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Astron. Astrophys. 341, 579-586 (1999)

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4. Results and analysis

Observational evidence points toward a relative simplicity of the kinematics, the density and the temperature structures of stellar envelopes, with an over-all symmetry prevailing (see e.g. Olofsson, 1996), although several AGB stars have been shown to have asymmetric winds (Plez & Lambert, 1994). Also, note the clear evidence of clumpy structure at some scale in e.g. Olofsson et al.(1998). The chemical structure is certainly quite complex, with a chemistry far out of equilibrium. A standard circumstellar envelope, with a non-rotating, smooth, and spherically expanding wind with a constant velocity is nevertheless a reasonable starting point for our analysis.

4.1. Are the bands optically thin?

In the following we will discuss the optical thickness of the environment from which the bands are emitted. The order-of-magnitude estimate made here will lead to an upper limit partly because the wind velocity will eventually shift the molecules out of the thermal broadening of the Q-branch lines due to Doppler shifts and thereby lower the estimated optical depth. For an estimate of this upper limit of the optical depth, [FORMULA], of the bands, we assume a spherically symmetric wind resulting from a constant rate of mass-loss. Within the region where CO2 is assumed to exist, its abundance relative to hydrogen, [FORMULA], is set constant. The optical depth is found by integrating the cross section per particle ([FORMULA], [cm2]) for the absorption times the number density of the absorber (N, [cm-3]) along the line of sight with an impact parameter p. For atomic or molecular scattering [FORMULA], with [FORMULA] being the band oscillator strength. From the continuity equation and the assumption of a stationary flow the number density of CO2 in all energy states is found. Thus, in cgs units,


where [FORMULA] is the outer boundary of the emitting sphere and [FORMULA]. The largest optical depth will be found in the [FORMULA] band since the lower lying energy state of this transition is the vibrational ground state, where all but a few molecules will most probably be found. An upper limit of the optical depth will therefore be estimated for an impact parameter [FORMULA] and assuming that all CO2 is in the ground state. An upper limit of the fraction of CO2 ([FORMULA]) is found by adopting a reasonable number of the CO2/CO ratio and assuming that almost all carbon is locked in CO. This is not a bad assumption in an oxygen-rich environment since the CO molecule has a large binding energy and is the most stable molecule in the circumstellar environment. The CO2/CO ratio is assumed be a few times [FORMULA], in accordance with the calculated abundances in the circumstellar chemical models of Willacy & Millar (1997). The width of the band ([FORMULA]) is approximately [FORMULA].

The results for the stars considered are tabulated in Table 3, which gives the optical depths for a standard wind extending out to around 1000 stellar radii. Obviously, all bands will be optically thin in this case. A wind extending out only to 3 stellar radii, would give around 20% lower optical depths under the same assumptions. If we consider a warm and quasi-static, molecular-forming region with small velocities, such as the one suggested by Tsuji et al. (1997), the densities close to the star are enhanced, which will lead to higher optical depths. A density enhancement close to the star by one or two orders of magnitude could make the [FORMULA] band optically thick. An increase of the density could be achieved if extra ejected matter eventually falls back on the star without raising the net matter transport outwards. Turbulent velocities will broaden the lines and the quasi-static layer will not shift the molecules out of the lines. Thus, there is a possibility that the absorption bands observed are formed in such high density regions. In view of their lower oscillator strengths and their excitation, the other bands are obviously optically thin also in this picture.


Table 3. Maximum optical depths,[FORMULA], for the [FORMULA] band for a spherical and stationary wind

4.2. Are the bands formed in LTE?

It is widely believed that most AGB stars pulsate with shocks travelling through their circumstellar regions. These dynamic regions are often assumed to show departures from LTE conditions. We will show that it may be questionable to assume that the bands are in LTE.

The highest probability for LTE to prevail, in an expanding wind model, is close to the star since the density decreases, at stationary outflow, with the square of the distance from the centre. From the equation of continuity for a spherically symmetric wind we get the number density of hydrogen molecules as a function of the radial distance from the star r, as [FORMULA] For R Crt with [FORMULA] and [FORMULA] and for R Dor with [FORMULA] and [FORMULA] (Loup et al., 1993), we get for the number density of H2 for this stationary model, adopting [FORMULA]: [FORMULA] and [FORMULA], respectively.

