA] line. We define this opacity as:
[FORMULA]) where [FORMULA] is the total luminosity in the disk at a given wavelength and [FORMULA] is the emergent luminosity. This opacity is computed for two different viewing angles of the model, [FORMULA] for face-on and [FORMULA] for edge-on. We have found a significant opacity for the face-on view, at 1000 Å, 2000 Å and for H[FORMULA], namely [FORMULA], [FORMULA], [FORMULA] for the whole galaxy.

The opacity is controlled simultaneously by the geometry of the molecular cloud ensemble and by the diffuse medium. If we ignore the extinction due to the diffuse component, we find an opacity of 0.51 at 1000 Å. This value is due to geometrical effects, mostly blocking of the UV radiation by molecular clouds, and it does not depend on wavelength. Thus we can write the opacity at any wavelength in the UV as [FORMULA] = 0.51 + [FORMULA], the second term accounting for the wavelength dependence of the extinction in the diffuse medium.

A global opacity of [FORMULA] corresponds to a fraction of approximatively 45% of the far UV stellar radiation leaving the galaxy disk, mostly above or below the main plane. Most of these photons have not been scattered because the probability of leaving the disc after a scattering event is low. This significant fraction of the radiation from massive stars leaking out of HII regions could contribute to the maintenance of the Reynolds layer of ionized gas. The derived face-on opacity at 2000 Å, 0.7, falls well within the range of opacities derived by Buat & Xu (1996). The mean extinction in their sample of nearby spiral galaxies is [FORMULA] 0.9 mag at 2000 Å. Though the opacity is not very large, the mean distance travelled by a UV photon before absorption is quite small, 440 pc, roughly equal to the HI disc thickness. As shown on Fig. 6, there are however photons travelling to much larger distances, 1 to 2 kpc, with a small probability (0.01). Conversely, many zones in the interarm receive very few UV photons. Due to the lower gas density, few OB associations are created in the interarm region. The numerous OB associations in the arms are too distant to contribute to the local radiation field since the arm/interarm separation is larger than 1 kpc in the disk.

[FIGURE] Fig. 6. Distribution of the distance from their parent OB association, travelled by UV photons before absorption. The median distance is 120 pc, and the mean distance is 440 pc.

The distribution of [FORMULA] values provide further information on the radiation field resulting from the OB associations (Fig. 7). Whereas most of the galaxy is exposed to a low UV radiation field, it is possible to find regions with high UV intensity ([FORMULA]) even at a moderate spatial resolution. The total dynamical range of the UV radiation field extends over more than 4 orders of magnitude. This huge variation can be explained by the close association of OB associations and molecular clouds: in a galaxy with a prominent spiral structure, OB associations are born in the spiral arms, where the gas density is the highest. This maximizes both the illumination of molecular clouds by UV radiation and the absorption of UV radiation by molecular gas, hence the heating of molecular gas. For the model galaxy, we find that 30% of the total number of cells with molecular gas are exposed to a strong or median radiation field ([FORMULA]). These cells are located in 40% of the molecular clouds. This figure is comparable to the clouds in Milky Way: Solomon et al. (1985) found that in the Galaxy, at a resolution greater than 10 pc, 25% of the molecular clouds are warm and associated with HII regions. Also, Williams and Mc Kee (1997) estimate that at least one OB star is present in half of the giant molecular clouds with masses larger than 105 [FORMULA]. The probability to find massive stars or clusters associated with a giant molecular cloud increases sharply with the cloud mass and reaches almost 1 for masses larger than 8 [FORMULA] 105 [FORMULA] (Williams & MacKee 1997). Our numerical results are in agreement with these facts.

[FIGURE] Fig. 7. Distribution of the UV intensity measured relative to the ISRF in the modelled galaxy, for cells filled with diffuse gas (dot-dashed line) and for cells filled with molecular gas (full line).

4.2. Far infrared emission

We now discuss the emergent radiation from the model galaxy and start with the infrared emission. As in the Désert et al. (1990) dust model, the luminosities in the IRAS bands are computed as [FORMULA], where [FORMULA] is the total observed flux density and D is the distance to the object. The infared colors are given as the flux density ratios, to compare with observed data.

The model galaxy has very similar emissions as NGC 6946 at 60-100 & 200 [FORMULA], with outputs of 5.1, 8.5 and 4.9 [FORMULA] 109 [FORMULA], corresponding to 114%, 128% and 144% of the luminosities observed at those wavelengths. The far infrared emission comes from both the molecular and atomic gas phases.

