4. Modeling the emission lines
4.1. Line data
In the observed spectrum (see Fig. 2) obvious strong emission bands are present at 13.48, 13.87, 14.98, 15.40 and 16.18 µm. Justtanont et al. (1998) identified these strong bands with ro-vibrational Q-branch bands of 12CO2 and 13CO2. Furthermore there is a forest of (weaker) P- and R- lines clearly visible over almost the entire observed wavelength range. The main spectral difference between 12CO2 and 13CO2 is the shift of the fundamental bending mode from 14.98 µm for 12CO2 to 15.40 µm for 13CO2. The presence of the fundamental bending mode of 13CO2 is necessary to reproduce the peak intensity at 15.40 µm (see Fig. 4). The other Q-branch bands do not show such clear evidence for the presence of 13CO2 but may show up as a shoulder in the wing of the corresponding 12CO2 band (e.g. around 16.2 µm).
We searched the HITRAN database (Rothman et al. 1987, 1992) for all known transitions of both 12CO2 and 13CO2 that fall in the observed wavelength range. Fig. 3 shows the vibrational level scheme of 12CO2 and 13CO2 and the main infrared active vibrational transitions involved. There are 3 normal modes : 2 stretching and 1 bending mode. The symmetric stretching mode is a type transition and hence is infrared inactive. The bending fundamental band (at 14.98 µm) is a perpendicular transition; this band has a strong Q branch band. More perpendicular transitions arise from the degenerate excited states, having subsequently slightly higher frequencies than the fundamental (see Fig. 3). The intrinsically strongest band is the antisymmetric stretching fundamental band at 4.25 µm. This is a type transition and hence only exhibits P and R branches. Furthermore there are combination bands, e.g. at 13.48 and 13.87 µm.
From the transition probabilities given in HITRAN we can calculate the Einstein A coefficient for each transition as Rybicki & Lightman (1979)
where and are the statistical weigths respectively for the lower and upper level considered and the frequency of the transition.
4.2. The model
We modeled the observed emission lines using a single layer, plane-parallel LTE model with a stellar background.
4.2.1. The background continuum
The stellar continuum is approximated by a blackbody with a given effective radius and effective temperature . Engelke (1992) derived an analytical expression to correct for H- free-free opacity in this wavelength range. We used this expression to determine and from the observed continuum in the short wavelength part of the SWS AOT1 low-resolution scan with complete wavelength coverage (2 - 45 µm). This observation was performed right after the AOT6 observation as part of the STARTYP program (PI S. Price). We derived = 264 and = 2447 K for a distance of 135 parsec and used these values as input for the corrected blackbody stellar background.
The main contribution to the continuum in the 15 µm region however comes from circumstellar dust. The SWS spectra show the silicate bands in emission and a strong 13 µm dust feature. Assuming that the dust emission is optically thin and that it is located further out than the CO2 emitting layer, the dust continuum is merely an additive term in the final spectrum and we can hence neglect this term by comparing the continuum subtracted model- and observed spectrum. We determined the continuum in the observed spectrum by fitting a cubic spline through the 13 µm feature and determining the onset of the 18 µm silicate band (see Fig. 2). This determination is rather ad hoc and may significantly influence the results. A discussion of this point is given in Sect. 4.4.
4.2.2. The CO2 layer
In front of the star we put a single circular slab of molecules for which the temperature T, the column density N and the radius is specified. This radius is larger than , causing the inner part of the slab to absorb the stellar radiation, while the whole layer produces emission. This is basically the same model as used by Yamamura et al. (1999a; 1999b).
Assuming the lines to be formed in LTE, the population distribution over the levels (both rotational and vibrational) only depends on T and can be calculated using the Boltzmann equation
with the statistical weight of the upper level, the energy of the upper level with respect to the ground state energy and where
is the partition function at temperature T, summing over all possible energy states. These parameters are all supplied by the HITRAN database.
Next we calculate the optical depth assuming that turbulent motions in the gas are the main source for line broadening. As the Einstein A coefficients are rather small, natural broadening will be negligible and hence we can use a Gaussian line profile. The optical depth is calculated on a high frequency resolution grid in order to accurately sample the optical depth profile. Taking stimulated emission into account, the optical depth at a frequency for a transition at rest frequency is given by
with and the Einstein coefficient for absorption and stimulated emission respectively and where the normalized gaussian line profile function is
with the Doppler width. Using the Einstein relations
and the Boltzmann equation
we can rewrite Eq. (6) as
The final profile is the sum of the profiles of all transitions considered. We used a turbulent velocity of 3 km/s; this is a typical value for the acceleration region Höfner & Dorfi 1997. If a molecular layer is desired with more than one chemical component, optical depths are calculated for each component separately and summed. In this particular case we want to include both 12CO2 and 13CO2 in the model if necessary (see Fig. 4).
