In astronomy we are faced frequently with problems in which the physical and geometrical properties of a radiating medium undergo random variations. In general, two possible approaches to such problems can be represented. For simplicity's sake one may deal with averaged characteristics of the radiating atmosphere and address a deterministic problem of the radiative transfer to give an approximate, in a certain sense, description of the real physical situation. The alternative approach is to tackle a much more complicated stochastic problem and use the statistical features of the observed radiation as additional information for the determination of the structural pattern of the atmosphere and its variations. Thus, we are led to a kind of inverse problem, whose solution may not generally be unique.
In this latter approach, we need in a suitable theory of radiative transfer through an atmosphere with randomly distributed inhomogeneities. In this connection, it is of particular interest to consider a multicomponent atmosphere, i.e., an atmosphere composed of a single type of structure, randomly distributed in space with or without an intervening ambient medium. The physical parameters characterizing each structural element are also assumed to undergo random variations, taking one or another value with some probability. The statistical properties of the radiation emerging from a multicomponent LTE atmosphere were studied by Jefferies and Lindsey (1988) in connection with the interpretation of total solar eclipse observations in the far-infrared. A model problem considered by these authors assumes two possibilities for the values taken by the random physical parameters, characteristics of the structural elements. Nevertheless, as we shall see below, the results remain to be valid for an arbitrary number of possibilities. The procedure developed in the mentioned work supposes that variations in the physical properties of two adjacent elements are non-correlated. More recently Lindsey and Jefferies (1990) extended their theory to handle transfer for structure scale lengths smaller than the scale size of the homogeneity, i.e., for the microscopic domain of inhomogeneities, when random variations of different elements are essentially correlated.
In this paper we shall limit ourselves by working in the macroscopic domain of inhomogeneities, so that no correlations exist between variations of different structural elements. Our treatment of the stochastic transfer problems is motivated by the study of EUV spectra of the solar quiescent prominences, whose fine structure may now be considered as wellestablished. Digital raster images of prominences obtained with the Harvard College Observatory (HCO) aboard ATM-Skylab represent a rich observational resource for investigating the statistical properties of the line formation in multicomponent media. In general, the spatial brightness variations of prominence are due to several factors (see Pojoga et al. 1998; hereafter Paper I). These include the random nature of the filling factor, the statistical variations in the line-of-sight number of structural elements, and instrumental errors. Perhaps most significant factor is physical inhomogeneities characterizing the structures, which is the primary subject of this paper. It is natural to expect that the study of prominence radiation in optically thin lines, such as C ii 1336 Å, C iii 977 Å and O vi 1032 Å, as well as in the extremely opaque lines of the Lyman series provides necessary information on the physical and geometrical characteristics of the fine structure of prominences. The procedure explored by Jefferies and Lindsey for LTE-atmospheres is obviously applicable only to optically thin lines with high temperatures of formation, while the formation of the Lyman series lines is controlled by the multiple scattering effects and needs the development of a suitable Non-LTE theory. This theory is essentially involved, as far as the physical conditions at a given point of a medium are now determined not only by the local values of thermodynamic parameters but also by the radiation field throughout the atmosphere. The Non-LTE atmosphere for conservative scattering was discussed by Nikoghossian et al. (1997; hereafter Paper II) by using Ambartsumian's (1960) procedure of addition of layers. The present paper is a further extension of results previously obtained for both the LTE and Non-LTE atmospheres. Particularly in the latter case, a new approach based on the concept of escape probability is explored. This enables one to advance as against Paper II and derive a closed-form analytical expression for the relative mean square deviation (RelMSD) for any number of structural elements.
The outline of this paper is as follows: we begin, in the opening Sect. 2, by reconsidering the LTE atmosphere. A somewhat more detailed derivation of basic equations and their physical consequences than in Paper II, are given. It is shown that the equations for the mean intensity and RelMSD remain valid for sufficiently general assumptions on the random character of the inhomogeneities. The results of numerical calculations are discussed. In Sect. 3 we construct the profiles of spectral lines formed in the LTE atmosphere with randomly varying characteristics. It is shown that the discrepancy between real line profiles and those obtained for an atmosphere with previously averaged values of physical parameters, may become substantial. The Non-LTE atmosphere with conservative scattering is considered in Sect. 4. The analytical expression for the RelMSD is derived for an arbitrary number of structural elements. The results obtained and their applications to prominences are discussed in Sect. 5.
© European Southern Observatory (ESO) 1999
Online publication: February 23, 1999