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Astron. Astrophys. 362, 1109-1121 (2000)

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4. Multi line analysis

With the aim to understand changes in excitation conditions of molecular material in a FUV illuminated cloud like Cepheus B we try to reproduce the observed data by several model calculations. Already rather simple LTE calculations and escape probability models set reasonably good constraints on the excitation conditions, and where these models fail they indicate the necessity for more sophisticated, and realistic modelling. As a further approach we therefore apply a PDR model, which provides a self-consistent treatment of the chemical and thermal balance of a cloud together with the radiative transfer of CO line emission in spherical clouds illuminated by an isotropic FUV field.

First, assuming LTE with a beam filling factor [FORMULA] of 1 we derive an estimate of the average kinetic temperature, density and mass in Cepheus B. We averaged the 12CO 3-2 and C18O 2-1 data seperately over the area of significant emission in C18O 2-1 ([FORMULA], [FORMULA], Fig. 4) and find from the optically thick 12CO a mean excitation temperature of 22 K. Using the C18O data, this temperature, and the canonical [C18O]/[H2] abundance ratio, the derived average LTE H2 column density per beam is [FORMULA] cm-2 and the average H2 volume density is [FORMULA]860 cm-3, assuming that the cloud is as extended along the line-of-sight as across the face of the cloud. The average 13CO optical depth is [FORMULA] in this center region and C18O is optically thin with opacities varying between 0.1 and 0.01. The total mass derived from C18O 2-1 is [FORMULA]. One should note that this is a rather accurate estimate, as the conversion from integrated C18O intensity to H2 column density is not strongly affected by the detailed excitation conditions, as long as the densities are above the critical density of the transition (e.g. Stutzki et al. 1993).

Since the above calculation only leads to the average density and does not take into account non-thermal excitation nor any difference in the kinetic temperatures of the 12CO and C18O emitting regions, we used an escape probability radiative transfer model (Stutzki & Winnewisser 1985) as a next step. It assumes a homogeneous, spherical symmetric cloud with constant kinetic temperature and volume density. While it is clear that real clouds are still much more complicated, such a model is commonly used and its results improve on the simple LTE analysis. Especially, the derived densities and temperatures refer to localized values. Collision rates were taken from Flower & Launay (1985) and models were calculated for kinetic temperatures between [FORMULA] K, H2 volume densities [FORMULA] cm-3 and CO column densities [FORMULA] cm-2.

4.1. Results of the escape probability analysis

We apply the model to interpret the spectral lines observed at four positions (Fig. 3), sampling different regions of Cepheus B. To circumvent the problem of varying beam filling factors, we try to find models which fit best the observed line ratios (Table 3). 12CO and 13CO 3-2/2-1 line ratios are susceptible to volume densities and temperatures, whereas the line ratios between different isotopomers respond to variations of column densities. Results of this analysis are summarized in Table 4.


[TABLE]

Table 3. Measured ratios of line peak temperature at 4 Positions and their mean value. Estimated line calibration errors are 15%, ratio errors around 20%. The last two columns show the ratio-mean and the ratio-range of the integrated intensities over the whole mapped region. The high upper limits in the range-column are seen only at edges of the cloud.



[TABLE]

Table 4. Results of the escape probability analysis of the peak line ratios (3-2/2-1) at the four selected positions: H2 column densities, local H2 volume densities and kinetic temperatures. See explanations in the text.


4.1.1. Volume densities

The ratio of 12CO 3-2/2-1 peak line temperatures lies between 0.8 and 1.5 at the four positions, indicating that these two transitions are thermalized. Indeed, the data are consistent with escape probability models of densities of about the critical density (Table 2) of a few [FORMULA] cm-3. This result is typical: depending on the molecule and transition observed, local densities derived from these lines are usually near their critical densities (e.g. Stutzki 1993). Local densities in Cepheus B are a factor of [FORMULA] higher than the average density derived assuming LTE, another result which is commonly found, and which indicates that the emitting regions are not homogeneous but exhibit significant substructure.

4.1.2. Column densities

H2 column densities, derived from line ratios and the escape probability models, vary between 0.4 and [FORMULA] cm-2 at the four positions. They are larger by a factor of upto 10 than the average column density per beam derived from integrated C18O intensities assuming LTE. This is a further indication of structure within the beam and low beam filling factors, as will be shown later. Subthermal excitation would lead to higher LTE column densities compared to the escape probability result and can thus not explain this discrepancy.

4.1.3. Temperatures

We find a temperature gradient across the surface of the cloud, as traced by 12CO, from more than 70 K at the hot core position ([FORMULA],[FORMULA]) down to 20 K at the south-eastern edge of the mapped region. All three isotopomers are consistent with a single kinetic temperature of more than 20 K at position ([FORMULA]) and more than 30 K at ([FORMULA]).

