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Astron. Astrophys. 334, 646-658 (1998)

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4. The physical conditions of the gas

4.1. Determinination of densities and temperatures

A Large Velocity Gradient (hereafter LVG) model was used to estimate physical conditions of the gas, namely density ([FORMULA])), kinetic temperature ([FORMULA]) and - in the case of SiO - molecular abundance (X (SiO)) (see Scoville & Solomon 1974 and De Jong et al. 1975 for the general properties). We used a modified version of the code of Henkel et al. (1980). The basic LVG assumption is that a of systematic velocity gradient large compared to random motions and monotonic. This allows us to treat the molecular excitation as a local problem. This is certainly an idealisation for Galactic center clouds. However, an application of an LVG code requires no detailed knowledge of the velocity field. In the case of CO lines, H2 densities determined under the assumption of LVG and microturbulence (the opposite extreme) do not differ by more than a factor of three (White 1977).

Our analysis is also based on the assumption that, for a given molecule, all transitions observed arise in the same volume. Since the excitation depends on critical density and optical depth, and thus is not identical for different transitions or different isotopomers, this assumption may not be strictly correct, especially if the medium is very clumpy (see, e.g., the discussion in Oka et al. 1998 for 12 CO). For [FORMULA], this is not a critical problem since the LVG calculations show that all transitions measured by us are optically thin, or, in the low temperature/high density scenario described below, reach, at most, [FORMULA]. Therefore, [FORMULA] line intensity ratios are not strongly affected by the possibly different locations of cloud `photospheres' from which the bulk of the photons are emitted. All transitions of [FORMULA] should trace all molecular gas at densities [FORMULA] [FORMULA], as long as [FORMULA] is not selectively dissociated by UV radiation. Away from the star forming regions Sgr A and Sgr B2, the UV radiation field in the Galactic center region is not likely to be strong (e.g. Nagakawa et al. 1995). In addition, because we are considering density peaks of the molecular gas (and dust), the destruction of C18 O should not be a problem; from the H2 column densities derived below, dust extinction is likely to provide sufficient shielding.

In our model calculations, the population [FORMULA], excitation temperature [FORMULA], brightness temperature [FORMULA], and optical depth [FORMULA], are determined for each rotational level up to a maximum [FORMULA] for [FORMULA] and [FORMULA] for SiO. Collision rates are taken from Green & Chapman (1978). We have chosen a constant velocity gradient of 5 [FORMULA], a value estimated from the velocity extent and diameter of Galactic center clouds as mapped by Bally et al. (1987) or Lindquist et al. (1995). The background continuum temperature was assumed to be 2.7 K.

4.1.1. C18 O

We assume that the abundance of [FORMULA] relative to H2 is [FORMULA]. This ratio is based on the standard Galactic center 16 O/18 O isotopic ratio of 250 (Wilson & Matteucci 1992, Wilson & Rood 1994) and a CO/H2 ratio of 10-4 (e.g. Frerking et al. 1982, Duvert et al. 1986 (dark clouds), Blake et al. 1987 (OMC-1)). According to the calculations of Farquhar et al. (1994), this ratio is stable against a possibly enhanced cosmic ray flux close to the Galactic center. For [FORMULA], we then calculate a grid by varying the H2 density, [FORMULA]) , from [FORMULA] [FORMULA] to [FORMULA] [FORMULA] and the kinetic temperature, [FORMULA], from 5 to 200 K.

We found the range of [FORMULA]) and [FORMULA] by comparing the observed line intensity ratios (Table 2) to the ratios predicted by the model (Fig. 2). It is important to use line intensity ratios, not just intensities, since intensities are affected by an a priori unknown beam filling factor, [FORMULA]. For the intensity ratios, we took [FORMULA] to be identical for all transitions of a given molecule.

[FIGURE] Fig. 2. Line intensity ratios from our LVG model calculations for [FORMULA] (upper panel) and 28 SiO and 29 SiO (lower panel). The contours are at:
[FORMULA] (solid): 0.3, 0.45, 0.6, 1.0, 1.5, 2.0, 3.0, 3.5, 3.75
[FORMULA] (dashed): 0.1, 0.2, 0.35, 0.5, 0.75, 1.0, 1.5, 2.0
[FORMULA] (solid): 0.04, 0.05, 0.1, 0.2, 0.5, 0.8, 0.9, 1.0
[FORMULA] (dashed): 1.2, 1.5, 2.0, 5.0, 10.0, 15.0, 19.0, 19.75, 20.0

