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Appendix A: Approximation for flux-averaged extinction

In a study of radiative transfer models in circumstellar dust shells (Gail et al., in prep.), it was found that the flux-averaged extinction coefficient can be approximated rather well by the expression given in Eq. (19). This approximation is based on the following considerations.

• 1.

  At the inner edge of the dust shell the radiation field is dominatedby stellar radiation. The spectral distribution of the flux,4πH_λ, can be approximated by a black body with the effective temperature of the star. The flux-averaged extinction coefficient can be approximated by the Planck-averaged extinction coefficient in this case.

• 2.

  Moving from the inside into the shell, the stellar contribution to the radiation field diminishes as exp(−τ_∗), where τ_∗ is the optical depth calculated from the inner edge with . At the same time, the contribution of the local dust emission to the total flux increases proportional to 1−exp(−τ_∗).

• 3.

  As long as the optical depth of the shell is not very high, the spectral flux distribution from dust emission can be approximated by a black body radiation field with temperature corresponding to the inner edge of the dust shell, T_ph. This results in a contribution to the flux-averaged extinction coefficient.

• 4.

  If the shell becomes very thick optically, the flux-averaged extinction coefficient should approach the Rosseland mean of the local radiation field, . The approach to the optically thick case can be modelled by the factor 1−f with the Eddington factor f, which is f ≈ 1 in an optically thin shell, and approaches f ≈ 1/3 for optically thick shells.

As a demonstration of the accuracy of the approximation, Fig. A.1 compares the result for κ_H based on a full calculation of radiative transfer, as outlined in Wetzel et al. (2013), using a powerlaw variation κ_λ ∝ λ^-1 as a simple model case. An inspection of the figure shows that the approximation for κ_H is fairly accurate.

	Fig. A.1 Approximation of the flux-averaged extinction coefficient κH in models for a dust shell with a powerlaw wavelength variation of the extinction coefficient for different mass-loss rates (in units M⊙ a-1). Solid line: Result of a complete model calculation of radiative transfer. Dashed line: approximation according to Eq. (19).
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© ESO, 2013