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Astron. Astrophys. 360, 562-574 (2000)

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6. Discussion

As mentioned above, the observed peak intensity is a good measure of the radial extent R in our slab models because the bands are optically thick. In fact, R is merely a measure of the emitting surface and offers no direct measure of where the CO2 layer is located. In this section we will assume that each of the CO2 layers at temperature T is located in a spherical shell around the central star and that [FORMULA] for every line of sight that crosses this shell. With these assumptions R is at the same time also a measure for the distance of this shell from the central star. We can then compare the results found for the different bands with temperature and density profiles derived with different models.

6.1. Temperature structure

Bowen (1988) studied heating and cooling of a periodically shocked Mira envelope. The radial kinetic gas temperature variation in the circumstellar shell is represented by a power law


where the exponent [FORMULA] may range from 0.4 to 0.7. Fig. 15 shows the temperature profile for [FORMULA]0.4, 0.5 and 0.7 with [FORMULA] K as in our LTE model. Temperature profiles obtained from hydrodynamical atmosphere calculations Höfner & Dorfi 1997 are similar to the [FORMULA] profile for the radial distances considered here. The temperatures derived from the CO2 bands are somewhat lower than expected from these models. This discrepancy may be due to [FORMULA] our conversion from R to radial distances, [FORMULA] non-LTE effects or [FORMULA] limitations on the hydrodynamical models.

The first possibility might be true for the 13.48 µm band that still has a relatively low optical depth. We recall that our conversion is based on the assumption that the CO2 layers are optically thick for every line of sight that crosses it. If this assumption does not hold for the other CO2 bands, this would imply that a significant part of the CO2 layer would be transparent and hence we would see emission from the hotter layers inside. Our consistency check described in Sect. 5 assures that this is not the case. Therefore we can rule out this possibility for these bands.

The rotational and vibrational molecular excitation temperatures are the same as the kinetic gas temperature only in LTE. Woitke et al. (1999) calculated vibrational and rotational excitation temperatures for CO2 in an oxygen-rich dynamical model atmosphere. They find that LTE holds only very close to the central star. Close to the photosphere, the vibrational excitation temperature drops below the kinetic gas temperature. At 2 [FORMULA] also the rotational excitation temperature decouples from the kinetic gas temperature. At 3.5 [FORMULA], the difference between the kinetic gas temperature and the rotational excitation temperature is already more than 300 K, and the discrepancy increases with increasing radius. Our results show a smaller discrepancy (at most 150 K) over a much larger range; moreover the discrepancy does not get larger with increasing radius as would be expected if non-LTE effects are the cause of the differences between our results and the calculated temperature profiles.

We therefore conclude that the hydrodynamical models we compared our results to are not yet adequate to describe the temperature structure of a semiregularly varying AGB star. A new generation of hydrodynmical models that include frequency dependent radiative transfer as described by Höfner 1999 may solve this discrepancy.

6.2. Density structure

From the derived radii R and column densities N we can calculate the number of emitting CO2 molecules [FORMULA]. To compare this with density laws derived for spherical models, we redistribute this number of molecules into a spherical shell from [FORMULA] to R and estimate the average H2 density in this shell as


Note that if we would adopt an inner radius larger than [FORMULA], the average densities would increase. Fig. 16 shows the local H2 densities calculated from the results in Table 1 adopting a CO2 abundance of [FORMULA] = 6 10[FORMULA]. This is the maximum CO2 abundance found in the chemical network calculations by Duari et al. (1999). The H2 densities estimated from our analysis are thus lower limits.

[FIGURE] Fig. 16. The local average H2 gas densities derived from LTE modeling (circles) assuming [FORMULA]= 6 10[FORMULA]. The dashed line is a density profile and the solid line an average density profile for a stationary outflow assuming v=1.5 km/s and [FORMULA]=2.3 10[FORMULA] [FORMULA].

The density structure can also be derived from other observations. Using conservation of mass and assuming a stationary outflow (v=constant), the local H2 density is given by


To compare this with our estimates, we have to calculate the average density in a spherical shell from [FORMULA] to [FORMULA], or


Fig. 16 shows the gas density [FORMULA] and the average gas density profile adopting the mass loss rate derived from the broad component in the CO line profile ([FORMULA] 10[FORMULA] [FORMULA], see Sect. 2) and assuming an outflow velocity in this region of v = 1.5 km/s. Combining the highest mass loss rate with the lowest outflow velocity assures that the derived density profile is an upper limit to the gas densities. For the radii derived from our analysis, this density profile is similar to the results of hydrodynamical model atmosphere calculations Höfner & Dorfi 1997; in these models the density gradient is steeper between [FORMULA] and [FORMULA] and locally there are density enhancements of an order of magnitude due to shocks.

When comparing the lower limits on the densities derived from the observations with upper limits on the densities from model calculations, we see that both are in reasonable agreement. However, if we take an inner radius for the CO2 layers that is larger than [FORMULA] and adopt a lower CO2 abundance, the densities derived from our analysis could easily increase an order of magnitude; if on the other hand we would adopt the mass loss rate of 1.7 10[FORMULA] and/or an outflow velocity of 10 km/s, the theoretical density profile would decrease with an order of magnitude and hence large discrepancies can easily be found. However, recent calculations in which frequency-dependent radiative transfer has been included in an existing hydrodynamics code Höfner (1999) show that the densities can increase dramatically when compared to hydrodynamical models treated in the grey approximation.

The critical density for thermalization of the vibrational bands of CO2 is of the order of 10[FORMULA] cm-3. The densities derived might be somewhat low for thermalizing the vibrational levels of CO2; however, as shown in the previous paragraph these densities are only lower limits and could easlily be increased an order of magnitude in which case the gas density is of the order of the critical density.

6.3. Evolutionary status

A good determination of the 12C/13C ratio would allow to pinpoint the location of this AGB star in the HR-diagram; the uncertain value of 10 for the 12C/13C ratio would indicate that no 3rd dredge-up has occurred yet and places this object at the early-AGB, which is supported by the short period and the low mass loss rate. A firm determination of the 12C/13C ratio however requires a multi-layer model.

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© European Southern Observatory (ESO) 2000

Online publication: August 17, 2000