The integrated intensity ratio of the [FORMULA] ([FORMULA]) to ([FORMULA]) transition, [FORMULA], from all sources is [FORMULA], with extreme values ranging from 0.5 to [FORMULA]. The ratio of the ([FORMULA]) to the ([FORMULA]) intensity, [FORMULA], is [FORMULA]. Averaged over all sources, the LVG model fits yield [FORMULA] [FORMULA] K for the [FORMULA] ([FORMULA]) transition, that is, [FORMULA] is subthermally excited. The beam averaged peak optical depth [FORMULA] is always [FORMULA]. The measured peak temperatures [FORMULA] indicate a beam filling factor [FORMULA] of [FORMULA] for [FORMULA]) [FORMULA] [FORMULA]. [FORMULA] is lower if [FORMULA]) is higher. From the structure seen in our maps of the [FORMULA] (1 [FORMULA] 0) beam in higher transitions, we estimate [FORMULA], averaged over all gas components, to be [FORMULA]. These maps, however, have a low signal-to-noise ratio.

From the model calculations (Fig. 2, upper panel), maximum values up to 4 in [FORMULA] and up to 2 in [FORMULA] are expected for hot ([FORMULA] [FORMULA] K) and dense ([FORMULA]) [FORMULA] [FORMULA]) gas. Such high ratios are reached in none of our sources. Possible combinations of [FORMULA]) and [FORMULA] are either a high [FORMULA] of [FORMULA] K and [FORMULA]) of (1 - 4) 103 [FORMULA] or a lower [FORMULA] ([FORMULA] K - 30 K) at a higher [FORMULA]) [FORMULA] [FORMULA]. We have carried out this analysis individually for all clouds. For all sources our [FORMULA] data unambiguously require gas with densities of [FORMULA] [FORMULA] in these Galactic center clouds to be cool.

From ammonia measurements (Hüttemeister et al. 1993b) it is known that both hot and cool gas is present within the area covered by our beam. Thus, it is likely that both scenarios possible from the [FORMULA] data are realized. A continuum of temperatures and densities is compatible with our data. Then [FORMULA] rises steadily as [FORMULA]) decreases, when the line of sight samples different parts of the cloud. Density peaks (indicated by strong CS lines) in Galactic center region GMCs not presently undergoing massive star formation can be regarded as `cool dense cores' in contrast to the hot dense cores generally encountered within GMCs in the disk.

4.1.2. SiO

It has been shown that the fractional abundance of SiO, X (SiO), can change by more than six orders of magnitude between quiescent cold material and hot, shocked dense gas (Ziurys et al. 1989, Martín-Pintado et al. 1992). Since SiO is readily detected in all our sources, X (SiO) cannot be low. However, we cannot assume any specific value. Therefore, we have used X (SiO) as a free parameter in the models, varying it from [FORMULA] to [FORMULA]. Grids have been calculated as a function of [FORMULA]) and X (SiO) for [FORMULA] [FORMULA] 25 K, 50 K, 75 K, 100 K, 150 K.

The average line intensity ratio between the SiO [FORMULA]) and ([FORMULA]) transition is [FORMULA] [FORMULA], for those sources where we detected SiO([FORMULA]). M+1.31-0.13 is exceptional and will be discussed further in Sect. 4.3. Since the signal-to-noise (S/N) ratio in the ([FORMULA]) line is not always high and the ([FORMULA]) SEST beam was not mapped in this transition, it is not clear whether source to source differences (apart from M+1.31-013) are significant. For the line ratios between the ([FORMULA]) transitions of 28 SiO and 29 SiO (excluding M+1.31-0.13), we find a mean ratio of [FORMULA] [FORMULA] with extreme values ranging from 7.4 to 15.9. Since these data, even in the 29 SiO transition, have very good S/N ratios and were measured with the same telescope and beamsize, there is no doubt that the source-to-source differences are significant.

Fig. 2 (lower panel) shows the line ratios [FORMULA] and [FORMULA] for [FORMULA] = 75 K from our LVG calculations. Lowering [FORMULA] to 25 K requires a slight increase of the H2 densities for a given line ratio, by a factor of [FORMULA]. Raising [FORMULA] to 150 K corresponds to slightly lower densities. In general, the effects of changing [FORMULA] on the line ratios are negligible: SiO line ratios are almost insensitive to cloud temperature.

To derive [FORMULA] and [FORMULA]) , we have analysed all sources individually. Excluding M+1.31-0.13, we find [FORMULA] ranging from 5 K to 10 K. [FORMULA]) changes with the assumed SiO abundance: A higher X (SiO) requires a lower H2 density (Fig. 2). Curves of constant [FORMULA] and [FORMULA] run parallel for a range of densities and SiO abundances. Typical ranges are X (SiO) [FORMULA] for [FORMULA]) [FORMULA] [FORMULA] to X (SiO) [FORMULA] for [FORMULA]) [FORMULA] [FORMULA]. Lower SiO abundances or higher densities are not possible, since then the curves diverge.

