Eqs. (11) to (14) form a complete set of algebraic expressions which allows for the calculation of the response of the radiation field to small harmonic (3D) perturbations of the grey opacity structure in radiative equilibrium. As argued in Sect. 2, the ratio of the perturbation amplitudes controls the stability of the dust formation zone. In the following, we apply this concept to discuss possible structure formation processes in stationary, dusty winds of AGB stars.
For the unperturbed solution we use the results of a spherically symmetric, self-consistent, grey, dust-driven wind model provided by Dominik et al. (1990), which is shown in Fig. 2. The model is determined by four parameters which have been chosen as follows: stellar mass , stellar luminosity , mass loss rate , and carbon-to-oxygen ratio . Besides the velocity, density, temperature and dust structure, the stellar temperature (K) and the final outflow velocity km/s) are results of the model calculation. This model has been used as the initial model for the more sophisticated iteration procedure required to construct a self-consistent, stationary, frequency-dependent model for the infrared carbon star IRC+10216 by Winters et al. (1994), who found the best agreement with the observations for the upper choice of parameters.
No special numerical technique was required to compute the integrals in Eqs. (12) to (14). A simple nested integration scheme according to Simpson's rule has been applied with angular and typically 40 to 200 spatial grid points along the discrete rays, using local spherical coordinates. In order to avoid spurious effects caused by an unintended perturbation of the stellar photosphere, the inner boundary is placed at , i.e. well inside the dust shell. The results are shown in the lower part of Fig. 2 for two exemplarily chosen perturbation modes.
Perturbations along the radial direction () are generally found to be characterized by . The behavior of these modes is similar to radial perturbations in spherical symmetry: A shell-like increase of the grey opacity does not affect the bolometric flux at any point (which is fixed by the stellar luminosity and the radius), but only hinders and dams the outflowing radiation. Thereby, the mean intensity is increased inside of the point of increased opacity (backwarming) 2. This dependence is evident from the monotonically increasing -relation in grey stellar atmospheres (e.g. Mihalas 1978, p. 55). According to the arguments given in Sect. 2, such perturbations are stable in the dust formation zone. The slight, periodical variations of as a function of distance arise from the phase-relation between the point of interest and the point of maximum opacity in the dust shell which mainly contributes to the local mean intensity at .
Perturbation modes perpendicular to the radial direction () behave in a different way. A perpendicular perturbation of the dust formation zone, located between about 2 R* and 4 R* in the model with corresponding optical depth , introduces a tendency to decompose the dust shell into several fragments (here equidistant slabs of size , see Fig. 3). In a medium of this kind, the flux preferentially escapes through the "holes" in the shell, which at the same time increases the mean intensity between and decreases the mean intensity inside the slabs. Accordingly, locations with increased opacity coincide with locations of decreased intensity .
Analysis of the full -curve in Fig. 2 reveals that actually both effects mentioned above are active in the case . At the inner edge of the dust shell () the backwarming effect dominates and . For larger radial distances (about outside of the point of maximum in Fig. 2, which approximately coincides with the sonic point in the model) the formation of flux tubes and in-between situated shadowed regions as sketched in Fig. 3 is more significant and . As discussed in Sect. 2, the dust formation will be accelerated in the shadowed, cooler regions and the opacity contrast will increase. Thus, the perpendicular perturbation mode is found to be self-amplifying beyond some critical distance (here ).
Fig. 4 illustrates the influence of the wavelength of the perpendicular -perturbation on the resulting amplitude. As apparent from the figure, is required to cause noticeable changes of the radiation field. Otherwise, the medium is essentially optically thin within the perturbations and the small-scale variations have no profound effect in the radiative transfer, since they cannot accumulate sufficient optical depth across one perturbation length . A saturation of can be noted at , possibly because such perturbation modes already cover the entire dust formation zone in the model.
© European Southern Observatory (ESO) 2000
Online publication: June 8, 2000