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

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2. The LTE atmosphere

Let us treat the number N of structural elements each of which is described by the power of the energy release, B, and optical thickness, [FORMULA]. The value of B is assumed to be constant within each individual element. As earlier in Paper II, we begin by considering the simplest situation, when the pair of quantities ([FORMULA]) takes randomly only two possible sets of values ([FORMULA] [FORMULA]) and ([FORMULA] [FORMULA]), each occurring with probability [FORMULA] and [FORMULA]. One may think of a more general problem when various elements take different values of [FORMULA] for the same value of B. Referring the interested reader to the paper by Jefferies and Lindsey (1988) for this problem, we note that all the results to be obtained can be easily extended to comprise this case as well.

Thus, suppose that we have a set of N radiating elements with randomly varying properties ([FORMULA] [FORMULA]). We are interested in the averaged characteristics of the emerging radiation such as the mean intensity [FORMULA] and the RelMSD [FORMULA]. As was emphasized in Paper II, the LTE atmosphere differs from the Non-LTE atmosphere essentially by the absence of reflected radiation from the structural components. This fact greatly simplifies the problem since the averaged characteristics of radiation emerging from a given medium are not affected as a result of the addition of new elements to the medium. This implies that the averaging process may be performed in parts, which allows to write directly

[EQUATION]

where [FORMULA] and [FORMULA], or the more general result

[EQUATION]

A more rigorous derivation of Eq. (1) might proceed along the following lines. Let a certain configuration of [FORMULA] elements occur with probability P. Adding a new element to the front part of such a medium we shall observe a radiation of intensity [FORMULA] or [FORMULA] with probability [FORMULA] and [FORMULA], respectively. Here we introduced the quantities [FORMULA] [FORMULA] characterizing the intensity of radiation emitted by an individual structural element of each type. Multiplying each value of intensity by the proper probability, and adding up the results we are led to Eq. (1). The same reasonings applied to [FORMULA] yields

[EQUATION]

where [FORMULA] and [FORMULA] [FORMULA]

An important comment to be made at this stage is that the physical considerations and ratiocinations underlying Eqs. (1) and (3), do not depend on the number n of the possible values taken by the pair of parameters ([FORMULA])[FORMULA] Therefore, the mentioned equations along with those to be obtained are valid for an arbitrary number of realizations of the components' physical properties. Moreover, Eqs. (1) and (3) remain true also for the continual analogue of the problem at hand, i.e., when B and [FORMULA] (hence J and [FORMULA]) are continuum-valued random quantities. Knowing the probability distributions of these quantities, one may easily find the proper averaged parameters (see below Eq. 12) being necessary for evaluation of the mean intensity and the RelMSD. For expository reasons, however, in what follows we continue studying the discrete problem, keeping in mind that the physical conclusions at which we arrive can be easily reformulated for the continuous distributions of B and [FORMULA].

Let us now establish several important equations for the mean intensity and the RelMSD that will be in use in further discussion. By applying successively the recurrence relation (1) we may write

[EQUATION]

[EQUATION]

where [FORMULA] and [FORMULA].

Eq. (5) allows us to find [FORMULA] in terms of [FORMULA] and write

[EQUATION]

The summation in the right-hand side of Eq. (6) can be easily performed to yield

[EQUATION]

where [FORMULA], and [FORMULA].

Thus, we finally have

[EQUATION]

which by virtue of Eqs. (4) leads to the requisite equation for the RelMSD:

[EQUATION]

where [FORMULA] [FORMULA]. Note also that Eqs. (3) and (8) allow us to write two equivalent recurrence relations for determining [FORMULA], which we present here in a more visualizable form

[EQUATION]

[EQUATION]

where [FORMULA]

We now have at our disposal all of the equations needed to discuss the statistical properties of radiation emerging from the LTE atmosphere. An important salient trait inherent in Eqs. (4) and (9) (see also Eqs. 10 and 11) is that the values of the mean intensity and the RelMSD for the multicomponent atmosphere with an arbitrary number of elements are determined by only a few parameters. These parameters for the general case of n realizations of ([FORMULA] are

[EQUATION]

[EQUATION]

[EQUATION]

It is seen that all the above parameters are the averaged characteristics of a single structural element. Thus we arrive at the conclusion that, in general, there must exist a great number of configurations, having a fixed number of components, that are characterized by the same values of the mean intensity and the RelMSD. Furthermore, the values of [FORMULA] and [FORMULA] for such media exhibit the same behaviour with varying N.

