4 Analysis

4.1 Modelling

 

 

Table 5: Model parameters

Model	Component 1			Component 2			Ratioa	Line Ratios
	$T_{\rm k}$	n(H2)	N(CO)/dV	$T_{\rm k}$	n(H2)	N(CO)/dV	Comp.	$\,{\rm ^{12}CO}^{b}$	$\13co^{c}$
	(K)	( $\,{\rm cm^{-3}}$)	( $\cm2/\,{\rm {km\,s^{-1}}}$)	(K)	( $\,{\rm cm^{-3}}$	( $\cm2/\,{\rm {km\,s^{-1}}}$)	1:2
NGC 6946
1	30	1000	$10\ 10^{17}$	150	1000	$0.3\ 10^{17}$	1:9	1.22 0.69 0.40	11 9.9 13
2	60	1000	$1\ 10^{17}$	30	10 000	$0.6\ 10^{17}$	6:4	1.12 0.73 0.41	11 10.0 13
3	150	500	$1\ 10^{17}$	30	10 000	$0.6\ 10^{17}$	8:2	1.18 0.73 0.44	11 9.8 13
4	100	1000	$1\ 10^{17}$	--	--	--	--	1.33 0.73 0.44	12 9.3 13
M 83
5	30	3000	$1\ 10^{17}$	100	3000	$0.06\ 10^{17}$	4:6	0.93 0.73 0.40	11 8.9 12
6	60	1000	$1\ 10^{17}$	60	100 000	$1\ 10^{17}$	9:1	1.14 0.76 0.49	10 9.5 12
7	150	500	$3\ 10^{17}$	60	3000	$0.6\ 10^{17}$	3:7	1.13 0.77 0.51	10 9.0 12
8	100	1000	$1\ 10^{17}$	--	--	--	--	1.33 0.73 0.44	12 9.3 13

Notes ^a Ratio denotes the relative contributions of the two components to the observed emission in the J=2-1 $\,{\rm ^{12}CO}$ line.
^b Model-calculated intensities of the J=1-0, J=3-2 and J=4-3 $\,{\rm ^{12}CO}$ transitions normalized to the J=2-1 $\,{\rm ^{12}CO}$ intensity.
^c $\,{\rm ^{12}CO}/\13co$ intensity ratios in the J=1-0, J=2-1 and J=3-2 transitions.

 

 

Table 6: Beam-averaged results

Model	Beam-Averaged Column Densities			Total Central Mass		Face-on Mass Density
	N(CO)	N(C)	$N({\it\h2})$	$M(\h2)$	$M_{\rm gas}$	$\sigma(\h2)$	$\sigma_{\rm gas}$
	( $10^{18} \cm2$)		( $10^{21} \cm2$)	(107 $M_{\odot}$)		($M_{\odot}$/pc-2)
NGC 6946; $N_{\rm H}/N_{\rm C}$ = 2500; N(HI) $^{a} = 1.3 ~ 10^{21} \cm2$
1	1.5	0.9	2.4	2.1	3.6	29	51
2	0.7	1.3	2.1	1.8	3.3	26	45
3	0.9	1.2	2.0	1.8	3.2	25	45
4	0.8	1.1	1.7	1.5	2.8	21	39
M 83; $N_{\rm H}/N_{\rm C}$ = 2500; N(HI) $^{b} = 0.6 ~ 10^{21} \cm2$
5	0.8	3.9	5.5	1.8	2.6	80	114
6	1.0	4.3	6.3	2.4	3.4	92	130
7	1.1	3.7	5.7	2.1	3.0	83	118
8	0.9	8.2	11.0	3.7	5.1	159	221

Notes: ^a Boulanger $\&$ Viallefond (1992); ^b Rogstad et al. (1974); Tilanus $\&$ Allen (1993).

