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Astron. Astrophys. 351, 10-20 (1999)

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4. Analysis and discussion

4.1. Modelling of observed line intensities

We have modelled the observed intensities and their ratios by assuming the presence of two molecular gas components of different temperature and density, a relatively cold component dominating the J=1-0 emission and a warmer component becoming progressively more important in the higher transitions. We have used the radiative transfer models from the Leiden astrochemistry group (Jansen 1995; Jansen et al. 1994); we included a background radiation field of [FORMULA] = 2.73 K. In these models kinetic temperature, molecular hydrogen density and CO respectively C column densities function as input parameters. A further constraint is provided by the chemical models discussed by Van Dishoeck & Black (1988) which show a strong dependence of the [FORMULA] column density ratio around molecular hydrogen column densities of about 10[FORMULA]. Above [FORMULA], virtually all carbon is in CO, whereas below [FORMULA] virtually all carbon is in C. The CI and CO intensities observed in position A, together with the sensitivity of the C/CO ratio to [FORMULA] column density provide rather stringent constraints on the models acceptable for at least the center of NGC 7331.

Although the available data do not allow a precise and unique determination of the physical condition of the molecular gas in NGC 7331, they serve well to constrain the parameter space of possible solutions. For instance, cold component CO column densities cannot exceed [FORMULA] [FORMULA] and warm component kinetic temperatures cannot be lower than 20 K without conflicting with observed 12CO and 13CO ratios. In Table 5 we list a few representative models. Note that the CO column densities listed in Table 5 are those of a single model cloud. We assume a homogeneous population of such model clouds, non-shadowing in position-velocity space, so that the actual galaxy beam-averaged column density is the sum of the model cloud column densities in the beam multiplied by their beam filling factor.


[TABLE]

Table 5. Model parameters for NGC 7331.
Notes:
a) Cold component assumed to consist of both low and high density gas as in D 478; see Israel et al. (1998).


In models 4 through 7 we vary the cold component input parameters, and assume a warm component of 30 K and density [FORMULA] = 1000 [FORMULA]. In models 2 and 3, we have changed the warm component temperature to 40 K and 20 K, and in model 1 we assume a more complex situation. In addition to the warm component, the cold component itself is structured into a high-density and a low-density contributor. It is a scaled version of the single-temperature, dual-density model ([FORMULA] = 10 K; [FORMULA] and 3000 [FORMULA]) applied to the M 31-D 478 cloud complex by Israel et al. (1998). Although a warm component must be included, its nature is unclear. Given the large linear beamsize (1.5 [FORMULA] 5.5 kpc) in the plane of the galaxy, this warm component may represent discrete, starforming cloud complexes at some distance from the center. In all models, J=1-0 [FORMULA] intensities are dominated by emission from the cold component with contributions of about 75%, 70% and 85% for positions A, B and C respectively. In contrast, the J=3-2 [FORMULA] intensities are all dominated by emission from the warm component. For the optically thin [FORMULA] transitions the situation is less clearcut: if we assume an intrinsic isotopic ratio of 100, emssion from the cold component contributes about 25% to the J=1-0 emission from positions A and B, whereas this fraction increases to about 45% if we assume an isotopic ratio of 50.

4.2. CO and C column densities

In order to relate neutral carbon and carbon monoxide column densities to that of molecular hydrogen, we have used [C]/[H] gas-phase abundance ratios estimated from the [O]/[H] abundance. From the data tabulated by Zaritsky et al. (1994) we determined for the central beam (position A) 12 + log (O/H) = 9.2, i.e. [O]/[H] = 1.5 [FORMULA] 10-3. Although high, such an oxygen abundance is normal for galaxy centers (Garnett et al. 1997; van Zee et al. 1998). Using results given by Garnett et al. (1999), notably their Figs. 4 and 6, we arrive at an estimated carbon abundance [C]/[H] = 2[FORMULA]1 [FORMULA] 10-3. As a significant fraction of all carbon will be tied up in dust particles, and not be available in the gas-phase, we adopt a fractional correction factor [FORMULA] = 0.33. Neglecting contributions by e.g. 13CO and ionized carbon, we thus find [FORMULA] = [2[FORMULA] + [FORMULA]] [FORMULA] 1700 [[FORMULA] + [FORMULA]] with a factor of two uncertainty in the numerical factor. Similarly, we find for the off-center positions B and C numerical factors of 2300 and 3000. The beam-averaged column densities in Table 6 have been obtained by scaling the model cloud column densities by the ratio of actual observed CO intensity to predicted model CO intensity.