Plotting the measured [FORMULA] 's for the 28 SiO([FORMULA]) transition in the LVG plots (e.g. Fig. 3, lower left panel), we find that a considerably higher [FORMULA] is needed to fit the observed line ratios. Thus we estimate that the beam filling factor [FORMULA] ([FORMULA] = [FORMULA] [FORMULA]) is [FORMULA] for the `typical' sources.

[FIGURE] Fig. 3. LVG Model calculations for two individual sources, demonstrating the range of physical conditions encountered. The observed line intensity ratios are plotted including their 1 [FORMULA] uncertainties. The [FORMULA] - curves are the observed [FORMULA] and the best fit temperature derived from the line ratios. The coresponding beam filling factor [FORMULA] is also given. For M+1.31-0.13, the observed and the best fit temperature are the same, thus [FORMULA]. [FORMULA] is the peak optical depth of the 28 SiO line (not beam averaged).

This analysis is biased toward more intense SiO lines, since only those were observed in transitions other than 28 SiO [FORMULA]. The strength in SiO toward these positions may be either due to a large amount of molecular gas, or to an abundance of SiO that is above average. This question will be addressed in the next section.

4.2. A joint view of C18 O and SiO

4.2.1. Column densities and SiO abundances

Further insight in the structure of the clouds is gained by combining what can be learned from [FORMULA] and SiO.

We calculate the total (beam averaged) column densities for [FORMULA] and 28 SiO, applying the same procedure to both molecules: First, we use the observed [FORMULA] and the best fit [FORMULA] from the LVG model for the ([FORMULA]) transitions (the lines with the best S/N ratio) to calculate the (beam averaged) optical depth [FORMULA]. Taking [FORMULA], we then derive the column density in the [FORMULA] level, [FORMULA]. The total beam averaged column density [FORMULA] is determined by dividing [FORMULA] by the fraction of the population residing in the [FORMULA] level, derived from the LVG model. (See Rohlfs & Wilson (1996) for a collection of the (standard) formulae we used.) For non-gaussian lines, the second moment of the line has been used as a measure of the line width instead of [FORMULA]. This has been combined with a [FORMULA] which reproduces the observed integrated intensity. This is correct as long as [FORMULA] does not change across the line profile. Since the line shapes in all transitions observed for a given molecule agree closely, this is a reasonable assumption.

For the sources where we observed only the SiO([FORMULA]) line, we have used average values for [FORMULA] and the fraction of the population in the [FORMULA] level. These are 7 K and 0.36, respectively. Total beam averaged column densities are given in Table 3.


[TABLE]

Table 3. Total beam averaged column densities of [FORMULA] ([FORMULA]) and SiO ([FORMULA]) and beam averaged abundances of SiO, X (SiO). [FORMULA] follows from [FORMULA], assuming X ([FORMULA]) to be [FORMULA]. Errors are given in parentheses.


The usually optically thin, easily excited transitions of [FORMULA] are known to be excellent tracers of H2 column density, [FORMULA], over a wide variety of H2 densities and kinetic temperatures. In particular, the integrated intensity of the [FORMULA] transition can be directly related to [FORMULA] (Gen- zel 1992). Our LVG modelling confirms that for a large range of temperatures the fraction of the total population in the [FORMULA] [FORMULA] level remains almost constant between [FORMULA]) [FORMULA] [FORMULA] and [FORMULA] [FORMULA].

In most cases, we find that column densities agree with the formula given by Mauersberger et al. (1992) and Genzel (1992) ([FORMULA] for the Galactic center isotopic ratios) to within 10%. Differences up to a factor of 2 are found if the excitation temperature is exceptionally low.

The knowledge of [FORMULA] and [FORMULA] allows us to derive a beam averaged SiO relative abundance, assuming that the [FORMULA] and SiO emission arises from the same gas. This is given in Table 3, and typically ranges from 0.5 10-9 to 5 10-9. Since [FORMULA] presumably traces all H2 with [FORMULA]) [FORMULA] [FORMULA], the true abundance of SiO cannot be lower than the value thus determined. It can be higher, if only part of the gas within the beam is abundant in SiO, which, considering the differences in line centers and line profiles between [FORMULA] and SiO (e.g. M+0.24+0.02, see Fig. 1), is likely.

