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Astron. Astrophys. 317, 859-870 (1997)

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2. Description of the computer code

A detailed description of the basic method used in the code and its equations is given by Yorke (1980a, 1980b). Briefly, the frequency-dependent radiative transfer equations are solved under the assumption of spherically-symmetric geometry simultaneously with the thermal balance equation for a dusty envelope. However, significant modifications to the original code have been made. First, we have changed the method of integration of the ray equations by carrying out the solution twice and taking the geometrical mean of the resulting specific intensities. As in the original version of the code, we assume that source function ([FORMULA]) is constant on the interval from radial grid point [FORMULA] to [FORMULA], but now the ray equations are solved first with the values [FORMULA] and then with the values [FORMULA]. The same method of integration was applied in the case of the optical depth ([FORMULA]), treating now the extinction coefficient ([FORMULA]), which is sum of the absorption and scattering coefficients, in the same way as the source function above. Such an approach significantly increases the accuracy of the ray equation integration.

We have also increased the number of impact parameters between stellar surface and the first radial grid point (see Fig. 4 in Yorke 1980b). In the original version of the code only one impact parameter (just missing the stellar surface) was used. Increasing in the number of rays within the inner radius of the dust shell is crucial in the study of detached dusty shells such as investigated in this paper because the inner radius of the shell ([FORMULA]) is far away from the stellar surface.

Next, we have changed the code by introducing a grain size distribution for each of the two dust components. In the present version we have introduced a size distribution described by a power law index (p) for all dust radii in the range from a minimum radius ([FORMULA]) to a maximum radius ([FORMULA]). In such a case the source function is given by (compare this to [FORMULA] given by Eq. 7 of Yorke 1980b):

[EQUATION]

Here the thermal emission by dust grains of i -th type with size a at a given distance r from the central source, with a temperature [FORMULA] found from the solution of the thermal balance equation, is characterized by the Planck function [FORMULA] and the mean intensity of the radiation field is represented by [FORMULA]. Symbols without index i are sums of the corresponding quantities over dust components. The extinction coefficient is defined as: [FORMULA] where [FORMULA] is the extinction cross section (sum of the absorption and scattering cross sections), and the size distribution of such particles is described by [FORMULA] where [FORMULA] is a normalization constant given by

[EQUATION]

Here [FORMULA] is dust-to-gas ratio, [FORMULA] is gas density at distance r from the star and [FORMULA] is the specific density of the material forming the i -th type of dust grain.

Finally, we have introduced temperature calculations of very small dust particles using the quantum approach. In the case of very small dust particles the absorption of an individual energetic photon produces a significant over-heating yielding a temperature much higher than the equilibrium value and, in consequence, the equilibrium temperature calculation gives a poor approximation to the grain emission. A set of temperature bins is defined, and transition rates into and out of each bin are calculated for the local radiation field, from which the temperature probability distribution can be calculated. The treatment follows that of Guhathakurta & Draine (1989) with the exception that we only consider the quantum effects of the radiation field and neglect those of electron collisions. In the case of post-AGB objects electron densities are negligible and such an approach seems to be justified.

The source function in this case (dust size distribution together with non-equilibrium heating) was constructed by dividing the integral over dust sizes in Eq. (1) into two components:

[EQUATION]

Here [FORMULA] is the maximum size of dust particles for which quantum effects are still important. This size could be different for each of the dust components and is a strong function of the local radiation field. [FORMULA] is the probability ([FORMULA]) that a dust particle has a temperature [FORMULA]. The index k represents the number of temperature bins and thermal emission from dust grains for which quantum processes are important is given by:

[EQUATION]

Tests of our code have been performed to check consistency with other existing codes. Excluding quantum treatment of dust temperature calculations and restricting the distribution of grain sizes to the single size, we have checked that our code reproduce exactly results of the DUSTCD code (see e.g. Leung 1975, 1976 and Egan et al. 1988).

We have also checked quantum heating calculations by comparing our model results with those published by Siebenmorgen et al. (1992) - their Figs. 1a and 1b. We were able to closely reproduce their probability density (defined as the probability that dust particle temperature belongs to the interval [FORMULA] - a somewhat different definition than the one adopted in this paper) and their results for the emission of small graphite grains.

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© European Southern Observatory (ESO) 1997

Online publication: July 8, 1998
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