[TABLE]

Table 6. CO and C in NGC 7331


The results of our model calculations are given in Table 6. In the table we give the predicted [CI] intensity [FORMULA], which can be verified observationally, the calculated beam-averaged column densities for both CO and C, the [FORMULA] column densities derived from these using the [FORMULA] ratios and [FORMULA] values given, as well as the implied CO to H2 conversion factor [FORMULA]. The neutral carbon intensities [FORMULA] were calculated under the assumption that a significant fraction (0.6-0.7) of the total atomic carbon column density is ionized and present in the form of [CII]. Changes in the input CO column densities do not strongly affect the resultant C column density: a substantially higher [FORMULA], for instance, implies a lower [FORMULA] ratio, yielding a relatively unchanged [FORMULA]. We have also performed the calculations for a ratio of 50. Generally, the ratio of 100 provides a better fit to the [FORMULA] data than the ratio of 50. As the end results for the two sets are moreover very similar, we have not included the latter in the table. The results of all models are given for position A, where we have also measured the [CI] intensity in a 10" beam. However, the models apply to measurements in the 21" beam observed or synthesized for CO. If atomic carbon is at a minimum in the center, [CI] intensities in a twice larger beam may be somewhat higher, perhaps by as much as 40[FORMULA]. Table 6 shows that model 1 yields a very good fit, whereas models 2 through 5 are marginally possible and models 6 and 7 are ruled out. Model 1 is not unique; various other combinations of somewhat different kinetic temperatures for both cold and warm gas and somewhat different densities, yield very similar results.

As models 6 and 7 are ruled out for position A and models 2 through 5 yield almost identical final results, we present only models 1 and 3 for positions B and C. The results for position C have relatively large uncertainties due to the weakness of its emission, and the lack of a J=1-0 [FORMULA] measurement. The results are not greatly different from those obtained at position A. Column densities decrease, and X factors increase somewhat with radius. At [FORMULA] = 10 K, the 3P2-3P1 [CI] transition at 809 GHz has negligible intensity, but this becomes comparable to the 3P1-3P0 492 GHz transition at [FORMULA] = 30 K. The presence of the warm component can therefore be verified by future observations of the 809 GHz [CI] transition, for which we predict an intensity of 15-30% of the 492 GHz intensity. For the [CII] emission we expect intensities of the order of [FORMULA] erg s-1 cm- 2 sr-1.

Beam-averaged neutral carbon to carbon monoxide column density ratios are [FORMULA] = 0.65[FORMULA]0.1 and [FORMULA] = 5.5[FORMULA]1.0 for models 1 and 3 respectively. The former is close to the typical values 0.2-0.5 found for M 82, NGC 253 and M 83 (White et al. 1994; Israel et al. 1995; Stutzki et al. 1997; Petitpas & Wilson 1998), but the latter is much higher and is only matched by the corresponding ratio of 3-6 found in Galactic translucent clouds (Stark & van Dishoeck 1994).

4.3. Molecular hydrogen and the I(CO) to N(H2) ratio

Although any explanation of the observed CO intensities requires the presence of both cold and smaller amounts of warm molecular gas in NGC 7331, the range of admissible parameters is not fully constrained. The [CI] intensity observed towards the center of the galaxy, however, strongly suggests a complex physical environment of the sort represented by model 1. This model is characterized by cold molecular gas (typical temperature [FORMULA] = 10 K) present at both high and low volume densities (typically of order a few hundred and a few thousand per cc respectively), in addition to a warmer component (temperature [FORMULA]) of high density. This is probably a simplification: in reality a range of densities and temperatures is likely to be present. As the large linear beamsize (1.5 kpc along the major axis, 5.5 kpc along the minor axis) only provides results averaged over a large radial range, the spatial distributions of the cold and the warm gas within the beam may well be different. Both kinetic temperature and mean molecular gas density in the centre of NGC 7331 are typically an order of magnitude below the values found in later-type starburst galaxies such as NGC 253 and M 82 (Israel et al. 1995; Wall et al. 1991).

