Table 1 lists the derived parameters and their uncertainties obtained from the for the different bands. The corresponding best fits can be seen in Figs. 10 to 14. For all the bands the derived parameters yield a maximum optical depth for each of the spectral windows. As the optically thin models are a subset of the parameter space we studied, the first obvious conclusion is that the observations are better explained by an optically thick model than an optically thin one.
Table 1. The derived parameters for all the bands using a single layer LTE model. The errors stated are the formal errors based on the analysis.
It can easily be seen from Eq. (14) that for the case of optically thick bands and hence in our model the observed peak intensity is a good measure of the extent R of the emitting region.
The whole spectrum cannot be reproduced at once by a single layer model. The temperatures derived for the individual bands are significantly different, reminiscent of a temperature stratification in the extended atmosphere. Q-branch bands arising from higher vibrational levels are then probing the hotter material closer to the central star, while the emission from lower vibrational levels arises from cooler material further out.
However, as can be seen from Fig. 5, any given band becomes stronger when increasing the temperature. When deriving the parameters for a given band, one might therefore expect to see a significant contribution from the hotter layers as those hotter layers always show stronger emission in the bands arising from lower vibrational levels (see Fig. 5). However, as a consequence of all the bands being optically thick, emission from the hotter layers is shielded from view at those wavelengths where a cooler layer shows prominent emission features. On the other hand, a contribution from the cooler layers might be present at those wavelengths in the spectrum where the hotter layers are visible if the cooler material is optically thin at those wavelengths.
As a consistency check we therefore performed a test calculation for the 13.48 and 13.87 µm bands with a two-layer model using the parameters from Table 1. The 13.87 µm band is indeed quite unaffected by the presence of the K layer, but the flux in the 13.48 µm band shows a significant contribution due to the K layer. The main effect is that the extent for the warmer layer decreases when compared to the single layer model and thus the values listed in Table 1 are probably too high.
All the bands are reasonably well reproduced by a single layer model except for the 14.98 µm band. Upon inspection of the peak position, it is obvious that an extra cool ( K) and optically thin emission component is necessary on top of the warmer component to reproduce the overall band shape. This component is probably the result of the dissociation of water into OH molecules by the interstellar UV field followed by a reaction with CO which could produce CO2 with a maximum CO2 abundance of typically 10 at a few hundred stellar radii where the temperature is of the order of 100 K Willacy & Millar 1997. Moreover, as the hot bands are consecutively shifted to the blue (see Fig. 3) and as these Q-branch transitions (fundamental as well as hot bands) have the highest Einstein A values in the observed wavelength range, it can be expected that this band is optically thick at all wavelengths. The shape of this band will therefore be the most sensitive to a temperature stratification and the temperature derived for this band with the single-layer model is probably somewhere in between the warmer and the coolest layers. A two-layer model with K, 10 and 100-150 for the second layer is indeed able to reproduce the observed spectrum much better. Also the other derived parameters are probably not very reliable for this band. As the Einstein A coefficients for this band are rather high, one could expect to find the same problem for the fundamental 13CO2 bending mode at 15.40 µm even though the 13CO2 column density is much lower. This band however is buried in the forest of P-branch lines of the 14.98 µm 12CO2 band and therefore this effect is not obvious. Nevertheless the observations clearly show that the CO2 layer is extended and that the temperature decreases with increasing radius (see Fig. 15).
5.2. Column densities
The derived column densities are 10 cm-2 for all bands except for the 14.98 µm band, where the column density is slightly higher. As discussed in the previous paragraph, this is a consequence of a temperature stratification and the presence of hot bands in the spectrum as well as the presence of a much cooler layer.
5.3. Isotopic ratio
On first sight there is not a single value for the 12C/13C ratio that is compatible with the results derived for the bands individually. However, the problem with the temperature stratification mentioned above will influence the 12C/13C ratio derived for both the 14.98 and 15.40 µm bands and hence the values for these bands are doubtful. The 16.18 µm band on the other hand is rather insensitive for the adopted 12C/13C ratio as can be seen from the uncertainties. As the 13.48 µm band has the lowest optical depth, this would be the best band to derive the 12C/13C ratio. However, this is the weakest band and unfortunately the quality of the spectrum in this region is not too good (see Fig. 10) due to fringe residuals, which hampers a good determination of the 12C/13C ratio. The best guess for the 12C/13C ratio is probably the one derived for the 13.87 µm band which is both very sensitive to the contribution of 13CO2 and not too optically thick. We find a 12C/13C ratio value of 10 for this band.
© European Southern Observatory (ESO) 2000
Online publication: August 17, 2000