At the other two positions, it is not possible to find a single temperature which is consistent with the emission of all three isotopomers: The parameters at ([FORMULA]) indicate a temperature gradient along the line of sight - from a warmer surface of more than 30 K, traced by the optical thick 12CO, to a colder interior with temperatures below 20 K, derived from 13CO and C18O. Similarly, line ratios at the hot core position ([FORMULA]) are not consistent with a constant kinetic temperature. In addition, a high temperature of at least 70 K is needed to explain the high 12CO 3-2/2-1 ratio of 1.5.

4.1.4. Beam filling factors

Beam filling factors can now be determined by comparing the observed peak line temperatures with the expected values from the escape probablity analysis, given the kinetic temperatures, column and volume densities derived above ([FORMULA]). In Table 5 we show the derived filling factors for the 2-1 transitions.


[TABLE]

Table 5. Beam filling factors [FORMULA] derived from the observed 12CO 2-1 peak line temperature and the modelled temperature resulting from the escape probability analysis


Filling factors [FORMULA] range between 0.1 and 1.1, the values for 12CO and 13CO are consistently larger than those for C18O. This may be due to the fact, that self shielding of C18O is less effective than for the more abundant isotopomers, leading to photo destruction of C18O at clump surfaces and thus a smaller emitting area within the beam.

Secondly, the C18O filling factors decrease systematically from positions inside the cloud, e.g. [FORMULA], to the cloud edge at ([FORMULA]), falling from 0.4 to 0.1. This is consistent with an enhanced FUV field and photo destruction in the outer regions of Cepheus B near the HII region, relative to positions at greater distances to the HII region.

4.2. Volume filling factor

For further description of the density structure in Cepheus B we compare the average density [FORMULA] derived from the LTE analysis with local densities [FORMULA] derived from line ratios using the escape probability model. The ratio of average over local densities, the volume filling factor, is about [FORMULA]=4%. Taking into account a possible interclump medium with the density of [FORMULA], the volume filling factor can be estimated, [FORMULA] and becomes even lower: the assumption of [FORMULA] cm-3 e.g. leads to [FORMULA]2%. Volume filling factors typically derived from multi-transitions studies range between a few% and 20% (e.g. Stutzki 1993) and provide indirect evidence for small scale structure.

Another way of putting this argument is considering the ratio of column densities over volume densities, [FORMULA], found via the escape probability analysis (Table 4). This ratio gives a clump diameter of [FORMULA] pc ([FORMULA]) on average, significantly below the spatial resolution of the observations of [FORMULA] or 0.42 pc. From the above analysis, we know that beam filling factors reach values as low as 0.1. Still, this means that about 20 clumps or more lie within one beam.

The high surface temperature [FORMULA] K, we derive at the hot core, and the small clump diameters [FORMULA] pc calculated from the N/nloc ratio are confirmed directly by recent 12CO 7-6 observations of Kramer et al. (in prep.) with the Heinrich Hertz telescope at [FORMULA] (0.05 pc) resolution. Spatial structure is found down to the resolution limit. In addition, the peak line temperature of [FORMULA] K indicates kinetic temperatures of at least 70 K. Optically thin emission or a beam filling factor of less than 1, would mean even higher local kinetic temperatures.

All this shows that only a small volume of the whole cloud is filled with molecular material in dense clumps. Such a strong clumping allows the FUV radiation of the HII region S155 and the hot core compact HII region to penetrate the cloud far more than it would be possible in a homogenous cloud.

4.3. The UV field

The UV field of O and B stars heats the molecular gas via the photoelectric effect on dust grain surfaces and via FUV pumping of H2 (e.g. Kaufman et al. 1999). Together with the cosmic rays, it is thus directly influencing the emission of CO lines which are one of the main coolants of the gas. The primary sources of UV radiation in the vicinity of the Cepheus B molecular cloud are the O7 star HD217086 and the B1 star HD217061 (Fig. 1, projected distances to the hot core 1.2 pc and 0.7 pc) which are members of the Cepheus OB3 association lying to the north and west of the cloud interface to the HII region S155, and the embedded B1 star in the hot core region. Three more B stars - also members of the OB3 association (see Fig. 3 in Felli et al. 1978, projected distances to the hot core 2.6 pc, 3.8 pc and 5.1 pc) - lie in north-eastern direction, and contribute weakly to the FUV field impinging on Cepheus B.