Our models suggest that a large fraction of the CO emission originates from cold gas of low column density. A smaller fraction originates in much denser gas, partly at higher temperatures. The cold gas in the center of NGC 7331 appears to be similar to that in cloud complexes such as D 478 in M 31 dark and the Taurus-Auriga complex in the Milky Way; most likely, it is highly fragmented and filamentary (Israel et al. 1998). The warm gas may be heated by the nucleus and by luminous stars in the inner spiral arms and the `molecular ring'. The presence of energetic photons in the inner part of NGC 7331 is betrayed by H[FORMULA]+[NII] emission (see Fig. 5 by Smith & Harvey 1996) and more directly by significant UV emission (Wesselius et al. 1982) unlikely to be dominated by the spiral arms because of their high dust content (Bianchi et al. 1998).

The evaluation of the models in Tables 5 and 6 assumes a radiation field [FORMULA], corresponding to [FORMULA] photons s-1 cm-2 which is consistent with the longer-wavelength UV data by Wesselius et al. (1982). Such a low radiation field density is also indicated by the strength of the 7.7 and 11.3 µm dust emission features (Smith 1998). The beam-averaged column densities in Table 6 are relatively insensitive to changes in the assumed [FORMULA], because in the cold diffuse gas, most carbon is already in C rather than in CO, whereas the much smaller filling factor of the dense gas greatly reduces the effect of changes in the [FORMULA] ratio on the beam-averaged neutral carbon column density. Moreover, we expect only limited variation (by a factor of 2-3) in the radiation field density over at least the inner 5 kpc because of the smooth distribution of H[FORMULA] emission as well as the far-infrared emission between 50 µm and 200 µm (Smith & Harvey 1996; Alton et al. 1998). The quiescence of the spiral arms is illustrated by the strong excess of 450 and 850 µm emission from cold dust (Bianchi et al. 1998). Thus, the spiral arms contain a relatively large amount of cold dust especially in comparison with the central region. We are therefore confident of the derived [FORMULA] values in Table 6.

As C and O abundances in NGC 7331 are 2-5 times higher than those in the Solar Neighbourhood, and radiation fields are not particularly intense, we expect CO in NGC 7331 to be relatively well-shielded, so that the CO to [FORMULA] conversion factor X should be lower than that in the Solar Neighbourhood, i.e. fewer [FORMULA] molecules per unit CO intensity. Indeed we find values of X lower than the value of [FORMULA] assumed for the Milky Way, which can also be compared to the relationship between X, radiation field intensity and metallicity found by Israel (1997). For positions A, B and C we take [O]/[H] abundance ratios of 1.5, 1.3 and 1.1 in units of 10-3 respectively (Zaritzky et al. 1994). From high-resolution far-infrared surface brightnesses (Smith & Harvey 1996), HI column densities (Begeman (1987) and Eq. (3b) from Israel (1997), we predict values X = 0.8, 0.7 and 1.0 in units of [FORMULA] cm[FORMULA] for positions A, B and C respectively. These are very close to the results from the preferred model 1, and a factor of two or more below the results for the other models. Neglect of the radiation field term in Israel's (1997) Eq. (3b), i.e. use of his Eq. (4) predicts in the same units X = 0.15, 0.25 and 0.4 for positions A, B and C, i.e. much lower than any of the model results. We conclude that the low values of X in the preferred model 1 are in good agreement with both the high abundances in NGC 7331 and the relationship between X, radiation field intensity and metallicity found by Israel (1997).

With respect to the value of X derived for position A it should be noted that the large linear beamsize includes both the center of NGC 7331 and more outlying regions along the minor axis. If the latter were to be characterized by an X value closer to that of position B, the actual central X value would be significantly lower. For instance, if we assign [FORMULA] to the outer half of the CO emission, the inner half would have [FORMULA], an order of magnitude less than the Milky Way value, and well below what is suggested by the high metallicity. Such a very low value would, however, not be unexpected. For the Milky Way centre, Sodroski et al. (1995) conclude to an X-factor 3-10 times smaller than the `standard' Galactic value. The COBE Galactic Centre data presented by Bennett et al. (1994) imply lower CO transition ratios somewhat similar to those in NGC 7331.

