## 3. Spectral line profilesAs a consequence of Eq. (4), we consider a particular problem that
concerns the profiles of spectral lines formed in an LTE atmosphere
with randomly distributed inhomogeneities. For simplicity of
exposition we consider the case of two species
() of the structural elements. Let
be the possible values of the source
function, and be the optical
thickness in the centre of the spectral line. We suppose that each
element radiates within a spectral line with the Doppler profile
for the normalized absorption
coefficient (here Our immediate objective is to compare the profiles of the spectral lines formed in an atmosphere with randomly varied physical properties (referred to as the case (c) in Figs. 5, 6) with those formed in atmospheres with given constant values of and (cases (a), (b), respectively). On the other hand, it is well-known that difficulties encountered in solving stochastic astrophysical problems often lead one to replace them by the proper deterministic problem with preliminarily averaged, in some sense, random physical parameters describing the medium. It is of interest from this point of view to compare the solution of the problem posed in such a way with the exact solution of the stochastic problem. With this in mind we give in what follows also the profiles of the line formed in an atmosphere with averaged physical characteristics, and (case (d)).
Figs. 5, 6 show the normalized line profiles,
, calculated for the four
formulations of the problem in case of
. It is seen that for relatively
small values of and
(Fig. 5), the profile obtained by
preliminarily averaging the random characteristics of an atmosphere
fits satisfactory the exact solution of the stochastic problem. Both
of these profiles lay between those corresponding to deterministic
problems. However, this is not the case when one of the possible
values of the optical thickness is large. This is illustrated by
Fig. 6, where two profiles relevant to the randomized problem are
given: for (case c1), and for
(case c2). We see that these
profiles differ fundamentally in their shapes from those obtained by
solving the deterministic problems. It is interesting to note that
profiles found by averaging random quantities may appear to be
erroneous quantitatively as well as qualitatively. Specifically, large
discrepancies with respect to the real situation may arise for
relatively small numbers of structures,
© European Southern Observatory (ESO) 1999 Online publication: February 23, 1999 |