The observed $\,{\rm ^{12}CO}$ and $\13co$ line intensities and ratios can be modelled by radiative transfer models such as described by Jansen (1995) and Jansen et al. (1994). The models provide line intensities as a function of three input parameters: gas kinetic temperature $T_{\rm k}$, molecular hydrogen density n(H₂) and CO column density per unit velocity (N(CO)/dV). By comparing model line ratios to the observed ratios we may identify the physical parameters best describing the actual conditions in the observed source. The additional filling factor is found by comparing model intensities to those observed. If $\,{\rm ^{12}CO}$ and $\13co$ have the same beam filling factor, a single component fit requires determination of five independent observables. As we have measured seven line intensities, such a fit is, in principle, overdetermined. In practice, this is not quite the case because of significant finite errors in observed intensities, and because of various degrees of degeneracy in the model line ratios. We found that a single-component fit could be made to the data of the two galaxies only if we allow CO J=1-0 intensities to be rather higher than observed. As we also consider a single temperature, single density gas to be physically implausible for the large volumes sampled, we reject such a fit.

It is much more probable that the large linear beams used sample molecular gas with a range of temperatures and densities. We approximate such a situation by assuming the presence of two independent components. As this already doubles the number of parameters to be determined to ten, a physically realistic more complex analysis is not possible. In our analysis, we assume identical CO isotopical abundances for both components, and by assuming a specific value (i.e. [¹²CO]/[¹³CO] = 40, cf. Mauersberger $\&$ Henkel 1993) reduce the number of parameters to eight. This leaves a single free parameter, for which we take the relative contribution (filling factor) of the two components in the emission from the J=2-1 $\,{\rm ^{12}CO}$ line. Acceptable fits are then identified by searching a grid of model parameter combinations (10 K $\leq T_{\rm k} \leq$ 250 K, $10^{2} \,{\rm cm^{-3}}\leq n(\h2) \leq 10^{5} \,{\rm cm^{-3}}$, $6~ 10^{15} \cm2 \leq N$(CO)/d $V \leq 3 ~ 10^{18} \cm2$; relative emission contributions 0.1 to 0.9) for sets of line ratios matching the observed set.

The CO line ratios observed for NGC 6946 and M 83 can be fit by various combinations of gas parameters or rather various regions in gas parameter space. We have rejected from further consideration all solutions where the denser component is also required to be the hotter, as we consider this to be physically unlikely on the scales observed. Although the various line ratios are very similar, in particular the isotope ratios for M 83 appear to be systematically somewhat lower than for NGC 6946. For this reason, and also in order to demonstrate the possible variation in model parameters, we have listed characteristic solutions for both galaxies separately. The quality of each solution can be judged by comparing the calculated model line ratios in Table 5 with those observed in Table 4. The single-component fit is included for comparison. The densest and coolest component has a fairly well-determined density of 3000 $\,{\rm cm^{-3}}$ and an even better determined column density N(CO)/dV = 6-1  $10^{16} \cm2$ irrespective of density. The low-density component ( $10^{2}-10^{3} \,{\rm cm^{-3}}$) must have higher column densities N(CO)/dV = 1-10  $10^{17} \cm2$ but its precise temperature is difficult to determine as long as the relative emission ratios of the components are a free parameter.

The observed C$^{\circ }$ and C⁺ intensities are modelled with the same radiative transfer model. For both we assume the CO-derived, two-component solutions to be valid as far as kinetic temperatures, $\h2$ densities and filling factors are concerned. We may then solve for C$^{\circ }$ and C⁺ column densities. The column density of the hotter component is usually well-determined, but that of the cooler component is more or less degenerate. Rather than a single solution, a range of possible column density solutions is found. These are constrained by requiring similar velocity dispersions (of about 3-5 km s^-1) for both hot and cold components and by requiring the resulting total carbon column densities to be consistent with the chemical model solutions presented by Van Dishoeck $\&$ Black (1988). These models show a strong dependence of the N(C)/N(CO) column density ratio on total carbon and molecular hydrogen column densities.

