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

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6. Numerical results for a single Lorentzian line

In order to obtain some insight into the behavior of radiative fluxes in a medium of high optical depth according to the above equations we consider the simple case (which, however, already contains the essential features) of a continuum [FORMULA] that does not dependent on [FORMULA] and a single spectral line of Lorentzian shape at [FORMULA] with damping constant [FORMULA] (in the [FORMULA]-scale, cf. Wehrse et al. 1998), i.e.


so that


In order check when the diffusion limit is reached we plot in Fig. 2 the value of the last integral in Eq. (38) as a function of s for [FORMULA] and some values of the line strength A; the other integral behaves in the same way. It is seen that there is hardly any longer a variation for [FORMULA], i.e. the diffusion limit is reached for [FORMULA] at latest. By additional line absorption it may even be shifted to much smaller values.

The dependence of the monochromatic flux [FORMULA] in the line center on the strength of the continuum [FORMULA] and of the line A (Fig. 3) for given constant w is basically the same as in the static case: an increase in the extinction leads to a decrease in the flux independent of the source of the extinction. This implies that lines have an influence on the flux only when they are are of sufficient strength. As is seen in Fig. 4, a reduction in the velocity gradient or in the damping width leads to a decrease in the monochromatic flux at the line center. However, the main variations occur only for small [FORMULA] and w values since for large w the line is essentially smeared out and the information on the intrinsic [FORMULA] is lost. This behavior holds only for the monochromatic flux at or close to the line center; in contrast, for the flux integrated over the line the situation is different since in moving configurations the influence of the line extends (cf. Fig. 5) much further in wavelength than in the static case. Fig. 5 also demonstrates that for a given distance ([FORMULA]) from the line center the [FORMULA] dependence of the flux may be quite complicated, since changes in the intrinsic line profile may or may not be compensated by Doppler shifts. Furthermore, it is seen that the line influence is - in accordance with Fig. 3 - under most conditions strongest at the line center.

[FIGURE] Fig. 3. Dependence of the monochromatic diffusion flux [FORMULA] on A and [FORMULA] for a single Lorentzian line on a continuum for [FORMULA], [FORMULA], and [FORMULA]. The behavior also reflects the effect of the motions on the wavelength-integrated flux and hence on the generalized Rosseland mean opacity.

[FIGURE] Fig. 4. [FORMULA] as a function of w for [FORMULA], [FORMULA], and [FORMULA] (top to bottom ) in the center ([FORMULA]) of a single Lorentzian line on a continuum with [FORMULA] and [FORMULA] (cf. also Fig. 3).

[FIGURE] Fig. 5. [FORMULA] as a function of [FORMULA] and [FORMULA] for [FORMULA], [FORMULA], and [FORMULA] for a single Lorentzian line on a continuum (cf. also Fig. 3).

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© European Southern Observatory (ESO) 2000

Online publication: July 7, 2000