Another way of verifying the derived [FORMULA] column densities is provided by the submmillimeter observations presented by Bianchi et al. (1998). For [FORMULA]m = 50 mJy and [FORMULA] K in a [FORMULA] beam (Bianchi et al. 1998), we derive a beam-averaged [FORMULA] = 6.6 (-1.2, +2.2). Furthermore assuming that the dust to gas ratio is proportional to metallicity, we modify the Galactic relation between total hydrogen column density and visual extinction (Bohlin et al. 1978) to [FORMULA] cm-2. This implies a column density [FORMULA] (corresponding to [FORMULA]). The similarly obtained result for position B is slightly lower. These results are thus in rather good agreement, given the various uncertainties, with [FORMULA] and [FORMULA] found for positions A and B using model 1.

Comparison of the models and the observations allows us to draw some general conclusions on the distribution of molecular hydrogen in NGC 7331. In model 1, relative amounts of cold/tenuous, cold/dense and warm/dense molecular hydrogen gas are 45[FORMULA], 30[FORMULA] and 25[FORMULA] for positions A and B. The results for position C seem to indicate a somewhat higher contribution by warm molecular gas. Using the beam-averaged [FORMULA] column densities and the model [FORMULA] volume densities, we find that the average line of sight within the beam contains cold/tenuous [FORMULA] over about 2 pc (Model 1, positions A and B) to 0.7 pc (Model 1, position C). Both the cold and the warm dense component have average line-of-sight extents a factor of 50 lower. However, the observed CO temperatures are much lower than the model excitation temperatures, indicating small beam-filling factors for the molecular material. Assuming individual lines of sight within the beam to be either empty, or homogeneously filled with molecular gas, we find for those line of sights that do contain molecular gas extents of about 20 pc (cold tenous gas), 2.5 pc (cold dense gas) and 25 pc (warm dense gas); these numbers are indicative of the maximum source size that can be expected.

Application of model 3 yields somewhat different results. Here, most of the molecular gas is in the cold/tenuous form rather than in the warm/dense phase: only 17[FORMULA] warm gas is required at position A, and about 5[FORMULA] at positions B and C.Average line of sight extents are 3.5 pc for cold gas at positions A and B, and half that at position C. After correction for beam filling, we find lines of sight extents of typically 115 pc for the cold molecular gas and l5[FORMULA] or less of that for warm gas. Only at position C a more uncertain extent of 20 pc is obtained.

The derived line of sight extents are much smaller than the length of the line of sight traversing the galaxy, which is about four times its thickness. Although the latter is not known, this length can be estimated at well in excess of a kiloparsec. Thus, only a small fraction of the volume sampled by the beam at each of the analyzed positions is filled with molecular material. This material is highly clumped or distributed in filamentary form.

4.4. Radial distribution of molecular gas

In the case of highly-inclined ring structures, major-axis position-velocity diagrams may give a misleading impression of the actual radial distribution of emitting material, because at the tangential points substantially longer lines of sight contribute to the emission. To determine the actual distribution of CO as a function of radial distance from the centre spectra, we have fitted an inclined axisymmetric disk model to the data in the velocity-integrated map (Fig. 3) by applying the Richardson-Lucy iterative scheme (Lucy 1974). In principle, with a priori knowledge of the (CO) velocity field, this technique can also be used to obtain radial distributions with a spatial resolution higher than that of the observing beam (Scoville, Young & Lucy 1983). Fig. 4 shows the fitted radial distribution of the velocity-integrated J=2-1 CO emission.

[FIGURE] Fig. 4. Deprojected radial profiles. Bottom: Face-on radial distribution of CO emission obtained with the Richardson-Lucy scheme (see Sect. 4.4). Vertical axis is [FORMULA]dV as would be observed perpendicular to the galaxy plane. Top: Face-on mass-densities of [FORMULA] and HI, in units of [FORMULA] pc-2. HI data were taken from Begeman (1987).

