Astron. Astrophys. 357, 180-196 (2000)

2. Time-dependent numerical hydrodynamics

We have applied two different computer codes to study the time-dependent hydrodynamics of circumstellar gas/dust shells. In the following we briefly describe their main features.

2.1. DEXCEL: two-component radiation hydrodynamics

This two-component (gas / dust) radiation hydrodynamics code is designed to model a stellar wind driven by radiation pressure on dust grains and subsequent momentum transfer to the gas component via dust-gas collisions. The coupled equations of hydrodynamics and frequency-dependent radiative transfer governing the structure and temporal evolution of circumstellar gas/dust shells are solved numerically, with a time-dependent inner boundary condition that accounts for the temporal changes of the stellar radiation (, ) and wind parameters (). In this way, the results from detailed stellar evolution calculations with a prescribed mass loss rate can be combined with a hydrodynamical model of the circumstellar envelope. In all models discussed in the following, dust is represented by single-sized grains composed of amorphous carbon with the usual optical properties (Rouleau & Martin 1991). For the mass loss `eruption' models (cases B and C , see below) we have assumed a grain radius of m, a dust condensation temperature of  K, and a dust-to-gas mass ratio of ; for the interacting winds models based on stellar evolution, the assumptions are only slightly different: m,  K, .

The numerical scheme adopted here for the solution of the system of the Eulerian hydrodynamics / radiative transfer equations in spherical geometry is fully implicit. Hence, the time step of the simulations is not restricted by the Courant condition, an important advantage for applications where it is necessary to cover long time intervals. On the other hand, the scheme is only first order in space, and as such suffers from considerable numerical diffusion. Although this code can work with a moving grid, this option was not used in cases A,B,C in order to avoid additional numerical diffusion.

All DEXCEL models used for this investigation have radial grid points spaced according to

In order to fully resolve the dust acceleration region, the parameter q was chosen such that the spatial resolution is higher in the innermost parts ( cm, ) than in the outer parts ( cm, ). Test calculations have shown that this resolution is more than sufficient for a correct modeling of the dynamics of the dust-driven wind.

Further details of the underlying physical assumptions and the numerical procedure being employed have been given by Steffen et al. (1998, henceforth SSS98).

We point out that this code was not designed for a detailed modeling of the thermal structure of the gas component, since we do not even solve the energy equation for the gas but simply assume that the gas temperature equals the radiative equilibrium dust temperature. Nevertheless, DEXCEL is a valuable tool for gaining basic insights into the fundamental dynamical processes occurring in a dusty AGB wind envelope. As far as possible, the results obtained with this code have been checked by recalculating the same problem with a completely independent code, which is described in the next section.

2.2. NEBEL: one-component, Godunov-type hydrodynamics

The one-component, explicit code NEBEL solves the Eulerian equations of hydrodynamics in spherical geometry together with the rate equations describing time-dependent ionization / recombination of astrophysical plasmas. The numerical scheme is based on a high-resolution second-order Godunov-type advection scheme ("wave propagation method", Le Veque 1997), including an approximate Riemann solver. This inherently conservative method adequately calculates the propagation and interaction of non-linear waves at each cell boundary. It is therefore particularly well suited to resolve even strong shocks, which can arise, e.g. due to interacting winds. The equation of state is that of an ideal gas of solar composition (including partial ionization). For more details about the NEBEL code, which was actually designed to compute the evolution of planetary nebulae, see Kifonidis (1996), Perinotto et al. (1998) and Kifonidis et al. (2000).

Note that in contrast to the two-component radiation hydrodynamics code described above, this code does not account for source terms in the momentum equation, i.e. the acceleration due to radiation pressure and gravity is neglected. However, there is a source term in the energy equation, describing the emission (and absorption) of radiation. For the purpose of the test calculations presented in this work, we have employed a simple radiative cooling function,

where is the cooling rate per unit mass in erg/g/s. This function was constructed from the cooling rate for H₂ as given by Woitke et al. (1996; their Fig. 5) and extrapolated to lower densities. Note that we use this cooling function for the purpose of illustration only. It probably underestimates the radiative cooling since other potential cooling agents are not accounted for. On the other hand, no source of gas heating is included, which implies that the gas would cool to very low values due to adiabatic expansion (). For this reason, and in order to make the gas temperature comparable with that in DEXCEL, we enforce a minimum temperature such that at every point , i.e. the local gas temperature cannot fall below the local dust temperature , which is estimated from the corresponding solution of the radiative transfer problem as obtained by DEXCEL (). As will be seen in examples presented in Sect. 3, the resulting gas temperature is then essentially equal to the minimum temperature throughout the model, except in the vicinity of the stronger shocks. The role of the cooling function is therefore limited to keeping the shocks roughly isothermal, and smoothing the shock fronts somewhat. For comparison, we have sometimes also considered the adiabatic case , where and T is allowed to drop below .

Since the NEBEL code does not account for radiation pressure and gravity, it has been applied only to regions of the wind where these external forces are negligible in comparison to the internal forces (thermal pressure, viscosity). From steady-state models of dust-driven AGB winds obtained with DEXCEL (Steffen et al. 1997), we find that for the parameters considered in the following the acceleration region is restricted to the inner parts of the outflow,  cm. In order to recompute a given problem with NEBEL, we use the values of gas velocity, density, and temperature at  cm, , computed before by DEXCEL as a time-dependent inner boundary condition for the NEBEL code. In this way, the results of the two codes may be compared in the outer wind region ( cm). It is impossible, however, to study the acceleration region ( cm) with NEBEL.

All NEBEL models used for this investigation have radial grid points, also spaced according to Eq. (1). The parameter q was chosen such as to achieve a distribution of mesh points complementary to that used with DEXCEL, giving a better resolution in the outer regions: in the innermost parts ( cm) , while in the outer parts ( cm) . With this setup, even the very thin shells discussed in this work are fully resolved at all radial distances, with at least 10 cells across the structure.

© European Southern Observatory (ESO) 2000

Online publication: May 3, 2000
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