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Astron. Astrophys. 359, 788-798 (2000)

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3. Limit of large w

We now consider the limiting case of large w when the Doppler shift occurring over a mean free path length of a photon in the continuum is much larger than the intrinsic line width, i.e. [FORMULA] (Eq. (3)).

According to Baschek et al. (1997), the spectral thickness is essentially a piecewise linear function, and the importance of deviations from the piecewise linear functions decreases with increasing [FORMULA]. This presentation is particularly useful if very many spectral lines have to be taken into account for the extinction coefficient, but is also valid for isolated narrow lines.

The simplest case is [FORMULA] over the entire spectrum, which corresponds to an extinction coefficient [FORMULA] which is independent of [FORMULA]. The resulting flux is


i.e. it does not depend on the velocity.

Whenever there is a break in [FORMULA], the integral in (5) has to be split. For example, if we consider a change in slope at [FORMULA] from [FORMULA] to [FORMULA] as well as a jump [FORMULA], i.e.


the resulting flux for any [FORMULA] is


Evidently, the use of the piecewise linear approximation speeds up actual calculations by a very large factor (cf. Baschek et al. 1997).

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© European Southern Observatory (ESO) 2000

Online publication: July 7, 2000