The UV radiation is the main heating mechanism of the dense and diffuse gas phases, with contributions of 4.2 [FORMULA] 109 [FORMULA] and 6.1 [FORMULA] 109 [FORMULA] at 60 & 100 [FORMULA]. The contribution to the FIR emission of the old stellar population, described by the ISRF, is a factor 3 lower, with 0.9 [FORMULA] 109 [FORMULA] and 2.4 [FORMULA] 109 [FORMULA] in the 60 & 100 [FORMULA] bands. The situation is different at 200 [FORMULA], where dust grains heated by the UV radiation or by the ISRF have almost equal contributions to the total luminosity: 3.0 [FORMULA] 109 [FORMULA] for the UV and 1.9 [FORMULA] 109 [FORMULA] for the ISRF. The contribution from the inner parts of clouds illuminated by the attenuated ISRF is only 0.5 [FORMULA] 108 [FORMULA]. As a whole, 72% of the far infrared luminosity can be attributed to UV heated gas, which is mostly molecular. The remaining 28% corresponds to dust heated by the ISRF, at locations far away from the OB associations.

The diffuse and dense phases have similar contributions to the total FIR emission, with a slight excess from the molecular clouds, 54% versus 46% from the diffuse gas. This significant contribution from the diffuse gas is due to the fact that it occupies a large fraction of the galaxy volume. Hippelein et al. (1996) also conclude from ISO observations of other nearby galaxies (M51, M101) that the neutral atomic gas has an important contribution to the far infrared emission. The contribution from the atomic gas may be underestimated because we do not take into account the compression of the diffuse gas in the spiral structure. Comparing with molecular clouds, we can estimate that, having atomic gas concentrated in the spiral arms would result in a brighter FIR emission, with a slighty warmer color temperature since the dust grains would be closer (in average) to the heating sources. A precise estimate of the magnitude of the effect is beyond the scope of this paper.

The global infrared excess for the model galaxy, IRE, is defined as the luminosity ratio IRE= [FORMULA], with [FORMULA] and [FORMULA]=13.6 eV. At the disk scale, the IRE takes the value 5.9, in agreement with observations of Galactic HII regions (Caux et al. 1985, Myers et al. 1986).

The diffuse and dense gas (atomic and molecular) have the following contributions to the total luminosity of the C+ 158 [FORMULA] line: 77% from the dense phase and 23% from the diffuse phase. Less than 104 [FORMULA] comes from HII regions. The total emission of the galaxy is 2.5 [FORMULA] 107 [FORMULA], a factor 2.5 lower than the measured value, 6.3 [FORMULA] 107 [FORMULA] (Madden et al. 1993). Compared to the 60-100 [FORMULA] far infrared emission, the C+ line represents 0.21% of the FIR (60-100 [FORMULA]) emission. This figure is comparable to the observed ratio for other spiral galaxies with 0.1-1% (Lord et al. 1996). Nevertheless, the value for NGC 6946 was found to be 0.6% (Madden et al. 1993), and in the Galaxy, Shibai et al. (1991) and Wright et al. (1991) have measured [FORMULA] = 0.7% LFIR with the same definition of LFIR as above.

4.3. Radial profiles

The 60 [FORMULA] radial profile is shown on Fig. 8a. There is a large decrease from the inner to the outer parts of the disk, about two orders of magnitude. In NGC 6946, the same behaviour has been observed by Tuffs et al. (1996) using ISO. Averaged over the model, the S60/S100 infrared color appears to be slightly different in the two phases: 0.32 for the diffuse phase and 0.38 for the dense phase. This FIR color decreases with increasing radius from 0.40 in the center to 0.23 at R [FORMULA] 5 kpc (Fig. 8b), in agreement with the maps by Engargiola (1991). The decrease is seen in both phases, with S60/S100 ranging from 0.35 to 0.23 for the diffuse phase, and from 0.42 to 0.28 for the dense phase.

[FIGURE] Fig. 8a. Radial profiles of the 60 [FORMULA] surface brightness. The thin solid line shows the combination of the two gas phases, while the triangles show the contribution of the diffuse atomic gas, and the squares the contribution from the molecular gas. The thick solid line shows the mean radial profile deduced by Engargiola (1991) from an IRAS image.