Observe, however, that the atmospheres of these stars extend far out due to pulsations and shocks, which will invalidate the assumption that the wind is in a steady state at small radii. Thus, from dynamic models of variable stars by, e.g., Bessel et al. (1996) and Höfner & Dorfi (1997), it is indeed seen that the densities at a few stellar radii from the star can be, in general, an order of magnitude higher than the ones given by the equation of continuity. Note also that observations indicate that the winds may be inhomogeneous and clumpy (e.g. Olofsson et al., 1998), which could increase the densities in the emission regions. Also the quasi-static layer discussed above will have higher densities, which could be enough for collisions to be dominant a few stellar radii out. However, from the simple considerations above we will see that we may nevertheless conclude that the assumption of statistical equilibrium may be needed instead of that of LTE, even a few stellar radii away from the star. This must be considered when discussing the kinetic gas temperature and the site of formation of CO2.

The statistical equilibrium equations, including radiative and collisional processes for a set of transitions, give the populations of the levels. These will also be given by the Boltzmann distribution if the density is sufficiently high in order for collisions to be dominant over radiation processes. The spectral lines are then formed in LTE and the gas is thermalised at the kinetic gas temperature. If the collisions are not dominant, it is not clear whether LTE is valid.

An order-of-magnitude estimate of the probability for collisional excitation is given by [FORMULA]. The cross section, [FORMULA], can be approximated with the geometrical area of a molecule, which is about [FORMULA]. The velocities of the molecules for the temperatures considered are of the order of [FORMULA]. For the Q-band at [FORMULA], with an Einstein coefficient for spontaneous decay of [FORMULA], the probability of radiative excitation ([FORMULA], [FORMULA] being the mean intensity of the star) is [FORMULA] for a typical M giant and extrapolating back to [FORMULA]. In order for the collisional processes to dominate we thus need [FORMULA]. Therefore, it seems that the densities in the circumstellar environments, are not high enough for the collisions to be completely dominant. Radiative processes may therefore be more important than collisional ones, which may lead to severe departures from LTE.

However, among the rotational energy levels in the vibrational ground state the levels are thermally populated. This conclusion can be drawn since the Einstein coefficient for spontaneous decay between the rotational states are several orders of magnitude smaller than the coefficients for vibrational decays. The relative population of the levels within the vibrational ground state are therefore given by the Boltzmann distribution and the local gas temperature. The population of the rotational levels of higher lying vibrational states are affected by radiative transitions between the different vibrational states.

In a situation dominated by radiative processes, the strongest transitions involved in a case of statistical equilibrium are shown in Fig. 3. These levels and the transitions between them were included in a Monte Carlo simulation of a radiative situation to discern to what extent the transitions will eventually spread out the populations of the rotational levels within higher lying vibrational states. In this situation the rotational levels of the ground state are given by the Boltzmann distribution and the probabilities of the transitions, i.e. the oscillator strengths, of an individual, rotation-vibration transition within a band are given by


where [FORMULA] denotes the projection of the electronic orbital angular momentum along the axis of the linear molecule and [FORMULA] are the Hönl-London factors which can be found in, e.g. Bernath (1995). These can be calculated for all the transitions considered.

The result of this investigation shows that the spreading out of the level populations is on a few percent level and hardly noticeable. This means that also in an environment dominated by radiative processes the relative rotational populations within the higher lying vibrational levels will map the Boltzmann distribution of the ground vibrational level. This leads to the conclusion that the widths of all vibration bands will reflect the underlying gas temperature. However, in general, the relative strengths or amplitudes of the bands will not be given by the thermal distribution for the vibrational levels, if the radiation processes dominate. The 13.9 and [FORMULA] bands originate from upper states of about the same energy and they are affected similarly by radiation excitation, which may, in any case, lead to similar band strengths.

4.3. The temperature of the bands

In this section we account for the excitation temperatures of the different bands for the stars showing emission. This is done by modelling the band widths of a thermally populated CO2 molecule and fitting its spectrum to the observed bands. The HITRAN molecular database (Rothman et al., 1987) and the high temperature analogue HITEMP database (Rothman et al., in prep.) are used to retrieve relevant spectral line data such as line positions, level populations and transition probabilities.

The widths of the bands depend on the population of the rotational levels within the upper vibrational state and thereby the excitation temperature. In a thermal model these are populated according to the Boltzmann distribution.