The far-UV luminosity L between 91 nm and 300 nm wavelength (13.6 eV and 6 eV), responsible for heating the cloud, is derived assuming that the star is a black body at the effective temperature corresponding to its spectral type (Panagia 1973). In addition, we assume that the distance R from the star to a position inside the cloud is given by the projected distance. Estimating the geometrical dilution as well as the blockage and scattering by molecular clumps inside the cloud, the average UV-flux S of each star at different positions inside the cloud can be calculated via (see Stutzki et al. 1988):

[EQUATION]

where [FORMULA] is a typical clump volume filling factor of 0.02 and [FORMULA] the typical clump diameter of 0.02 pc. [FORMULA] is the distance between the position inside the cloud and the cloud surface, in the direction of the exciting star. The typical scale length, [FORMULA], of the UV penetration is thus 1 pc.

Table 6 shows the derived fluxes [FORMULA], as the sum of the fluxes of all 6 stars, in units of the average interstellar radiation field, [FORMULA] erg s- 1 cm-3 (Habing 1968). The FUV flux drops by three orders of magnitude, between the immediate interface region at the hot core, where fluxes rise to [FORMULA], and the south-eastern outskirts of the mapped region which are weakly illuminated by fluxes only slightly above the average interstellar field.


[TABLE]

Table 6. Summed FUV fluxes [FORMULA] of all 6 stars penetrating the clumpy cloud and distance [FORMULA][pc] to the hot core.


The position [FORMULA] represents the north-eastern edge of Cepheus B, where the line ratios of 12CO 3-2/2-1 rise again (Fig. 4) - maybe due to heating by three B stars which contribute a flux of 35.

4.4. PDR modelling of CO lines

The above discussion shows that a homogeneous cloud model cannot consistently describe the observed line ratios of all three CO isotopomers at all positions, although such a model still provides first estimates on column densities, densities, clump sizes, and temperature gradients. Apparently contradictory line ratios appear to be a widespread phenomenon in photon illuminated clouds. Castets et al. (1990) interpreted similar findings in Orion A by temperature gradients increasing towards the cloud surfaces. Such gradients are naturally expected for UV irradiated clouds. Gierens et al. (1992) modelled the line ratios found in Orion A using a spherical symmetric PDR code including analytic temperature and density gradients. Störzer et al. (2000) present a more realistic cloud model which provides a self-consistent treatment of the chemical and thermal balance of a cloud together with the radiative transfer of CO line emission in spherical clouds illuminated by an isotropic FUV field. The model assumes the density to increase radially inward following a power law. Main input parameters determining the model are the average H2 volume density, the average H2 column density and the intensity of the FUV field. The clump diameter of the model is proportional to the ratio of column density over volume density, the thermal and chemical structure of the clump models are iteratively calculated from the balance of heating and cooling. However, this PDR model does not take into account the effects of several small clouds of varying sizes and masses within a single telescope beam nor internal heating sources. We thus expect the model to consistently describe most line intensities and ratios observed in Cepheus B, but not all details, thus refining our first analysis.

We compared the 4 measured peak line ratios, 12CO 3-2/2-1, 13CO 3-2/2-1, 12CO/13CO 2-1, 12CO/C18O 2-1 (Table 3), and the observed 12CO 2-1 line temperature (Fig. 3) with the model calculations, assuming an impinging FUV field of [FORMULA] (cf. Table 6). Note that the analysis of Störzer et al. (2000) indicates that the magnitude of the impinging FUV field does not lead to large variations of line ratios. We take into account a 20% error on the line ratios. This analysis was not done for position [FORMULA], because we did not measure all ratios there.

4.4.1. Position [FORMULA]

This position exhibits the strongest emission in 12CO 2-1 within the mapped region. The 12CO and 13CO 3-2/2-1 line ratios of 0.8 (Table 3) are - within the errobars - in the range of the expected ratios for thermalized emission in the LTE case and allow to derive the local H2 density. The PDR model indicates densities of less than [FORMULA] cm-3. Low H2 column densities are indicated by the rather high inter-isotopic line ratios 12CO/C18O and 12CO/13CO 2-1 of 32 and 3.4 (Table 3). The PDR model predicts H2 column densities per clump of [FORMULA]cm-2 corresponding to an optical extinction AV of [FORMULA] mag. Both, the upper limits on n and N, are consistent to within a factor of 2 with the homogeneous cloud model (Table 4), which also leads to a consistent description.

The PDR model fails however to reproduce the observed 12CO peak line temperature of 26 K, it merely predicts line temperatures of [FORMULA] K. At least 5, small overlapping clumps within the beam may reproduce the observed intensity since the optical depths are not high for the low column densities found at this position. Small clumps show the observed high 12CO/C18O ratios due to two effects: photo-dissociation of C18O in the surface regions leading to a smaller C18O emitting region and lower kinetic temperatures in the clump interiors (cf. Zielinsky et al. 2000).

The next two positions are of particular interest, since the escape probability model does not allow to describe consistently all observed line ratios found at these positions.