The very strong [CII] intensity observed in M 83 exceeds that expected from the CO-derived solutions. It thus implies the presence of ionized carbon in high-density molecular volumes poorly represented by CO emission. Consequently, in Model 6 (Table 5) we ascribed essentially all [CII] emission to a gas with density $n(\h2) = 10^{4} \,{\rm cm^{-3}}$ at temperature $T_{\rm kin}$ = 60 K whereas in Model 7 we assumed $n(\h2) = 3000 \,{\rm cm^{-3}}$ at $T_{\rm kin}$ = 150 K.

In order to relate total carbon (i.e. C + CO) column densities to those of molecular hydrogen, we have estimated [C]/[H] gas-phase abundance ratios from [O]/[H] abundances. Both galaxies have virtually identical central abundances [O]/[H] = 1.75 10^-3 Zaritsky et al. 1994; Garnett et al. 1997). Using results given by Garnett et al. (1999), notably their Figs. 4 and 6, we then estimate carbon abundances [C]/[H] = $1.45\pm0.5~10^{-3}$. 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 $\delta_{\rm c} = 0.27$ (see for instance van Dishoeck $\&$ Black 1988), so that $N_{\rm H} = [2N($H₂) + N(HI)] $\approx$ 2500 [N(CO) + N(C)] with a factor of two uncertainty in the numerical factor.

The results of our model calculations are given in Table 6, which presents beam-averaged column densities for both CO and C (C$^{\rm o}$ and C⁺) and the $\h2$ column densities derived from these. Table 6 also lists the total mass estimated to be present in the central molecular concentration (R < 300 pc) obtained by scaling the $\h2$column densities with the J=2-1 $L_{\rm tot}/\int T_{\rm mb}$dV ratio from Table 3, and the face-on mass densities implied by hydrogen column density and the galaxy inclination. Beam-averaged neutral carbon to carbon monoxide column density ratios are N(C $^{\rm o})/N$(CO) $\approx\ 0.9\pm0.1$ for both NGC 6946 and M 83, somewhat higher than the 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).

4.2 The center of NGC 6946

Notwithstanding the significant differences between the model parameters, the hydrogen column densities, masses and mass-densities derived in Table 6 are very similar. The [CI] and [CII] line and the far-infrared continuum (Smith $\&$ Harvey 1996) intensities suggest that they predominantly arise in a medium of density close to $10^{4} \,{\rm cm^{-3}}$ subject to a radiation field log $G_{\rm o}$ = 1-1.5 (cf. Kaufman et al. 1999). Emission from the molecules CS, H₂CO and HCN has been detected from the CO peaks in Fig. 2 (Mauersberger et al. 1989; Hüttemeister et al. 1997; Paglione et al. 1995, 1997); their intensities likewise indicate a density $n(\h2) \approx 10^{4} \,{\rm cm^{-3}}$ which is only provided by models 2 and 3 which we consider to be preferable. Note that the single-component CO fit (model 4), which we have already rejected, also does not fit the C$^{\rm o}$ and C⁺ intensities predicted by the PDR models (Kaufman et al. 1999). The high-density component probably corresponds to the molecular cloud complexes that are the location of the present, mild starburst in the center of NGC 6946 (Telesco et al. 1993; Engelbracht et al. 1996). It represents about a third of the total molecular mass, and contributes a similar fraction to the observed J=2-1 CO emission. The low-density component has a temperature in the range $T_{\rm kin} = 60$-150 K, and a density of order $n(\h2) \approx \,{\rm cm^{-3}}$. This is conformed by a reanalysis of the midinfrared $\h2$ measurements by Valentijn et al. (1996). The J=2-0 S(0) $\h2$ line intensity at 28 $\mu$m is entirely consistent with these values for an ortho/para ratio of two (P.P. van der Werf, private communication). However, in order to also match the observed J=3-1 S(1) line strength at 17 $\mu$m, a small amount of high-temperature molecular gas with $T_{\rm kin} \approx 500$ K, $n(\h2) \approx 5000 \,{\rm cm^{-3}}$ need be present as well, but with a mass no more than a few per cent of the mass given in Table 6. Our measurements are insensitive to such a component.

