Before turning into planetary nebulae, low to intermediate mass stars
(1 -8 
) are found to evolve along the Asymptotic Giant Branch (AGB). In this
phase the time scales of evolution are determined mainly by mass loss since
this process dominates over nuclear burning. High mass loss rates between
10-6 and 10-4 /yr are very often detected, mostly by
excess continuum radiation in the infrared, originating from thermal
emission of circumstellar dust, and by the presence of molecular
rotation lines from different molecules seen in emission at submm, mm, and cm
wavelengths.
It is now widely accepted that the mechanism responsible for
such high mass loss rates during the AGB evolution is based on the efficiency
of radiation pressure on dust grains. Pushed away by shocks generated by the
Mira-type stellar pulsations, the outflowing gas cools down until, at some
distance from the star, heavy elements can condense. The dust grains belong
to one of two different types: (1) silicate-type grains, found around so
called ``Oxygen stars'' with an abundance ratio 
, and (2) carbon-based
grains around ``Carbon stars'' with 
. The newly formed dust particles
scatter and absorb stellar photons, in this way extracting momentum and
energy from the radiation field. The absorbed stellar light is reemitted in the
infrared while
the acquired momentum is transferred to the gas by collisions between dust
particles and gas molecules (see e.g. Gilman 1972;
Salpeter 1974; Kwok 1975;
Goldreich & Scoville 1976).
Following these pioneering works, the effect of
radiation pressure on dust particles has been studied in more detail by many
authors (see e.g. Tielens 1983). So far, however, models used for the
interpretation of observed spectra usually assume a prescribed density
distribution, typically 
(see e.g. Justtanont
& Tielens 1992; Danchi et al. 1994;
Hashimoto 1994, 1995;
Groenewegen 1995), ignoring the fact that the problem of momentum transfer
(hydrodynamics) is inherently coupled with the problem of radiative transfer:
Radiation pressure on dust determines the outflow velocity and hence the
density structure; at the same time, the density structure determines - via
radiative transfer effects - the spectrum of the photons and hence the
effective radiation pressure. Over the last years, this circular problem
has been addressed by Netzer & Elitzur (1993),
Habing et al. (1994)
and Ivezić & Elitzur (1995), and it has been found that several different
factors influence the efficiency of momentum transfer to the gas
and that it is important to describe the gas/dust shell as at least a two fluid
system allowing for a relative motion of gas and dust.
For simplicity, however, all of these papers consider only steady state solutions, although it is well known that the stellar parameters and the mass loss rate can undergo significant variations on rather short time scales when so called ``thermal pulses'' occur on the upper AGB.
We have developed a new code which is suitable to treat
time-dependent radiative hydrodynamics, and we have performed
calculations similar to those presented by Netzer & Elitzur (1993), with the
aim of first testing our code in the simpler case of constant mass loss rate
before applying it to time-dependent calculations of dusty AGB shells.
These will take into account the evolutionary changes of the stellar
parameters and the mass loss rate of the star at the center of the circumstellar
shell over a time interval of several 100 000 years, covering the final stages
of evolution on the AGB, including several so called ``thermal pulses'' (with
typical interpulse time scales of some 10 000 yrs) but ignoring the dynamical
effects of the Mira-type stellar pulsations (with typical periods of some 100
days). For first examples of such time-dependent calculations see
Schönberner et al. (1997a,b) and Steffen et al. (1997).
One additional goal of this work was to investigate the contribution of the
gas pressure to driving the outflow, an effect usually neglected when modeling
AGB winds. We have studied the influence of this factor on the outflow velocity
and on the efficiency factor 
, the ratio of momentum flux in the wind
to momentum flux of the stellar photons.
In Sect. 2 we describe all the equations and assumptions incorporated into our code as well as some simple calculations performed for gray dust opacity in order to check the numerical solution against the analytical one. The main results of our computations for a variety of stellar parameters are presented in Sect. 3, analyzing in detail the properties of the gas and dust outflow velocity and the corresponding spectral energy distributions as a function of mass loss rate. In Sect. 4 we perform an extensive comparison of our results with those presented by Netzer & Elitzur (1993) and by Habing et al. (1994). Finally, the conclusions are drawn in Sect. 5.