The fitted profile corresponds to the face-on radial distribution of [FORMULA] = [FORMULA]. The CO luminosity starts at [FORMULA] = 4 K [FORMULA] in the center, reaches a minimum at [FORMULA] 1.75 kpc, and the reaches a maximum at [FORMULA] 3.5 kpc after which it drops smoothly to [FORMULA] = 2 K [FORMULA]. The ring-to-disk intensity contrast ratio is about 0.6. The molecular `ring' is clearly discernible, but it does not dominate the CO distribution in the galaxy. Both the major-axis CO distribution and the fitted radial CO profile are different from those of M 31, where most of the CO is found farther out in the spiral arm `ring' at R=9 kpc, and very little CO occurs at the centre (Dame et al. 1993).

In order to determine the radial distribution of interstellar gas in NGC 7331, we have converted the CO radial profile to a radial distribution of [FORMULA] mass densities, using the J=1-0/J=2-1 CO ratios and X values from Tables 4 and 6, and combined these with the radial HI profile published by Begeman (1987). Both the radial [FORMULA] and HI profiles are also shown in Fig. 4. In the inner 4.5 kpc, the [FORMULA] mass dominates that of HI by about 40[FORMULA]. Beyond this, HI becomes increasingly dominant. The radial distribution of molecular gas reaches its peak (at R =3.1 kpc) well before that of HI (at [FORMULA] 10 kpc). Although the radial J=2-1 CO profile exhibits a relatively low contrast between the ring feature and the underlying disk emission, the radially increasing transitional ratios and X-values serve to enhance the contrast in [FORMULA]. The central region is not empty, but the [FORMULA] mass density inside the ring is only 75[FORMULA] of that in the ring; due to the lack of HI in the center, the relative mass density of all hydrogen is even lower with 55[FORMULA] of the ring value. The face-on [FORMULA] mass distribution increases from [FORMULA] = 6 [FORMULA] pc-2 to 8 [FORMULA] pc-2 at R = 3.1 kpc. The total hydrogen radial mass-density distribution increases from a central value [FORMULA] = 8 [FORMULA] pc-2 to [FORMULA] = 14 [FORMULA] pc-2 at R = 3.5 kpc and then drops slowly. The gaseous fraction (including helium) of the total mass was estimated from the rotation curves given by Rubin et al. (1965) and Begeman (1987), assuming a spherical bulge and circular velocities. Inside the ring, the gas-to-total mass ratio [FORMULA]/[FORMULA] is about 1[FORMULA]. In the ring, it rises to 1.5[FORMULA], and then slowly climbs to 3[FORMULA]. From Begeman's (1987) data, neglecting [FORMULA], we find in comparison a global ratio [FORMULA]/[FORMULA] of 3.2[FORMULA]. Even with the dominant contribution by H2, the gas in the inner part of NGC 7331 is only a minute fraction of the total mass.

Finally, it is of interest to compare the radial gas distributions to the radial far-infrared profiles representing interstellar dust. The 100 µm far-infrared profile (Smith & Harvey 1996) is rather flat in the center, reaching a very minor maximum at about 2 kpc, after which a steep decline sets in. The 450 µ and 850 µm profiles show radially increasing intensities that reach a peak at about 3 kpc, the 850 µm profile peaking a little farther out than the 450 µm profile (Bianchi et al. 1998). The [FORMULA] mass-density peaks at R = 3.1 kpc, just about coincident with the 850 µm maximum. At the location of the molecular `ring' peak, 100 µm intensities are about 80[FORMULA] of those in the center. However, the total gas mass-density peaks slightly farther out at R = 3.8 kpc. Thus, emission from warm dust peaks well inside the molecular ring, and both molecular gas and cold dust peak inside the radius of highest gas mass-density. Beyond the ring peak, the decline of the 450 µ and 850 µm emission more or less follows that of the [FORMULA] mass density profile. We conclude that the interstellar dust is hottest in the central region, where gas mass-densities are lowest. The mean dust temperature defined by the far-infrared ratios smoothly decreases from the center reaching a shallow minimum at about R = 3 kpc, i.e. at the H2 peak , beyond which it appears to increase slightly.

