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Astron. Astrophys. 337, 149-177 (1998)

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2. Time-dependent hydrodynamical model calculations

Our model calculations are based on the following assumptions: The central AGB star's luminosity, [FORMULA], effective temperature, [FORMULA], and mass loss rate, [FORMULA], vary as a function of time according to the stellar evolution calculations by Blöcker (1995). The mass of the star, [FORMULA], changes in agreement with the prescribed mass loss rate ([FORMULA]; note that [FORMULA] 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 [FORMULA] which, due to mass loss, is reduced to a final mass [FORMULA] when the end of the AGB is reached.

At some distance from the star the outflowing gas has cooled to the dust condensation temperature ([FORMULA]) and dust is assumed to form instantly at this radius [FORMULA], 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 [FORMULA], [FORMULA], and [FORMULA], the location of the dust formation point [FORMULA] 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 gas and dust under the assumption of spherical symmetry. Details about the numerical methods used are described by Yorke & Krügel (1977), and by Yorke (1980a; 1980b). The code has been modified extensively to adapt it for the modeling of dusty stellar outflows. A moving grid is employed to keep the dust formation point close to the innermost point of the numerical mesh. The basic equations to be solved in this context are given below.

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):

Equation of motion for the gas component:

[EQUATION]

where u and w are the gas and the dust velocity (relative to the central star), respectively, [FORMULA] is the radial velocity of the moving grid, [FORMULA] is the gas density, p is the gas pressure, [FORMULA] is the total mass inside radial coordinate r, [FORMULA] measures the thermal velocity dispersion of the gas particles at temperature T (for the molecular weight we assumed [FORMULA], corresponding to gas of solar composition with hydrogen in atomic form; the choice of µ is not critical, see Paper I) and [FORMULA] is discussed below. Note that [FORMULA] means the time derivative of f at a fixed position of the moving grid. All other symbols have their usual meaning. The first term on the right hand side describes the acceleration due to the gas pressure gradient (gas pressure term), the second term corresponds to the gravitational force and the third term accounts for the dynamical coupling between gas and dust (friction term).

The last term, [FORMULA], 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 u cannot drop significantly below a minimum velocity [FORMULA]. On the other hand, we require [FORMULA] to vanish whenever the gas outflow velocity u exceeds the limit [FORMULA]. The free parameters [FORMULA] and [FORMULA] ([FORMULA]) control the magnitude and range of [FORMULA]. The values adopted here for [FORMULA] and [FORMULA] are [FORMULA] km/s and [FORMULA] km/s ([FORMULA], see Sect. 2.5).

We have constructed the following analytical expression for [FORMULA]:

[EQUATION]

where

[EQUATION]

and

[EQUATION]

Here

[EQUATION]

is the local escape velocity at distance r form the star. Notice that there is no particular physical model behind the analytical form of [FORMULA].

Evidently, [FORMULA], so [FORMULA] just balances the gravitational attraction if the gas velocity equals [FORMULA] at least in the inner region, [FORMULA], where [FORMULA] and the second factor [FORMULA]. The factor [FORMULA] was introduced to ensure that [FORMULA] vanishes in the outer part of the shell, [FORMULA], where [FORMULA], irrespective of the actual gas velocity u. For a stellar mass of [FORMULA] our choice of [FORMULA] and [FORMULA] gives [FORMULA] cm and [FORMULA] cm. It can be shown that for these parameters the terminal outflow velocity will always be larger than [FORMULA].

In simulations performed with [FORMULA], we sometimes found conditions where the net acceleration of the gas was negative due to the weak coupling between gas and dust during periods of low mass loss, resulting in a gas flow back towards the star (the same behavior was also found by Vassiliadis & Wood 1992). Such situations cannot (presently) be handled in a self-consistent way by our code and lead to unrealistic density structures. We circumvented these problems by introducing [FORMULA].

Equation of continuity for the gas component:

[EQUATION]

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.

Equation of motion for the dust component (single grain):

[EQUATION]

where [FORMULA] and a are the mass and radius of a dust grain, respectively, c is the speed of light, and

[EQUATION]

is the flux weighted extinction cross section (including absorption and isotropic scattering: [FORMULA]). 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.