For those sources where the SiO([FORMULA]) transition is weak and which were only observed in this SiO transition, the H2 column density obtained from [FORMULA] is not systematically lower than for the more intense sources. On average, in these clouds X (SiO) is lower by a factor of [FORMULA] 3. Source-to-source variations are significant, even though the properties in dense gas as traced by CS (Hüttemeister 1993) and total gas as traced by [FORMULA] do not differ. This agrees with the variation found in the survey of Martín-Pintado et al. (1997). Note, however, that we find SiO with an abundance that is high when compared to typical disk GMCs in all sources.

4.2.2. Density and temperature

We now combine the LVG models for [FORMULA] and SiO to derive a consistent solution for [FORMULA]) and [FORMULA]. This is done in terms of a two component model, where part of each [FORMULA] emission line is ascribed to a cool component, with the remaining part arising from a hot component.

From the NH3 studies of Hüttemeister et al. (1993b), we know that, for a typical cloud, roughly 25% of the neutral gas mass is contained in the hot ([FORMULA] K) component, which, from our [FORMULA] analysis, must have a low density. Thus, the remaining [FORMULA] % of the total gas mass is at a density of [FORMULA] [FORMULA]. The hot, thin and the cool, dense component are roughly in pressure equilibrium. Let us assume that the SiO emission arises in only the cool, dense component. The corresponding relative abundances of SiO, X (SiO), are 25% above the beam averaged values given in Table 3. X (SiO) lies within the range allowed by the SiO line ratios (Fig. 2) and is consistent with an H2 density of typically (1 - 4) 104 [FORMULA]. The beam filling factor [FORMULA] in this dense component must be the same for both the SiO and the part of the [FORMULA] emission that arises in this component.

We can estimate the main beam brightness temperature of the dense, [FORMULA] emitting gas to be 0.75 of the total [FORMULA] intensity. Using the beam filling factor determined for the dense gas from SiO lines, the brightness temperature of the [FORMULA] (2 [FORMULA] 1) line is [FORMULA] (75%) [FORMULA] 0.75 [FORMULA] [FORMULA]. This simple relation can be used since the [FORMULA] ([FORMULA]) line has an optical depth [FORMULA]. If the solution is self-consistent, the H2 density indicated by this value of [FORMULA] and the observed line ratios must agree with the density derived for SiO. To within a factor of 2, we find that this is indeed the case. Thus, it is possible that the SiO emission originates in gas having [FORMULA]) [FORMULA] [FORMULA]. This gas must be relatively cool, which is a surprising result in the light of the clear association of the SiO emission with hot shocked gas in the Galactic disk. The intrinsic optical depth [FORMULA] of the SiO([FORMULA]) then ranges from 4 - 8.

Can the SiO emission be associated with the hot, thin gas component? To check this, we used the same procedure as before: From the assumption that the SiO emission arises in [FORMULA] % of the gas, we get a value for X (SiO) and a corresponding H2 density. The beam filling factor [FORMULA] for SiO and, consequently, for the part of [FORMULA] originating in the same gas does not change. Now, [FORMULA] (25%) [FORMULA] [FORMULA] [FORMULA]. The additional requirement that this gas should be hot yields a H2 density that is, for many clouds, lower by an order of magnitude than what is derived for the same component from SiO. Thus, this is not a consistent solution. If the SiO arises in a still smaller portion of the total gas, the discrepancy becomes even worse. Note that we do not claim that the hot, thin component is devoid of SiO. We consider it likely that the SiO abundance in this component is similar to what we determine for the dense component. However, the SiO excitation in this component is extremely subthermal, leading (1) to lower line intensities than those we observe (specifically, no emission from the (5 [FORMULA] 4) transition) and (2) to different line intensity ratios.

Finally, we can rule out the presence of a hot, high density ([FORMULA]) [FORMULA] [FORMULA]) component as the origin of the SiO emission: We know from the line ratios [FORMULA] and [FORMULA] that X (SiO) must decrease with increasing density, independent of [FORMULA]. Thus, a hot, high density component containing the bulk of the SiO would have to contain a significant fraction of the total H2 column density and mass. Such a component should be visible in [FORMULA]. Thus we conclude that it does not exist in these clouds.

Therefore the only consistent scenario that accounts for both the SiO and [FORMULA] results requires that most of the SiO emission arises from a cool (20 K - 30 K) gas component with [FORMULA]) [FORMULA] [FORMULA].