The formal solution of the radiative transfer equation is given by
where denotes the background intensity and the source function. In LTE, and in our single layer model this is constant. Hence the solution of the radiative transfer equation becomes
with D the distance to the star. Finally, the spectrum is convolved with the instrumental profile which is assumed to be a Gaussian with a FWHM determined by the resolution and rebinned to the SWS resolution.
4.3. The parameters and their influence
The model as described above has essentially 3 free parameters for a given chemical composition : T, N and . Changing any of the parameters can considerably change the output spectrum.
4.3.1. The temperature T
The temperature T of the CO2 layer is the most important parameter in the model. Changing the temperature redistributes the molecules over the different energy levels and this has several consequences for the resulting output spectrum, as illustrated in Fig. 5.
At low temperatures, most of the molecules are in the ground state. Only a very small fraction of the molecules can get excited to even the lowest vibrational level, and hence only the fundamental bending mode at 14.98 µm is present in the spectrum. Increasing the temperature of the CO2 layer causes a redistribution of the CO2 molecules in which the higher vibrational energy levels become more populated. The most obvious effect on the resulting spectrum is the appearance of bands arising from these higher levels, as can be seen from Fig. 3 and Fig. 5. A secondary effect of this vibrational redistribution is the contribution of hot bands to the spectrum. As the hot bands of the fundamental bending mode (at 14.98 µm) have wavelengths consecutively to the blue, higher temperatures cause the peak of the band around 15 µm to shift to the blue (see Fig. 5).
Also the higher rotational levels within a given vibrational level will become more populated at higher temperatures. The Q-branches will consequently become broader (see the inset in Fig. 5), which is also true for the P- and R- branches; furthermore the peak of these P- and R- bands will move away from the Q-branch.
A final remark concerns the global intensity of the output spectrum. Although the optical depth in the lines generally decreases with increasing temperatures (as higher vibrational levels become more populated), the output intensity is generally higher for models with a higher temperature. This is mainly because the source function increases with increasing temperature, but also because more transitions become excited.
4.3.2. The column density N
Changing the column density N can also severely affect the output spectrum (see Fig. 6). Increasing the column density will increase the optical depth at any given wavelength. As long as all lines are optically thin, this will only scale the entire output spectrum to higher fluxes.
The first lines to become optically thick are the Q-branch lines around 15 µm. Increasing the column density will then start deforming the output spectrum as weaker lines gain more in intensity than the optically thick lines. This results in three types of changes, often similar in appearance to increasing the temperature T.
First of all, the Q-branch band itself becomes broader. When the cores of the lines that contribute to the band are optically thick, their wings will start producing a significant contribution to the output flux. Moreover, the higher energy transitions also gain in relative importance to the output flux. As can be seen from Fig. 6, the peak position of the 15 µm band shifts slightly to the blue, as also the hot bands start contributing to the output spectrum.
A second effect is that the relative intensities of the P- and R- branches increase slightly with respect to the Q-branch, due to the same effect. And finally, whole regions of the spectrum can become optically thick and thus produce a "continuum"-like deformation in the spectrum. This is especially the case in the regions next to the 15 µm band, where there is a dense forest of lines arising from different vibrational levels; when the column density is sufficiently high, they all contribute significantly to the spectrum, also in their wings, and this produces the deformation. In the limit of infinite column densities, for all frequencies and the output spectrum is a Planck function at temperature T as can be easily seen from Eq. (14).
4.3.3. The size of the emitting region,
The size of the circular CO2 slab determines in the first place whether the spectrum shows absorption lines or emission lines. If the slab has the same size as the star, a pure absorption spectrum results (see Fig. 7) unless . Enlarging will cause the part that geometrically extends beyond the star to produce emission that gradually fills up the absorption lines, and eventually results in net emission. This happens first at longer wavelengths where the background continuum has a lower intensity and hence the absorption is weaker; in such cases the result is a spectrum where the red part can be seen in emission while the blue part is still in absorption. The turnover to a net emission spectrum happens in a very small range of . When is much larger than this critical radius, the output spectrum is dominated by the emission and increasing the radius will then just scale up the output spectrum.
In summary we conclude that :
Although this is a rather simple model, it reproduces quite well the observed band profiles (see Figs. 10 to 14). The next section describes how these fits are obtained.
4.4. Determining the model parameters
4.4.1. Quantifying the quality of the fit
In order to derive the model parameters that provide the best fit to the observations, a quantitative comparison between the model spectrum and the observed spectrum needs to be made. This is done by calculating
where and are the continuum subtracted flux values for the observations and the model respectively; are the uncertainties on the data points (assuming that the uncertainties on the model spectrum are negligible); the sum runs over all data points in the wavelength range considered and is the number of degrees of freedom with m the number of free parameters.