We thus conclude that the total mass of molecular gas within R = 0.5 kpc from the nucleus of NGC 6946 is $18\pm3$ million solar masses; this is about 1.5 per cent of the dynamical mass, so that the total mass of the inner part of the galaxy must be completely dominated by stars. No more than a quarter of all hydrogen is HI; most is in the form of $\h2$. Between 15 and 25$\%$ of all hydrogen is associated with ionized carbon and almost equal amounts with neutral carbon and CO. Madden et al. (1993) reach very similar conclusions from C⁺ mapping of NGC 6946, but find different masses. Part of this difference arises in our use of two components rather than a single component. Another important difference between this and other studies is our use of the gas-phase carbon abundance rather than an assumed conversion factor to obtain hydrogen column densities and masses. For NGC 6946, this results in effective conversion factors of the order of X = 1  10¹⁹ $\h2$ mol $\cm2$/ $\,{\rm {K\,km\,s^{-1}}}$, which is more than an order of magnitude lower than traditionally assumed values. The difference greatly exceeds the uncertainty of a factor of two or three associated with the carbon abundance, illustrating the danger of using "standard'' conversion factors in centers of galaxies where conditions may be very different (higher metallicities, higher temperatures) from those in galaxy disks.

The observed CO temperatures are typically a factor of 15 lower than the model brightness temperatures, implying that only a small fraction of the observing beam is filled by emitting material. We find small beam-filling factors for the molecular material of order 0.06 - not very dependent on choice of model. However, velocity-integrated intensities are a factor of two or three higher than that of a model cloud, implying that the average line of sight through NGC 6946 contains two or three clouds at various velocities.

4.3 A GMC in the bulge of NGC 6946

Figure 6: J=2-1 emission profiles at offset positions; vertical scale is in $T_{\rm mb}$. Left: NGC 6946 at $\Delta \alpha , \Delta \delta$ = +10'', +10''. Right: M 83 at $\Delta \alpha , \Delta \delta$ = +7'', -7''. In both cases, the $\13co$ profile, multiplied by a factor of seven, is included as the lower of the two curves

The eastern extension seen in our J=3-2 and J = 2-1 maps is caused by a discrete cloud at $\Delta \alpha , \Delta \delta$ = +12'', +6''. At this position, CO profiles show a strong, relatively narrow spike asymmetrically superposed on the weaker broad profile from the more extended emission (Fig. 6). This spike can also be discerned in J=1-0 CO profiles published by Sofue et al. (1988) and in the J=3-2 and J=4-3 CO profiles by Nieten et al. (1999). By subtracting the broad emission, we have attempted to determine the parameters of this cloud. We find a deconvolved size of about 400 pc along the major axis and $\leq$160 pc (i.e. $\leq$260 pc deprojected) along the minor axis. Its deprojected distance to the nucleus is about 350 pc. Peak emission occurs at $V_{\rm LSR} = -20~\,{\rm {km\,s^{-1}}}$, and the linewidth is $\Delta V(FWHM) = 30~\,{\rm {km\,s^{-1}}}$. These results suggest that the object is a molecular cloud complex in the bulge of NGC 6946, comparable to the Sgr B2 complex in the Milky Way. Although the subtraction procedure is not accurate enough to obtain good line ratios, these do not appear to be very different from those of the major central concentration. They indicate a total mass $M(\h2) \approx 3~ 10^{6}$ $M_{\odot}$ for the complex. Most of this mass should be at a kinetic temperature of about 10 K, but about 15$\%$ of the total mass should experience a temperature of order 100 K.

4.4 The center of M 83

Not surprisingly in view of the very similar CO line ratios, the radiative transfer solutions for M 83 do not differ much from those for NGC 6946 (Table 5). The major difference is found in Table 6, and is caused by the much stronger [CII] emission. With model 5, the [CII] intensity can be reproduced using the CO derived gas parameters, but only if in the hot 100 K component essentially all (94$\%$) carbon is in the ionized atomic form C⁺; very little CO can be left. Use of the CO two-component parameters requires solutions with implausibly high C⁺ column densities for models 6 and 7. As already mentioned in the previous section, we have instead assumed that the [CII] emission from M 83 mostly samples conditions inbetween those of the two components, i.e. those at the interface of hot, tenuous and colder, denser gas.

