The input parameters used for the calculation of the steady state models
presented in the previous section were chosen such that they match exactly those
used by Netzer & Elitzur (1993, hereafter NE) who list the results of similar
dust-driven wind calculations for a variety of parameters in their Table 3
(missing information about input data, e.g. 
, u1, were kindly
communicated to us upon request by these authors). When comparing our results
with those of NE one has to be aware that NE ignored the gas pressure term
in Eq. (1 (click here)) and the thermal velocity dispersion
of the gas molecules, 
, in the friction term (cf. Eqs.(1 (click here)),
(2 (click here)) and (4 (click here))), which is equivalent to assuming zero temperature for
the gas. For the purpose of a direct comparison, we changed our code such that
the above mentioned terms can be ignored and recomputed all cases for vanishing
gas pressure and (model names ending with _n). The results of
these calculations should be directly comparable to the data given in NE's
Table. 3.
The comparison for different Oxygen stars with dust composed of astronomical
silicates in their circumstellar shells (cf. panels C to F of NE's Table 3),
each with a sequence of mass loss rates, is presented in Fig. 15 (click here). We
notice that neither the shape of the relation 
(
) nor the values of the
terminal gas velocities (except for the lowest mass loss rates) are in
reasonable agreement. In particular, all of our models show a maximum outflow
velocity at intermediate mass loss rates and a distinct decrease of \
towards higher mass loss rates, while the NE models exhibit monotonic increase
of with mass loss rate. Moreover, note that we were not able to find a
steady state solution for the lowest mass loss rate in panel C without the
support of gas pressure. In this case we found the gas to fall inwards; the
coupling between dust and gas was not sufficient to drive the wind.
A similar comparison for Carbon stars with graphite dust in their circumstellar
shells (cf. panels G to I of NE's Table 3) is
shown in Fig. 16 (click here). Here the shape of our relation () is in
qualitative agreement with the results of NE. However, in all cases our
terminal gas velocities computed without gas pressure are significantly lower
than those given by NE. For the highest mass loss rate in panel I we were not
able to find a steady state solution without the support of gas pressure.
To find out the reason for these differences, a number of models from NE's Table 3 were recomputed by Z. Ivezić (private communication), using exactly the same opacities as used in the present work, and using a code similar to that of NE, but with an improved version of the radiative transfer. The results found from these test calculations are very similar to ours.
For completeness and future reference, we have recomputed all the Carbon star
models assuming the
dust grains to be composed of amorphous carbon instead of graphite (such models
were not considered by NE). The results are displayed in Figs. 9 (click here) to
11 (click here) and are listed in more detail in Tables 9 to 11.
The terminal outflow velocities, , as a function of mass loss rate are
shown in Fig. 17 (click here). They are qualitatively very similar to those
obtained with graphite dust (cf. Fig. 16 (click here)). Quantitatively, the
velocities are somewhat higher, especially for high mass loss rates. For models
J, we have computed additional models assuming a more realistic molecular weight
of 
(molecular hydrogen) instead of the standard assumption 
(atomic hydrogen). As expected, the results are
between those for and those obtained ignoring the gas
pressure terms (
). We note that also for the gas
pressure leads to a noticeable increase of the expansion velocity.
A similarly detailed comparison with the work by Habing et al. (1994) (hereafter HTT) turns out to be impossible due to the lack of information about a variety of input parameters used in their computations. So we have to restrict our comparison to a few qualitative remarks.
In their Fig. 5, HTT show the variation of the gas velocity as a function of the
prescribed mass loss rate for an Oxygen star model with 
and
. These models are roughly comparable with our models C
( and 
). The velocities obtained by
HTT are quite similar to ours (considering only models computed without gas
pressure). Existing differences may be related to different assumptions about
the stellar mass, the dust condensation temperature, and the dust opacity which,
according to Fig. 10 of HTT, is about a factor of two higher than ours
(different grain size?). Nevertheless, the qualitative agreement with our
results is much better than in the case of NE. In contrast to NE, both the
results of the present work and those of HTT indicate that the outflow velocity
attains a maximum near 
/yr and decreases again towards
higher mass loss rates.
We have also tried to reproduce the 
relations plotted in
Figs. 6 and 7 of HTT. Our results, shown in the left-hand column of
Fig. 18 (click here), demonstrate that there is no unique
relation neither for Oxygen nor for Carbon stars, in contrast to what is
suggested by the work of HTT. Rather, depends strongly on
the stellar parameters. Compare, for example, the relations for models K and L
(bottom panel), which differ only in the adopted stellar mass. Similarly, the
relations for models D and E (top panel) are significantly different, indicating
that depends sensitively on the stellar luminosity. The
relations are expected to depend also on the adopted dust-to-gas ratio and on
the grain properties.
HTT's relation for Oxygen stars is much steeper than the
steepest one found in our set of models (ignoring gas pressure). It seems
impossible to pin down the reason for this discrepancy since there is no precise
information about the parameters used by HTT to produce the results shown in
their figure. Similar remarks apply to the comparison for Carbon stars with
amorphous carbon dust.
In Fig. 18 (click here) we also present the corresponding 
relations (right-hand column), where 
the flux-averaged
optical depth defined as
It can be shown that, at least in the case of no dust drift and no gas pressure,
(for details see Ivezić & Elitzur, Eq. (5)). We note that the relations shown in Fig. 18 (click here) can rarely be approximated by a linear
relationship and that their slope depends on the stellar parameters just like in
the case of . The reason for this variation is that for the
AGB stars considered here, 
is not much larger than unity
(cf. Eq. 26 (click here)). Only the supergiants (models E) realize the limiting
case 
.
Of course the 
relations become significantly steeper when the flow
is supported by gas pressure. This is also demonstrated in Fig. 18 (click here),
where for models D, H and K the relations computed with gas
pressure terms are plotted for comparison.
In summary, we conclude that neither the nor the
relations are particularly useful for deriving
additional information about the physical parameters of dusty stellar outflows
from their observable properties.