[FIGURE] Fig. 8b. Radial profiles for the ratio of 60 and 100 [FORMULA] fluxes, global and for the two phases.

The radial profile of the intensity of the C+ [FORMULA] - [FORMULA] 158[FORMULA] line (Fig. 8c) shows a much flatter gradient than the FIR emission. This is due to the logarithmic dependence of the line intensity on the incident radiation field in PDRs. As shown in Figs. 8c and 8d, the diffuse atomic gas is the main source of C+ emission at large distance from the nucleus, for radii larger than 4 kpc. It is thus possible to determine the intrinsic [FORMULA] luminosity ratio from the two gas phases, using the data at [FORMULA]kpc for the molecular gas and data at [FORMULA]kpc for the atomic gas. We find that [FORMULA] is equal to 0.10% in the diffuse phase and to 0.25% in the dense phase.

[FIGURE] Fig. 8c. Radial profiles of the C+ emission: global and for the two phases.

[FIGURE] Fig. 8d. Radial profiles of [FORMULA]: global and for the two phases.

4.4. Maps

We show on Fig. 9 face-on maps of 100 [FORMULA], C+ and UV(912-2000 Å) emissions. Edge-on maps at the same wavelengths are shown in Fig. 10 for comparison. Compared to the C+ observations of the edge-on galaxy NGC 891 (Madden et al. 1994), there is an overall agreement. In particular, the scale height in C+ is predicted to be larger than the scale height of the CO emission, due to the contribution of the diffuse neutral and ionized media which have a larger scale height (Fig. 11).

[FIGURE] Fig. 9a-c. Face-on views of: a  100 [FORMULA] emission, at 48 pc resolution. The gray scale ranges from 1 to 103 [FORMULA] pc-2. We have overlaid contours of the same map convolved with a 750 pc beam: the levels are at 10, 30, 60, 100, 200 [FORMULA] pc-2. b  C+ line, at 48 pc resolution. The gray scale ranges from 10-2 to 1 [FORMULA] pc-2. Overlaid contour levels from 0.1 to 0.6 by 0.1 [FORMULA] pc-2 for the same image convolved with a 750 pc beam. c  emergent UV surface brightness (resolution: 48 pc). The gray scale ranges from 10-2 to 105 [FORMULA] pc-2.

[FIGURE] Fig. 10a and b. Edge-on views of NGC 946, at 48 pc resolution. The linear scale is not identical for both axes. a  100 [FORMULA] IRAS band, with contour levels at 103, [FORMULA], [FORMULA] and 104 [FORMULA] pc-2. b  C+, at the same resolution, accounting for the line opacity. The gray scale ranges from 0 to 50 [FORMULA] pc-2. Contour levels from 10 to 50 by 10 [FORMULA] pc-2.

[FIGURE] Fig. 11. Average vertical profiles through the disk for the C+ (dot-dashed line) and CO(1-0) (full line) emissions. The CO(1-0) profile presents a smaller scaleheight (40 pc) than the C+ profile (100 pc). We have assumed that the CO(1-0) emission is proportional to the molecular gas column density.

In the face-on C+ map, there is a large hole in the interarm regions in the NW, at a similar position to the hole detected by Madden et al. (1993) with the KAO. This hole is due to the lower density of molecular gas and of OB associations in the interarm regions. Therefore few UV photons illuminate this region and the radiation field is very low. The map shows many details and a large contrast between arm and interarm regions. We have smoothed the image from the model to the resolution of the KAO observations (50" beam = 1.2 kpc at the distance of NGC 6946). The contrast between the brightest regions and the disk drops by a large factor. This resolution effect may explain the low dynamical range found in the observed data. If PDRs are the main source of C+ 158[FORMULA] radiation in galaxies, we predict that the emission should have more contrast at higher spatial resolution. This could be tested by maps of external galaxies made with the future Stratospheric Observatory For Infrared Astronomy (SOFIA).

The edge-on maps at 100 µm and in C+ are fairly symmetrical with respect to the center. The edge-on C+ map shows however a hole in the central region ([FORMULA]pc) which does not appear on the 100 µm map. This hole is largely due to the large opacity for these lines of sight ([FORMULA] = 0.4).

4.5. Sensitivity of the model to input parameters

The model results are of course sensitive to the input parameters, therefore we have run different models deviating from the standard model by one parameter.

