*Astron. Astrophys. 359, 788-798 (2000)*
## 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.
(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 . 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 over
the entire spectrum, which corresponds to an extinction coefficient
which is independent of
. The resulting flux is
i.e. it does not depend on the velocity.
Whenever there is a break in , the
integral in (5) has to be split. For example, if we consider a change
in slope at from
to
as well as a jump , i.e.
the resulting flux for any is
Evidently, the use of the piecewise linear approximation speeds up
actual calculations by a very large factor (cf. Baschek et al.
1997).
© European Southern Observatory (ESO) 2000
Online publication: July 7, 2000
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