Astron. Astrophys. 360, 562-574 (2000)
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
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
![[EQUATION]](img165.gif)
where the exponent may range
from 0.4 to 0.7. Fig. 15 shows the temperature profile for
0.4, 0.5 and 0.7 with
K as in our LTE model. Temperature
profiles obtained from hydrodynamical atmosphere calculations
Höfner & Dorfi 1997 are similar to the
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 our
conversion from R to radial distances,
non-LTE effects or
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 also
the rotational excitation temperature decouples from the kinetic gas
temperature. At 3.5 , 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
. To compare this with density laws
derived for spherical models, we redistribute this number of molecules
into a spherical shell from to
R and estimate the average H2 density in this shell
as
![[EQUATION]](img174.gif)
Note that if we would adopt an inner radius larger than
, the average densities would
increase. Fig. 16 shows the local H2 densities
calculated from the results in Table 1 adopting a CO2
abundance of = 6
10 . 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]](img186.gif) |
Fig. 16. The local average H2 gas densities derived from LTE modeling (circles) assuming = 6 10 . 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 =2.3 10 .
|
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
![[EQUATION]](img188.gif)
To compare this with our estimates, we have to calculate the
average density in a spherical shell from
to
, or
![[EQUATION]](img190.gif)
Fig. 16 shows the gas density
and the average gas density profile
adopting the mass loss rate derived from the broad component in the CO
line profile (
10
, 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 and
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 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 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
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.
© European Southern Observatory (ESO) 2000
Online publication: August 17, 2000
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