## 2. Time-dependent hydrodynamical model calculationsOur model calculations are based on the following assumptions: The central AGB star's luminosity, , effective temperature, , and mass loss rate, , vary as a function of time according to the stellar evolution calculations by Blöcker (1995). The mass of the star, , changes in agreement with the prescribed mass loss rate (; note that is defined here as a positive quantity). The evolutionary track used for the examples presented in the following is for a star with an initial mass which, due to mass loss, is reduced to a final mass when the end of the AGB is reached. At some distance from the star the outflowing gas has cooled to the dust condensation temperature () and dust is assumed to form instantly at this radius , which must very closely coincide with the sonic point (critical point). Since at a given distance from the star the dust temperature is a function of , , and , the location of the dust formation point can vary considerably over a thermal pulse cycle on the upper AGB, typically by a factor of 2. After their formation the dust grains of single size are accelerated away from the central star by radiation pressure, drift through the gas and drag along the gas component due to the frictional coupling provided by dust-gas collisions. A radiation hydrodynamics code, originally developed to investigate
the dynamical evolution of protostellar clouds, is employed to solve
the time-dependent equations of hydrodynamics and frequency-dependent
radiative transfer for a two-component 'fluid' consisting of
## 2.1. Equations of radiation hydrodynamics
The structure and time evolution of the stellar outflow is governed by
the following set of partial differential equations (basic equations
of hydrodynamics):
where The last term, , is our simplistic approach
to represent the time-averaged outward acceleration of the gas due to
shock waves generated by the Mira-type pulsations of the central AGB
star. It is introduced as an artificial acceleration term supporting
the outflow, designed to ensure that the gas velocity We have constructed the following analytical expression for : Here is the local escape velocity at distance Evidently, , so just
balances the gravitational attraction if the gas velocity equals
at least in the inner region,
, where and the second
factor . The factor was
introduced to ensure that vanishes in the outer
part of the shell, , where
, irrespective of the actual gas velocity
In simulations performed with , we sometimes
found conditions where the net acceleration of the
In addition to the usual terms on the left hand side, the term on the right hand side describes the change of the gas density (as seen by an observer attached to the moving grid) due to the grid motion relative to the density distribution in the inertial frame of reference.
where and is the flux weighted extinction cross section (including absorption
and isotropic scattering: ). The first term on
the right hand side of Eq. 7 describes the acceleration of a dust
grain by radiation pressure, the second term corresponds to the
gravitational deceleration by the central mass (usually much smaller
than the radiative acceleration), and the third term again accounts
for the dynamical coupling between gas and dust. Interaction between
dust particles ("dust pressure") and the effect of shock waves on dust
are neglected.
where is the number density of dust grains.
Hence, the total amount of dust in the computational domain is given
by the fluxes through the model boundaries. Condensation or
evaporation of dust is not considered. For a given dust density distribution, the thermal structure of the
dust shell is determined by the radiation field of the central star,
which, for simplicity, is assumed to radiate as a blackbody with
. The energy equation for the dust component
stipulates the condition of radiative equilibrium at any time:
where is the frequency-integrated radiative energy flux. The temperature of the dust component is related to the equilibrium radiation field by where is the angle-averaged specific intensity and is the Planck function at the local dust temperature . Note that does not only represent the direct stellar radiation but also includes the thermal emission and diffuse scattered radiation from the dust shell. Details about the method used for the numerical solution of the equation of radiative transfer in spherical geometry are given in Paper I and in Yorke (1980b). We presently do not solve the energy equation for the gas
component. Instead, we simply take
an assumption which needs to be modified for applications where the gas temperature is a critical quantity (e.g. for the calculation of molecular emission line profiles). Finally, we would like to point out that the numerical scheme
employed for the solution of the hydrodynamical equations is fully
implicit. This means that the time step of the simulations is
## 2.2. Dust opacities
The solution of the set of equations given above depends on the
geometrical and optical properties of the dust grains and their
abundance. In this study we use
## 2.3. Initial condition, radial grid
The time-dependent simulations start from a steady state solution (see
Paper I) computed for the stellar parameters and mass loss rate
corresponding to the initial time . The distance
of the inner edge of the where is the dust condensation temperature (see Table 1) and given through Eq. 11. For the initial model, the computational radial grid starts at , the inner edge of the dust shell, and ends at an outer radius cm ( pc). All models used for this investigation have radial grid point spaced according to By choosing an appropriate value we have concentrated the grid points in the inner part of the model such that the spatial resolution in the innermost parts (acceleration region) is four times better than for an equidistant logarithmic grid covering the same radius range with the same number of points. In order to take into account the presence of the ## 2.4. Controlling the grid motionSince the stellar parameters and the mass loss rate change with time, the position of the dust condensation radius, , will follow these variations. Our calculations employ a grid that moves with a time-dependent velocity, , relative to the central star such that The grid velocity is uniform throughout the grid, i.e.
is independent of The time constant controls the response time of the grid motion to changes of the dust condensation radius. If changes on time scales much longer than , the moving grid can closely follow the position of the dust condensation radius. Variations on time scales much shorter than are essentially ignored. We have chosen yrs, corresponding to the shortest time scales encountered in the variations of the stellar parameters during a thermal pulse. This choice also ensures that the resulting grid velocities are always small compared to the gas outflow velocities. If necessary, the hydrodynamical time step, , is reduced to restrict the grid motion over to be always less than the distance between any two adjacent grid points. The alternative to employing a moving grid is to use a fixed grid which is large enough to cover the whole range of possible dust condensation radii. In this case one would have to deal with more or less extended dust-free regions in the inner parts of the computational domain, requiring special treatment. This complication is avoided in the moving grid approach, which we found to be more economic (in terms of the number of grid points required) and more straightforward. ## 2.5. Boundary conditions
The boundary conditions at essentially
determine the nature of the solution. At the The gas density at the inner boundary is then given by The mass loss rate as a function of time through the inner boundary of the dust shell, , may, in principle, be specified in an arbitrary way. In this work it is enforced in accordance with a particular stellar evolution sequence as described below. The dust density is related to the gas density via the dust-to gas ratio as The function controls the degree of dust condensation and is defined as with and . This means
that we assume the process of dust condensation to become inefficient
below mass loss rates of , which via Eq. 18 may
be translated into a critical gas density of g
cm Although, due to the finite value of the time constant , the actual dust temperature at the innermost grid point can deviate somewhat from , it is assumed that dust condensation, and hence dust acceleration, always begins at . For the computation of the radiative transfer, we apply a
time-dependent At the outer boundary, ,
, and
can be computed consistently from the
hydrodynamical equations in the case of a © European Southern Observatory (ESO) 1998 Online publication: August 6, 1998 |