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Astron. Astrophys. 342, 785-798 (1999)

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3. Spectral line profiles

As 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 ([FORMULA]) of the structural elements. Let [FORMULA] be the possible values of the source function, and [FORMULA] 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 [FORMULA] for the normalized absorption coefficient (here x is the dimensionless frequency denoting the displacement from the centre of the line measured in the Doppler widths). Because of the absence of scattering, and hence effects of the frequency redistribution, the averaging process may be obviously performed for each frequency separately so that applied to our problem, Eq. (4) can be rewritten as follows

[EQUATION]

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 [FORMULA] [FORMULA] and [FORMULA] [FORMULA] (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, [FORMULA] and [FORMULA] (case (d)).

[FIGURE] Fig. 5. The normalized profiles of the LTE spectral line for the four formulations of the problem discussed in the text: (a) homogeneous atmosphere with [FORMULA] const, and optical thickness [FORMULA]; (b) homogeneous atmosphere with [FORMULA] const, and optical thickness [FORMULA]; (c) atmosphere with randomly varied values of B and [FORMULA]; (d) the profile obtained as a result of preliminarily averaging the random quantities. The profile for the case (d) fits satisfactory the real profile of the spectral line (c).

[FIGURE] Fig. 6. The same as in Fig. 5 for the values of parameters indicated. The real profiles (c1 - [FORMULA], c2 - [FORMULA]) differ fundamentally from those obtained by considering the deterministic formulation of the problem.

Figs. 5, 6 show the normalized line profiles, [FORMULA], calculated for the four formulations of the problem in case of [FORMULA]. It is seen that for relatively small values of [FORMULA] and [FORMULA] (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 [FORMULA] (case c1), and for [FORMULA] (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, N. Fig. 7 demonstrates such an example for the intensities [FORMULA] emerging from an atmosphere, in some conventional units. The roughness of the result corresponding to case (d) is striking.

[FIGURE] Fig. 7. The intensity of radiation emerging from an atmosphere for the same cases as in Fig. 5. It is seen that averaging the random characteristics of an atmosphere may lead to essentially erroneous results.

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© European Southern Observatory (ESO) 1999

Online publication: February 23, 1999
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