The observations of the 15.0 and the [FORMULA] Q-bands of carbon dioxide in R Dor and R Crt are shown in Figs. 4 and 5. These spectra show the residual emission after the fringes, due to the spectrometer, have been removed. Also plotted in the figures are the bands of a thermal model for [FORMULA] smoothed to the resolution of the observations. The emission predicted by the model is plotted with the observed continuum as a base line and is scaled to fit the [FORMULA] band.

[FIGURE] Fig. 4. The full line shows the two observed Q-bands of R Crateris and the dashed line represents a thermal model of [FORMULA] scaled to the [FORMULA] transition

[FIGURE] Fig. 5. As in Fig. 4 but for R Doradus. The forests at 14.5 and [FORMULA] are the R and P branches of the [FORMULA] transition in the model of [FORMULA]. These lines are not resolved in the resolution of the observation

For R Dor the excitation temperature as estimated from the [FORMULA] band width is [FORMULA], while the [FORMULA] band is fitted best with [FORMULA]. The band ratio between the [FORMULA] and the [FORMULA] bands is 1.7.

Also for R Crt the [FORMULA] band suggests [FORMULA], whereas the [FORMULA] band is better fitted with a thermal model with a temperature of [FORMULA], cf. Fig. 6. The observed band ratio is for this star 1.0.

[FIGURE] Fig. 6. The observed [FORMULA]-band for R Crt is plotted in full line. Arbitrarily scaled thermal models for temperatures of [FORMULA], [FORMULA], [FORMULA], and [FORMULA] are also plotted in the figure. The maximum of the model spectra is shifted towards the blue with increasing temperature. The relative errors in the observations are on the order of a few percent

It is also clear from Figs. 4 and 5 that the band intensity ratios can not be explained by a simple, optically thin, one-component LTE-model. Our observations of the [FORMULA] band of R Dor in the high resolution mode also show the individual Q-lines. These observations are unfortunately very difficult to reduce due to cosmic hits. However, from the observations, it can be concluded that the intensity distribution of the different rotational lines within this vibrational band suggest a temperature several hundred degrees below [FORMULA], cf. Fig. 7. At [FORMULA] the maximum line strength lies around the transition Q24, and at [FORMULA] at Q10. Also, the Fabry-Pérot observations smoothed to a resolution corresponding to the one used in the other spectra, can not be fitted with a [FORMULA] model, cf. Figs. 6 and 7.

[FIGURE] Fig. 7. The high-resolution Fabry-Pérot observations of R Dor. These observations are very difficult to reduce due to cosmic ray hits, and spurious features are a problem. Also plotted in the figure is the spectrum smoothed to the resolution corresponding to the other observations

4.4. The number of emitting molecules

The number of molecules needed in a certain vibrational state in order for an emitting optically thin envelope to radiate a given measured flux, [FORMULA], can be calculated. With the Einstein coefficient for spontaneous decay, [FORMULA], the total number of molecules in the upper level, [FORMULA], is given by


This number should be compared to the total number of available CO2 molecules in all vibrational states, which can be estimated from


where [FORMULA] is the abundance of CO relative to hydrogen, [FORMULA] the mass-loss rate, [FORMULA] the terminal wind velocity and [FORMULA] and [FORMULA] are the outer and inner radii of the emitting shell, respectively. As before, we adopt a number ratio of CO2 molecules to CO molecules of a few [FORMULA].

Using these expressions we can compare the number of excited CO2 molecules needed for the measured flux to the total amount of CO2. The measured fluxes in the 13.9 and [FORMULA] bands are presented in Table 4. The table also gives the number of excited molecules required for the measured flux as well as the total number of molecules in an emitting shell, 3 stellar radii wide, and a large emitting envelope. The extension of the CO2 shells for R Dor, R Crt and R Cas used for the results in the last column in Table 4, are found by scaling the results for TX Cam from the models of Willacy & Millar. This column therefore gives the number of carbon dioxide molecules in a large emitting shell extending from 50 to 1000 R* for R Crt and R Cas and from 20 to 300 R* for R Dor.


Table 4. Number of excited CO2 molecules required for the measured flux and the total number of molecules for two different shell sizes. See text

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© European Southern Observatory (ESO) 1999

Online publication: December 4, 1998