4.5. Origin of bulge molecular gas and the ring

Various mechanisms for the occurence of ring morphologies have been suggested in the literature. Interaction between magnetic fields and the gas distribution were proposed and tested by Battaner et al. (1988). Inner Lindblad resonances may be responsible (Kormendy & Norman 1979), and a ringlike feature may also result from evacuation of gas from the central regions by stellar winds (cf. Faber & Gallagher 1976; Soifer et al. 1986, Muñoz-Tuñon & Beckman 1988). Finally, as Young & Scoville (1982) suggested for NGC 7331, a ringlike appearance may also be caused by the nuclear bulge having used up the originally present molecular gas in the center.

In NGC 7331, the solid-body rotation curve rises rapidly out to R = 3.5 kpc after which it flattens and reaches a broad maximum at R = 6 kpc (see also von Linden et al. 1996). The ring is thus located just at the radius where rigid rotation is lost, but well within the radius of maximum rotational velocity ([FORMULA] = 0.6 [FORMULA]). This is unlike M 31, where the molecular ring is found at a radius twice that of peak rotational velocity (Brinks & Burton 1984; Dame et al. 1993). Both the molecular ring and the boundary of the solid-body rotation region are also just at the radius at which the light of the disk becomes dominant over that of the bulge (cf. Begeman 1987). Although the radius of the mostly nonthermal radio continuum ring radius is more difficult to determine because of its inhomogeneous structure, it appears to coincide more or less with the molecular ring, also well inside the radius of maximum rotational velocity (Cowan et al. 1994).

On the same reasoning as used by Young & Scoville (1982), we may rule out the presence of an inner Lindblad resonance in NGC 7331 as an explanation for the observed molecular ring structure, because an ILR can only occur well outside the region of solid-body rotation (e.g. Kormendy & Norman 1979). Our observations clearly show that there is no pronounced CO hole in the centre of NGC 7331, although the derived distributions of both [FORMULA] and total interstellar gas do show a significant central depression . von Linden et al. (1996) have suggested that the ringlike distribution in NGC 7331 is caused by the dynamical action of a weak central bar. However, their simulations yield both a ring more massive than observed, and center more devoid of gas than observed, casting doubts on the proposed central bar.

In the case of M 31, Soifer et al. (1986) suggest that the amount of interstellar matter observed in the center of M 31 could have accumulated from late-type stellar mass loss in the bulge, and is kept low by continuous gas removal by supernova explosions and star formation. Could this also be the case in NGC 7331? It has been suggested by Prada et al. (1996) that the bulge of NGC 7331 is counter-rotating. Since the bulge gas is rotating in the normal sense, this would seem to preclude a stellar origin for the gas. However, spectroscopy by Mediavilla et al. (1997) and Bottema (1999) does not confirm the suggested counter-rotation. The total mass of the interstellar gas inside R = 2 kpc is 1.6 [FORMULA] 108 [FORMULA]. According to the reasoning outlined by Soifer et al. (1986), stellar mass loss in the bulge would accumulate this amount in 4 [FORMULA] 108 years. Removal of the same amount of material from the bulge requires the energy output of [FORMULA] Type I supernovae, n being the fraction of energy available for the acceleration of interstellar material. For a Type I SN rate of 4 [FORMULA] 10-13 [FORMULA] yr-1 (Lang, 1992) the timescale for removal is 6.5/n [FORMULA] 107 years. Only if the fraction of supernova energy actually available for removal exceeds 17[FORMULA], will the interstellar gas be evacuated from the bulge faster than bulge stars can manufacture it. More generally, with the assumptions from Soifer et al. (1986), the ratio of evacuation to deposition timescales is [FORMULA] = [FORMULA]. The tabulation of NGC 7331 rotation velocities by Begeman (1987) then suggests relatively efficient evacuation in the inner 1 kpc ([FORMULA], and much less efficient evacuation at the edge of the bulge (R = 4 kpc; [FORMULA]). The radial decrease of the ratio of far-infrared emission to [FORMULA] mass-density found above may be related to this finding. It thus appears that the relatively small amounts of interstellar gas in the bulge of NGC 7331 ([FORMULA]) also may well be the result of mass loss from the bulge stars themselves, rather than the result from a net inflow of molecular material from greater radii.

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Online publication: November 2, 1999
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