In the following we present the resulting steady state solutions for different stellar parameters and mass loss rates. Three sets of models are considered, the first assuming oxygen-rich chemistry with silicate dust (Figs. 2 (click here) to 5 (click here), Tables 2 to 5), the other two being based on carbon-rich chemistry with graphite dust (Figs. 6 (click here) to 8 (click here), Tables 6 to 8) and grains of amorphous carbon (Figs. 9 (click here) to 11 (click here), Tables 9 to 11), respectively. The figures show the resulting steady state velocity field (gas and dust component) and the corresponding combined spectral energy distribution of the star and the dusty outflow for a representative sample of models. Details about the stellar parameters used and other input data are given in the figure captions. Additional information can be found in the tables which list the basic results for all the steady state models investigated here. For each set of input parameters, two runs were carried out, one including the effects of gas pressure in the equations of motion, the other assuming no gas pressure. Note that for some examples, the adopted mass loss rates are so high that a significant fraction of the stellar mass is expelled during the flight time of a gas parcel through the wind shell. In these cases, steady state models are certainly not adequate.
From the analysis of the present sample of steady state models we can draw the following conclusions about the properties of dust-driven winds around late type stars:
a) The acceleration of the dust and gas takes place in a narrow region close
to the dust formation point (
). Although in
general 
is not constant but depends on r, this is in
qualitative agreement with Eq. (17 (click here)) which suggests that 90% of the
terminal velocity is reached at 
.
b) For low mass loss rates, the drift velocity of the dust relative to the
gas, 
, may amount to several times the gas velocity and
reduces the effective dust opacity (per unit volume) by a factor u/w. In
general, 
(used by e.g. Le Sidaner & Le Bertre 1996) is a
reasonable approximation only for low to moderate mass loss rates, while
becomes valid for very high mass loss rates.
c) For high mass loss rates, i.e. large (flux-averaged) optical depths, the
momentum flux in the outflow, 
can exceed the momentum flux of the
photons, 
. The ratio of these quantities, 
, is listed in Tables 2 to 11 and can be seen to exceed unity in
a number of cases. In fact, several stars with 
are
known and it is now clear that there is no physical reason preventing the
outflow from crossing the 
border (see e.g. Nezter & Elitzur
1993; Habing et al. 1994; or
Ivezić & Elitzur 1995 for a discussion of
this issue).
d) As expected, the outflow velocity, 
, depends on stellar mass,
luminosity, grain properties and dust abundance. But in contrast to our
simple gray model (cf. Eq. 17 (click here)), depends also on the stellar
effective temperature, 
, and on the mass loss rate. This is
because the extinction cross section of realistic dust grains is strongly
wavelength dependent (see Fig. 1 (click here)), and hence the flux-averaged
opacity, 
, the quantity determining the
net radiative acceleration, depends on the spectral energy distribution of
the stellar radiation () and on the degree of reprocessing
of the stellar radiation by the dust shell, which increases with the optical
depth of the dust shell, i.e. with the mass loss rate.
e) The terminal outflow velocity, , is of the order of the escape
velocity at the dust formation point, 
. Since
(see also Eq. 17 (click here)) with the effective given by
(essentially the average in the acceleration region),
outflow velocities in the range 1 to 3 times imply effective
s in the range 2 to 10.
f) It has often been argued that the acceleration due to the gas pressure
gradient may be neglected because the outflow is highly supersonic
(e.g. Habing et al. 1994). However, this condition is not really fulfilled
by the relatively slow outflows around AGB stars where the ratio
hardly exceeds 10.
The gas pressure term 
can safely be neglected
if it is much smaller than the advection term 
over most of the
acceleration region. For the wind shells investigated here,
the ratio of pressure term to advection term can be evaluated analytically
assuming a velocity structure according to Eq. (17 (click here)) with a
gas temperature falling like r-1/2. The result is
where x=r/r1 is the dimensionless radial coordinate, and 
is the ratio of initial to final gas velocity,
which approximately equals the inverse ratio of outflow velocity to sound speed
at the inner radius of the shell. Near x=1, Y is close to 1 since the flow
starts with 
at the dust formation point r1. For small
, Y decreases sharply as x increases and becomes
negligible for x > 1.2, so the overall effect of the gas pressure term is
small. However, the situation is different for larger 
where
Y attains a minimum near x=1.7 and increases again for larger x. In the
case 
, for example, Y is larger than 0.35 everywhere,
so the gas pressure term is not at all negligible. Indeed, comparison of the
runs with and without the effects of gas pressure shows that, in general, the
gas pressure term leads to significantly higher outflow velocities
(see Figs. 15 (click here), 16 (click here), 17 (click here) and tables). Only if
the radiative acceleration is strong enough to produce highly supersonic
outflows (
30 km/s) the gas pressure term becomes unimportant. This is
what we see in our models with the highest luminosities (``supergiant'' models
E and F, see Tables 4 and 5 as well as Fig. 15 (click here)).
g) For decreasing mass loss rate, the coupling between dust and gas
diminishes and the drift velocity increases monotonically. Consequently,
the effective decreases because of the factor u/w
(cf. Eqs. (15 (click here)), (16 (click here))). The
minimum mass loss rate is reached for 
. No dust
driven wind is possible below this critical mass loss rate. We have
refrained from calculating these minimum mass loss rates from our models
because we found it to be very sensitive to the assumed gas pressure (see
models C6_m and C6_n, Table 2 or models J6_m and J6_n,
Table 9). Lower mass loss rates can be
sustained when gas pressure supports the outflow. In addition, it must be
kept in mind that some kind of shock wave pressure is generated by the
stellar pulsation which will reduce the minimum mass loss rate even
further.
h) It has been noted above that the outlow velocity is independent of
for gray dust (cf. Eq. 17 (click here)), so there is no maximum possible
mass loss rate. However, the situation is different for realistic dust
opacities which strongly decrease with wavelength (cf. Fig. 1 (click here)).
