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Astron. Astrophys. 359, 780-787 (2000)

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7. Concluding remarks and outlook

Another equivalent way to interpret Eq. (41) is obtained by introducing the mean extinction coefficient at [FORMULA] over the interval [FORMULA],

[EQUATION]

instead of the spectral thickness. Then

[EQUATION]

so that the radiative quantities such as the flux [FORMULA] or the radiative force for moving media can - in the deterministic case - completely be described either by the spectral thickness or by the set of the mean extinction coefficients [FORMULA] for the relevant ranges in [FORMULA] and ws.

In order to elucidate the connection to the static diffusion, we consider the very special case that the mean extinction coefficient does not depend on the interval ws, i.e. [FORMULA]. Then, using (54), the integration over depth in Eq. (41) can be performed, and with Eq. (49) leads to

[EQUATION]

This result resembles the usual static Rosseland mean except that now for the moving medium the mean extinction coefficient [FORMULA] over the interval ws replaces the "ordinary" monochromatic extinction coefficient [FORMULA].

In conclusion, we have derived this paper general expressions for the radiative flux and acceleration in arbitrarily shaped, optically very thick, and differentially moving media far from the surfaces. These expressions are basically rather simple but the integrals involved can be evaluated analytically only in very special cases. In addition, the dependencies on the input parameters are not immediately evident. We therefore have presented here only numerical results of the monochromatic flux for a single Lorentzian spectral line on a continuum which show that there is in fact a quite intricate interplay of the parameters. We have restricted the discussion of our examples on monochromatic quantities in order investigate the relative importance of the line core and the near and far wings.

In astronomical applications strictly monochromatic radiative quantities are only rarely of interest, they more or less serve as the basis for calculating the more important wavelength-integrated quantities.

In Paper II we show that in the limits of small and large velocity gradients much more insight can be gained. In particular, it is demonstrated that for [FORMULA] Rosseland's result for the static case is regained. In addition, for isolated narrow Lorentzian lines on a flat continuum the wavelength integrals can be obtained analytically. When there are many overlapping lines, convenient expressions can be obtained for large velocity gradients, whereas for small w the integration over [FORMULA] has to be carried out numerically. Since many wavelength points have to be considered and the integrand requires numerical differentiations of the extinction coefficient such calculations are quite demanding in programming and CPU time. A more satisfactory approach, however, is to describe the extinction coefficient by a Poisson point process (cf. Wehrse et al. 1998) since it allows to derive the expectation values for the flux and the acceleration in terms of the mean line separation and of the shapes and strengths of the lines as well as of the continuum. The formalism will be presented in a subsequent paper.

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© European Southern Observatory (ESO) 2000

Online publication: July 7, 2000
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