4.4.2. Position [FORMULA]

With increasing distance to the HII region, the C18O line temperatures rise and peak with 2.3 K at ([FORMULA],[FORMULA]), at a distance of 1.7 pc to the hot core, indicating the position of highest column density. Also, in contrast to position (0,0), the 12CO 3-2/2-1 ratio differs significantly from the corresponding 13CO ratio indicating that both isotopomers trace different physical regions of the cloud. Indeed, the escape probability analysis (Table 4) indicates that the 12CO emission arises in (surface) regions of at least 30 K kinetic temperature while the 13CO emission stems from colder (interior) regions. Since the PDR model takes into account a temperature gradient, it allows to model consistently the data at this position and clearly improves on the esc.prob. models. The best fit to the PDR model is obtained for densities of about [FORMULA] cm-3 and H2 column densities of about [FORMULA] cm-2. For these parameters, the model calculates a peak 12CO 2-1 line intensity of 14 K, exactly what is observed, indicating a beam filling factor of unity for this line.

4.4.3. Position [FORMULA]

This position lies in the immediate vicinity of the hot core. Its influence is evident in the high 12CO 3-2 peak temperatures of 23 K (Table 3). However, C18O intensities are low at this position. The 13CO 3-2/2-1, 12CO/13CO 2-1, and 12CO/C18O 2-1 peak line temperature ratios (Table 3) are similar to the ratios found at position [FORMULA]. The PDR model leads to H2 densities of less than [FORMULA] cm-3 and H2 column densities of less than [FORMULA] cm-2. The observed 12CO 2-1 line peak temperature is 16 K, the model gives [FORMULA] K, indicating again that several small clumps lie within the beam, and their added emission produces the observed higher line temperatures.

However, in contrast to position [FORMULA], the large 12CO 3-2/2-1 ratio of 1.5 cannot be reproduced by the same model clump, neither can it be reproduced by any of the models presented by Störzer et al. (2000). In these models, calculated ratios rise to about 1.1 for very high volume densities of [FORMULA] cm-3 and low column densities of [FORMULA] cm-2, which are the lowest column density analyzed by Störzer et al.. Column densities lower than that would render the 12CO lines optically thin. In this limit ratios of 1.5 could be explainable by excitation temperatures of more than 40 K.


[TABLE]

Table 7. Results of the PDR analysis derived from four peak line ratios and 12CO 2-1 peak line intensities. Remarks: (1) The analysis at position ([FORMULA]) ignores the high 12CO 3-2/2-1 ratio which cannot be reproduced. (2) Clump diameters are derived from the upper limits found for N and n.


4.5. Atomic carbon

Additionally, we observed the [FORMULA] transition of atomic carbon at two positions, at ([FORMULA]/[FORMULA]) position and at [FORMULA], near the edge to the HII region S155. Fig. 6 presents the [CI ] spectra in comparison with 13CO spectra. The line shapes look similar in the wings, but the core emission of CI is less intense than in 13CO, maybe a signature of CI selfabsorption. Table 8 presents line parameters and ratios between the three transitions. The 13CO and [CI ] ratio of integrated intensities [FORMULA] do not vary significantly between the two positions. Regarding ([FORMULA]/[FORMULA]) as representative for the bulk of the cloud and ([FORMULA]/[FORMULA]) as representative for the cloud edges, this may suggest a similar spatial distribution for CI compared with 13CO.

[FIGURE] Fig. 6. Spectra of CI (492 GHz) and [FORMULA] at two positions in Cepheus B


[TABLE]

Table 8. Parameters of the CI and [FORMULA] spectra: peak temperatures, integrated intensities and FWHM


With the CI data, the abundance ratio of molecular to atomic carbon can be estimated. Assuming optically thin [CI ] emission and LTE allows to derive a lower limit of the CI column density (Frerking et al. 1989):

[EQUATION]

with [FORMULA] taken from the [FORMULA] 3-2 transition.

To compare gas phase abundances of carbon relative to CO we calculate LTE column densities for CO by multiplying [FORMULA] (optically thin) column densities with the abundance factor of 67 (Table 9). At the two positions observed we find that the abundance of atomic carbon is 17% and 30% of the CO column densities. As expected the CI /CO ratio is rising at the cloud edges, because more molecular material is dissociated there due to the impinging FUV field. This phenomenon is also confirmed by the decreasing FWHM ratio of CO to atomic carbon. Because of the unknown CI opcacity these abundance ratios are only lower limits. Tauber et al. (1995) derived 0.17 as lower limits for the Orion bar, and Plume et al. (1999) find values between 0.32 and 0.47 for a sample of clouds.


[TABLE]

Table 9. CI and [FORMULA] column densities


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Online publication: October 30, 2000
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