Whichever model is preferred, typically 50$\%$-65$\%$ of all carbon in the center of M 83 must be in ionized form. Because of this, and the rather low HI column density observed towards the center of M 83, molecular hydrogen column densities must be quite high, of order 5-7  $10^{21} \cm2$. Although only a relatively small fraction of all $\h2$ is related to CO emission, the conversion factor is nevertheless higher for M 83 than for NGC 6946: $X = 0.25 ~ 10^{20}\cm2/\,{\rm {K\,km\,s^{-1}}}$, but still well below the Galactic standard value.

The models are consistent with densities $10^{3} \,{\rm cm^{-3}}$ subject to radiation fields log $G_{\rm o} = 2$ implied by comparing the CO, [CI] and [CII] line and far-infrared continuum (Smith $\&$ Harvey 1996) intensities with the PDR models given by Kaufman et al. (1999). Few density estimates from other molecules exist. Paglione et al. (1997) estimate $n(\h2) \leq 10^{3} \,{\rm cm^{-3}}$ from HCN J=3-2 and J=1-0 measurements, whereas the beam-corrected ratio I(CO)/I(HCN) = 9 (J=1-0) from Israel (1982) suggests $n(\h2) \approx 3~ 10^{4} \,{\rm cm^{-3}}$ (see Mauersberger & Henkel 1993, their Fig. 4).

An important difference between NGC 6946 and M 83 is that the strong [CII] emission characterizing the latter cannot be explained by assuming that only relatively modest amounts of carbon monoxide have been photodissociated into atomic carbon. The considerably stronger starburst in M 83 (Gallais et al. 1991; Telesco et al. 1993; Turner $\&$ Ho 1994) has apparently created a PDR-zone in which large amounts of high-temperature, high-density ionized carbon gas have largely replaced efficiently eroded CO clouds, so that a significant fraction, of order 80%, of the molecular hydrogen in this PDR-zone is effectively not sampled by CO emission. Dense, [CII] emitting gas is thereby a major contributor to the total gas content of the center of M 83. The actual contribution is somewhat uncertain because of the uncertainty in [CII] gas temperature. If we take $T_{\rm kin} = 250$ K and $n(\h2) = 10^{4} \,{\rm cm^{-3}}$ instead of the actual values adopted, the resulting masses for models 6 and 7 in Table 6 would be about 60$\%$ of the listed values.

We conclude that the total amount of molecular gas in the center of M 83 ($20\pm10$ million solar masses) is very similar to that in NGC 6946 ($18\pm3$ million solar masses). As in the case of NGC 6946, this is of order 1-2 per cent of the dynamical mass, so that the mass of gas is negligible with respect to the stellar mass. About 6$\%$ all hydrogen is HI; the remainder must be in the form of $\h2$. About half of all hydrogen is associated with ionized carbon; the other half is mostly associated with CO. We thus confirm the predominant role for C$^{\rm o}$ that was already found by Crawford et al. (1985) and Stacey et al. (1991). As for NGC 6946, we note that the total molecular mass found in the central region (R < 0.5 kpc) is much less than suggested by others on the basis of assumed conversion factors.

In M 83, observed CO temperatures are typically a factor of 7.5 (Model 5) to 15 (Model 7) lower than the model brightness temperatures, indicating beam-filling factors for the molecular material of order 0.12-0.06, i.e. larger than for NGC 6946. At the same time, the velocity-integrated intensity is a factor of two to five larger than that of a model cloud, implying the presence of that number of clouds in an average line of sight through M 83. Although the central gas masses in NGC 6946 and M 83 are very similar, the face-on mass density in the center of M 83 is more than double that of NGC 6946.