Equation of continuity for the dust component:

[EQUATION]

where [FORMULA] 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 [FORMULA]. The energy equation for the dust component stipulates the condition of radiative equilibrium at any time:

Energy Equation for the dust component:

[EQUATION]

where [FORMULA] is the frequency-integrated radiative energy flux. The temperature of the dust component is related to the equilibrium radiation field by

[EQUATION]

where [FORMULA] is the angle-averaged specific intensity and [FORMULA] is the Planck function at the local dust temperature [FORMULA]. Note that [FORMULA] 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

Energy Equation for the gas component:

[EQUATION]

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 not restricted to the Courant time step. This is a very important advantage in the present application where it is necessary to cover long time intervals.

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 "astronomical" silicates or amorphous carbon, with properties as specified in Table 1 (see also Fig. 1 of Paper I). The corresponding opacity data for "astronomical" silicates were kindly provided by B. Draine (for details see Laor & Draine 1993), and were taken from Rouleau & Martin (1991) in the case of amorphous carbon. In the hydrodynamical calculations we use a grid of 169 wavelength points distributed unevenly between 0.01 and 3100 µm.


[TABLE]

Table 1. Dust grain properties and abundances used in this work. The corresponding optical properties are shown in Fig. 1 of Paper I.


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 [FORMULA]. The distance of the inner edge of the dust shell from the central star, [FORMULA], is given by the condition

[EQUATION]

where [FORMULA] is the dust condensation temperature (see Table 1) and [FORMULA] given through Eq. 11. For the initial model, the computational radial grid starts at [FORMULA], the inner edge of the dust shell, and ends at an outer radius [FORMULA] cm ([FORMULA] pc). All models used for this investigation have [FORMULA] radial grid point spaced according to

[EQUATION]

By choosing an appropriate value [FORMULA] 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 interstellar medium (ISM), we modified the initial steady state solution in the outer parts where the gas density drops below [FORMULA] g cm-3. Here we require [FORMULA], [FORMULA], and [FORMULA]. An example of an initial velocity and density structure, including ISM, is shown in Fig. 3 (left hand panels). The role of the ISM is further discussed in Sect. 4.1.1.

2.4. Controlling the grid motion

Since the stellar parameters and the mass loss rate change with time, the position of the dust condensation radius, [FORMULA], will follow these variations.

Our calculations employ a grid that moves with a time-dependent velocity, [FORMULA], relative to the central star such that

[EQUATION]

The grid velocity is uniform throughout the grid, i.e. [FORMULA] is independent of r and the initial distribution of grid points given by Eq. 14 is only translated as a whole. Based on the approximation [FORMULA], i.e. [FORMULA], [FORMULA] is computed as

[EQUATION]

The time constant [FORMULA] controls the response time of the grid motion to changes of the dust condensation radius. If [FORMULA] changes on time scales much longer than [FORMULA], the moving grid can closely follow the position of the dust condensation radius. Variations on time scales much shorter than [FORMULA] are essentially ignored. We have chosen [FORMULA] 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, [FORMULA], is reduced to restrict the grid motion over [FORMULA] 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 [FORMULA] essentially determine the nature of the solution. At the moving inner boundary we adopt a constant initial velocity which is identical for the gas and the dust component and is taken to be 3 km/s, close to (but not smaller than) the local isothermal sound speed:

[EQUATION]

The gas density at the inner boundary is then given by

[EQUATION]

The mass loss rate as a function of time through the inner boundary of the dust shell, [FORMULA], 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 [FORMULA] as

[EQUATION]

The function [FORMULA] controls the degree of dust condensation and is defined as

[EQUATION]

where

[EQUATION]

with [FORMULA] and [FORMULA]. This means that we assume the process of dust condensation to become inefficient below mass loss rates of [FORMULA], which via Eq. 18 may be translated into a critical gas density of [FORMULA]g cm-3.

Although, due to the finite value of the time constant [FORMULA], the actual dust temperature at the innermost grid point can deviate somewhat from [FORMULA], it is assumed that dust condensation, and hence dust acceleration, always begins at [FORMULA].

For the computation of the radiative transfer, we apply a time-dependent stellar radiation field at the inner boundary of the dust shell, with a spectral energy distribution corresponding to that of a blackbody at [FORMULA], and with a net radiative flux given by

[EQUATION]

At the outer boundary, [FORMULA], [FORMULA], [FORMULA] and [FORMULA] can be computed consistently from the hydrodynamical equations in the case of a supersonic outflow considered here. We assume an external, isotropic ("interstellar") radiation field to be incident on the outer boundary, characterized by a blackbody temperature [FORMULA] K.

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© European Southern Observatory (ESO) 1998

Online publication: August 6, 1998
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