As the mass loss rate is increased the optical depth of the dusty envelope
is increased as well, and the maximum of the spectral energy distribution
of the radiation, as seen by a dust particle in the acceleration region of
the shell, is shifted to longer wavelengths. As a consequence, the flux
averaged extinction cross section (cf. Eq. 5 (click here)) is decreased and the
effective is reduced (cf. Eqs. (15 (click here)), (16 (click here))).
This mechanism puts an
upper limit to the mass loss rate, the quantitative value of which depends
on the stellar parameters. Again, we find this limit to be sensitive
the the assumptions about the gas pressure. For example, model I1 can drive
a mass loss of 
/yr only if supported by gas pressure
(see Table 8).
i) Test calculations have confirmed that the assumption of complete
momentum coupling is an excellent approximation. Under this assumption,
the drift velocity, is given by (see e.g. Tielens 1983)
j) We note that for the parameters used in this study, the resulting gas outflow velocities are in the range 5 to 40 km/s, covering the range of observed outflow velocities.
Along with the velocity structure, Figs. 2 (click here) to 11 (click here) show the corresponding stellar input spectrum (dot-dashed) and the emergent spectral energy distribution (SED) of the star plus circumstellar dust shell (solid line). From the analysis of this sample of spectra we infer the following properties.
a) The shape of the SEDs changes monotonically with mass loss rate, i.e. with
the optical depth of the dust shell. Oxygen stars with low mas loss
rates show the characteristic silicate features (
and 
m) in emission, while the overall shape of the SED closely corresponds to
that of the central star. For intermediate mass loss rates, the SED is
shifted to the red through processing of photons by the dust shell, and self
absorption begins to become important, noticeable by a central absorption in
the main silicate feature which is, however, still seen in emission (see model
D3_m, center of Fig. 3 (click here)). For even higher mass loss rates,
the stellar SED is no longer visible (IR objects) and the silicate features
turn into strong absorption.
b) Analogous changes occur for Carbon stars, although the lack of features in the opacity of carbon-based grains makes it more difficult to detect subtle differences in the spectra of models with slightly different mass loss rates.
c) For given dust properties, the shape of the SED is determined, to a first approximation, only by the total optical depth of the dust shell (at some reference wavelength), and is essentially independent of the stellar luminosity (see also Ivezić & Elitzur 1995).
d) In principle, the sequences of models presented here can be used to estimate
the mass loss rates of Oxygen and Carbon AGB stars by finding the model which
most closely matches the observed SED together with the observed (gas)
outflow velocity. The real mass loss rate may differ from that of the model
identified in this way if the star to be analyzed has a different luminosity
or dust-to-gas ratio than assumed in the model. However, it can be shown that
in this case the scaling relation
can be used to derive the actual mass loss rate if 
and 
are known independently (see e.g. Eq. (1) in Groenewegen 1995).
e) To provide a reference for comparison with the time-dependent
calculations to be presented in a forthcoming paper (for first results see
Schönberner et al. 1997a,b; Steffen et al. 1997),
we have calculated the positions of all
of our steady state models in two different IRAS two-color-diagrams (
m)/
m) versus m)/
m) and
m)/m) versus m)/m),
see Figs. 12 (click here) to 14 (click here)). The IRAS colors were computed by
convolution of the emergent synthetic spectrum with the IRAS band profiles,
taking into account the wavelength dependence of the spectra assumed by the
IRAS team (i.e. 
, see the IRAS Explanatory
Supplement 1985). The models with the lowest mass loss rates are found to the
left of these diagrams, while the models with the highest mass loss rates are
located at far right. We note that for fixed dust properties, all models fall
on a simple color-color relation with (or optical depth) as the only
parameter, in agreement with the results presented by Ivezić &
Elitzur (1995, see their Fig. 6). These diagrams also demonstrate the
effect of the gas pressure on the computed IRAS colors, which is clearly
noticeable for most of the AGB stars while it is negligible for the
supergiants.
f) For a number of cases we have computed ``simple'' comparison models
in order to illustrate the difference between synthetic spectra computed
from these models and those resulting from our self-consistent hydrodynamical
approach. The simple models are constructed by assuming a constant
outflow velocity equal to the terminal gas outflow velocity of the
respective hydrodynamical model (i.e. 
), and ignoring
dust drift (i.e. 
). The comparison of the SEDs is shown in
Figs. 3 (click here), 6 (click here), and 9 (click here). We were surprised to see
the relatively close agreement between both sets of models. Obviously, the
effects of assuming a constant outflow velocity and neglecting the dust
drift velocity cancel to some degree. We conclude that ``simple'' models
may be used for the analysis of observed SEDs without introducing large
systematic errors, provided the adopted constant outflow velocity equals
the observed one.