So far, we have assumed that the [FORMULA] abundance in all components considered is known and constant. If this is not the case, i.e. if [FORMULA] is not a reliable H2 column density tracer, the line of argumentation given above cannot be maintained. In this case, it becomes possible to claim that the SiO emission arises in a hot, dense component not seen in [FORMULA]. This is, however, only possible if [FORMULA] is selectively underabundant in the component where the SiO abundance is high. Since selective [FORMULA] dissociation due to lack of shielding is efficient only in diffuse, low density gas (if the UV field is strong enough), it is difficult to find a scenario causing [FORMULA] to be underabundant in dense, warm gas.

4.3. A detailed investigation of two sources

M+0.83-0.18 In Fig. 3 (left panels), we illustrate the procedure of deriving the physical cloud parameters described above for the source M+0.83-0.18. This is representative of the type of cloud where SiO emission arises in a cool, moderately dense component. The SiO line ratios in this source (lower left panel in Fig. 3) are well fit by [FORMULA] [FORMULA] K. Then, X (SiO) is 3.9 10-9 if SiO arises from 75% of the total gas. The observed [FORMULA] is 0.48 K, while a [FORMULA] of 7.5 K is required to fit the range of [FORMULA]) and X (SiO). This results in the lowest value for [FORMULA] of all sources, 0.06, and the highest optical depth [FORMULA], 8. Applying the same [FORMULA] to the [FORMULA] data (upper left panel), we have [FORMULA] ([FORMULA] @ [FORMULA] 0.75 [FORMULA] /0.06 = 5.1 K. If we take the resulting [FORMULA]) from the SiO analysis, 4 104 [FORMULA], and transfer this to [FORMULA], we find that [FORMULA] is only [FORMULA] K. If we require [FORMULA] to be identical to the value we have measured for the cool component from NH3, 21 K (Hüttemeister et al. 1993b), the density obtained for [FORMULA] is 2 104 [FORMULA]. Considering all the assumptions, there is satisfactory agreement. Could the results be consistent with the `high temperature scenario' described in the last section? We find that [FORMULA]) required from an analysis of the SiO data is [FORMULA] [FORMULA], while the [FORMULA] data at the appropriate [FORMULA] yield [FORMULA] [FORMULA], clearly a far less consistent solution.

M+1.31-0.13 Now we analyze the one source which is very different from all the others in our sample: M+1.31-0.13 (right panels of Fig. 3). This is the only source where the ([FORMULA]) transition of 28 SiO was not detected, even though the ([FORMULA]) line is strong. It also is the one source where [FORMULA] approaches the terrestrial isotopic ratio of 20, indicating that [FORMULA] for 28 SiO([FORMULA]) cannot be large. M+1.31-0.13 has two velocity components. We analyze the (slightly) more intense [FORMULA] [FORMULA] [FORMULA] component, but the [FORMULA] [FORMULA] [FORMULA] component shows the same characteristics.

From the SiO line ratios, we derive an extremely low excitation temperature, 3.2 K. The large column density for SiO depends strongly on the exact value of [FORMULA]. Therefore, we can give only an estimate of [FORMULA] in Table 3. The corresponding X (SiO) is [FORMULA], by far the highest in the sample. In addition, for this source we find [FORMULA] [FORMULA] [FORMULA], i.e. [FORMULA]. As expected, this results in a lower [FORMULA], [FORMULA], in the 28 SiO([FORMULA]) line. For the H2 density, we find [FORMULA]) [FORMULA] [FORMULA]. This is a factor of [FORMULA] lower than for all other sources.

Since our calculations give [FORMULA] and [FORMULA] for SiO, we conclude that in this source SiO traces all the gas, as does [FORMULA]. Hence, [FORMULA] and [FORMULA] [FORMULA] [FORMULA] is also valid for [FORMULA]. The resulting [FORMULA]) from [FORMULA] is [FORMULA] [FORMULA], identical to the result obtained from SiO. As expected, [FORMULA] is high, [FORMULA] K. Thus, for this one source, we have found a fully self-consistent solution of an entirely different type: The SiO arises in thin, hot gas and has an abundance that is far above average, even for the clouds that are rich in SiO.

Is there evidence that M+1.31-0.13 lacks the dense and cool gas component? Remarkably, our NH3 data show that this cloud is the only one in the entire sample in which the [FORMULA] inversion transition at 200 K above ground is more intense than the (1,1) transition at 23 K above ground. Thus, in this cloud a cool, dense gas component containing the bulk of the gas is not present. Most, perhaps all of the gas is hot and thin. Relating this to the high SiO abundance, it is likely that SiO does indeed form at high temperatures in the Galactic center region. In M+1.31-0.13, the SiO formation process is either still ongoing or has occured very recently. The SiO has not yet had time to recondense onto dust grains, and the cloud itself has not had time to form dense cool cores.

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© European Southern Observatory (ESO) 1998

Online publication: May 15, 1998

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