For a good fit, should be low, of order unity. If is much larger however, this does not necessarily mean that the adopted model is a bad fit to the observations; a large value can indicate a doubtful error assignment or the presence of systematic errors in the data.
We first minimized using a model with only 3 free parameters, T, N and R. It turns out that the best fit in this case yields unreasonably high optical depths while the remains quite large. A systematic error in the determination of the continuum in the observations causes the line contrast to change in such a way that high optical depths are required to reproduce the line shape and the relative intensities; this is related to the broadband deformation for high column densities as described in Sect. 4.3. We therefore allowed the continuum to shift up and down in intensity (while keeping the shape as determined from Fig. 2) and introduced this shift as an extra free parameter in the model and minimized the again. The new values are much lower while the derived continuum shift is small compared to the line intensities. In order to assess the significance of this parameter, we used the F-test of an additional term on this parameter (Bevington & Robinson, 1992, chapter 11). We calculated
where is the change in when comparing the best fit with and without adding the extra parameter and corresponds to the best fit value with the extra parameter added. This ratio is a measure of how much the additional term has improved the value of the reduced chi-square and should be small when the function with the added parameter does not significantly improve the fit over the function without the added term. Using the associated F distribution function, one can evaluate the probability of obtaining the observed value of when the added parameter would be superfluous. In our case, the probabilities are lower than 10 for all bands, indicating that introducing the continuum shift as an extra parameter is justified. Therefore we included this continuum shift for the entire minimization procedure. Allowing also for a change in the slope of the continuum on the other hand did not significantly improve the .
4.4.2. Searching parameter space
The model parameters that provide the best fit to the observations are those for which has a minimum. We defined a grid in T between 100 K and 900 K in steps of 50 K, and a logarithmically equidistant grid in N in the range 10 cm-2 - 10 cm-2. At each of the grid points radiative transfer is calculated for the given T and N and with the dollar; circ; lcub; lcub;25 rcub; rcub; dollar; downhill simplex method (Press et al., 1992, Sect. 10.4) we determined at the same time the value of and the continuum shift that minimizes the to a fractional tolerance of 10.
As the observed spectrum of EP Aqr shows Q-branch bands arising from vibrational levels at a very different energy, it is likely that these bands are formed at different locations and hence different excitation temperatures. Instead of fitting the whole spectrum we therefore determined the parameters for each of the bands individually by defining a spectral window of 0.4 µm around the bands, centered at the band head. This includes the whole Q branch and typically 10-15 P and R branch transitions per vibrational transition.
After the first run we introduced the 12C/13C ratio as another free parameter and calculated the corresponding values. This significantly improves the for all bands except for the 13.48 µm band which is apparently less sensitive to the contribution of 13CO2. We therefore also defined a grid in 12C/13C ratio values of 1, 5, 10, 15, 20, 25, 30, 40, 50, 60, 70, 100; these values were chosen to adequately sample the range of values typically found in AGB stars Dominy & Wallerstein 1987; Smith & Lambert 1985, 1986; Sneden et al. 1986.
4.4.3. The model parameters and their uncertainties
Once the is calculated on the grid points, the values of the 12C/13C ratio, T and N and the uncertainties on these parameters can be derived in the following way (Bevington & Robinson, 1992, Sect. 11.5). Fig. 8 shows a two-dimensional projection of the hypersurface onto the (12C/13C ratio, T) plane. For each (12C/13C ratio, T) grid point, there are 19 values that were calculated for each of the grid points in N. For this projection, we only consider the minimum of these 19 values. The values for the 12C/13C ratio and T that provide the best fit to the data thus correspond to the minimum in this projection. A 1 contour on the derived parameters in this projection is then found by evaluating for which (12C/13C ratio, T) values . Projecting the outer edges of this contour onto the 12C/13C ratio axis gives the 1 uncertainties on the derived 12C/13C ratio; projection on the T axis yields the uncertainties on the temperature. In a similar way one can also determine the uncertainties on N by making a projection onto the (T, N) plane (see Fig. 9). However, as we only have the values for the predefined grid points, the accuracy is at best limited to the grid spacing.
For R and we have no predefined grid and so the uncertainties on these parameters have to be estimated in another way. Both parameters depend on T and N; around the minimum however, they are affected mainly by N as the grid spacing is logarithmic for this parameter. We therefore changed N to within the derived 1 uncertainties while keeping the T at its best value and then determined the R that minimizes the ; the resulting R values corresponding to the extreme N values were subsequently taken as the uncertainties on R. As it is straightforward to calculate the change in when changing N and also here we took the extremes as uncertainties on . We note however that the corresponding to these extremes deviates by more than one from the minimum value; therefore these formal uncertainties are probably rather conservative.
© European Southern Observatory (ESO) 2000
Online publication: August 17, 2000