Because of the poor knowledge of the albedo in UV, we have run a model with a lower albedo of dust grains, [FORMULA]= 0.4. We find that the opacity increases to 1.0 at 1000 Å & 0.90 at 2000 Å. The 60 [FORMULA] emission from the dense phase increases by 5%, while the 100 and 200 [FORMULA] emissions both decrease by 10%. This difference in far infrared emission is due to the moderate increase of the opacity which leads to a warmer dust temperature. The effect on the emission from the diffuse phase is negligible.

A more extreme case is for a null albedo, suppressing any scattering effect. In that case, we maximize the UV opacity and the FIR emission. The opacity increases to 1.25 at 1000 Å and 1.01 at 2000 Å respectively. As a result of this larger absorption, the 60-200 [FORMULA] emission increases by 47% as compared to the standard model.

In another run, we have kept the total mass of molecular gas constant, but used a lower mean density, 20 H2 cm-3 instead of 50 H2 cm-3, to increase the clouds sizes. The volume filling factor is then 3.8%. These larger clouds block more light, and 30% only of the molecular cells are heated, instead of 40% in the standard model. As a result, the 60-200 [FORMULA] luminosity decreases by 10%, to [FORMULA] [FORMULA].

We have also investigated the effect of the number of OB associations: we have kept the same star formation rate but have gathered adjacent associations to form more powerful sources. As a consequence, nOB decreases from 12000 to 3000. Then a smaller fraction of the cloud population is heated, 15%, as compared to 40% in the standard model. But because these cells are heated by more powerful OB associations, the far-infrared emission is larger and reaches [FORMULA] [FORMULA]. Thus the FIR emission depends slightly on the number of associations. The C+ emission decreases to [FORMULA] [FORMULA], because of the smaller number of illuminated clouds.

If we now increase the SF efficiency, from 5 to 10%, so as to double the UV luminosity, the production rate of Lyman continuum photons increases by 80%. In that case, the mean UV opacity is 0.78. The FIR 60-200 [FORMULA] luminosity increases by 55% to 28.7 [FORMULA] 109 [FORMULA]. This shows that the FIR emission is not a linear function of the UV luminosity in our model. This non-linear behaviour arises because the opacity is largely controlled by geometrical effects. With a larger star formation activity, HII regions are very large and can destroy molecular clouds efficiently. Thus the mass of molecular gas decreases in the model with a higher SFR. This is the main reason for the non-linear behaviour. This result has been established with the same number of OB associations, while an increased SFR will probably lead to more associations in the disk. However we have previously shown that the FIR emission does not depend strongly on the number of OB associations.

We have investigated the effect of the atomic density on the size of HII regions, because we probably overestimate the diameter of HII regions, using a mean atomic density and neglecting the dust absorption. If the local gas density is multiplied by two, the volume of the Strömgren sphere is 4 times smaller than in the standard model. The 60-200 [FORMULA] luminosity of dense molecular gas increases by 10% to [FORMULA] [FORMULA]. This is explained by the reduced destructive effect of HII regions on molecular clouds, and then the larger chance for photons to be absorbed by molecular gas. The respective contributions from the diffuse and dense gas to the FIR(60-200 [FORMULA]) are now 34% and 66%.

This last test shows that the distance between clouds and OB associations has a strong influence on the UV reprocessing by dense gas. For the standard model, we have calculated the mean distance between an OB association and the nearest cloud edge, d[FORMULA], and have found a value of 35 pc, the mean distance between clouds centers is 37 pc. To have a larger separation between clouds and OB associations, we have increased vescape to 30 km s-1. We obtain d[FORMULA]= 39 pc. The FIR(60-200 [FORMULA]) emission from the dense phase decreases by 15% because of the smaller solid angles of the clouds viewed from the associations. As for the FIR (60-200 [FORMULA]) from the diffuse phase, it slightly increases by 6%.

We have shown that part of the UV opacity is due to geometrical effects. Indeed the UV opacity is lower when the molecular clouds are distributed uniformly in the disk. We have used an earlier epoch of the simulation, when the distribution of gas clouds is axisymmetric. We have kept the same value for the other parameters (number of OB associations, star formation rate, etc.). In that case, the clouds occupy a larger fraction volume of the galactic disk, and the mean distance between clouds increases. Because of this larger mean distance between clouds, the opacity at 1000 Å decreases to 0.47.

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© European Southern Observatory (ESO) 1998

Online publication: September 30, 1998