It is of particular interest to study the asymptotic behaviour of fluctuations when [FORMULA]. Eqs. (4) and (9) show that while [FORMULA], the RelMSD goes generally to the nonzero limit, viz.,

[EQUATION]

in contrast (as will be seen later) to the Non-LTE atmosphere. Let us now consider several special cases when one of the physical parameters is the same for all components of an atmosphere:

(i) Let [FORMULA] be [FORMULA], common for all structural elements. Then [FORMULA], [FORMULA] and we obtain from Eq. (9)

[EQUATION]

where [FORMULA] As might be expected, the fluctuations stem from the differences in values of B, and fall off with [FORMULA], tending to the nonzero limit [FORMULA]. The larger the optical thickness of the components, the greater [FORMULA] and the faster the passing of [FORMULA] to its asymptotic plateau, [FORMULA]. For [FORMULA] sufficiently small such that [FORMULA], Eq. (14) simplifies to

[EQUATION]

(ii) Now we suppose that all components of a medium are characterized by the same value of [FORMULA] i.e., the atmosphere is homogeneous, so that fluctuations in the observed radiation are due to variations in the total optical thickness. It is obvious that only in this case the arrangement of the elements for a given proportion of various species is not essential. Taking into account that now [FORMULA] and [FORMULA], we obtain

[EQUATION]

It follows from Eq. (16) that for a homogeneous atmosphere the RelMSD tends exponentially to [FORMULA] when [FORMULA]. One may show that this is the only case in which fluctuations vanish with increasing N.

(iii) Also of interest is the situation in which all structural elements are emitting equal amounts of energy [FORMULA] so that the fluctuations are due only to the difference in absorption of the emerging radiation. Now [FORMULA] and [FORMULA] which represent the minimal values of these quantities for a given set of [FORMULA], as compared to other cases. Thus the fluctuations in the observed intensity are also the lowest. In the special case in which the equality of [FORMULA] follows from the equality of [FORMULA] (and hence [FORMULA]), we are led back to the homogeneous atmosphere. In this particular case, when [FORMULA] are also equal, [FORMULA] is obviously zero, otherwise [FORMULA] is zero only for [FORMULA], and increases monotonically with an increase of N to its asymptotic value, [FORMULA], resulting from Eq. (13).

In order to make an impression on the run of [FORMULA] with N for any values of [FORMULA] and [FORMULA], we consider the results of calculations concerning the simplest problem of [FORMULA]. As was stated above, the conclusions at which we arrive may be readily generalized to cover more complicated problems. Particular attention will be paid to the behaviour of [FORMULA] with respect to N for [FORMULA] which may be regarded as the discrete and extremely schematic model of the continuous-valued problem with symmetrical probability distributions characterizing the physical properties of an atmosphere. The values of [FORMULA] for [FORMULA] fall typically between those evaluated for large and small probabilities (see Fig. 1) (here we exclude the non-interesting situations in which p is close to zero or unity, which collapse to the homogeneous problem). The only exception shown in Figs. 2-4 concerns the case in which [FORMULA] approaches [FORMULA] and this will be discussed below.

[FIGURE] Fig. 1. The function [FORMULA] of N for various [FORMULA] and indicated values of other parameters in the case of [FORMULA], [FORMULA]. The less probable the appearance of the brightest component, the larger values of [FORMULA]. The function [FORMULA] for [FORMULA] occupies some intermediate position.

Depending on the values of the parameters given by Eq. (12), the function [FORMULA] can exhibit a broad variety of different behaviours. It may decrease or increase monotonically with [FORMULA] or exhibit an initial decrease followed by an increase for greater values of [FORMULA] To facilitate further discussion, we note that the symmetry of the problem with respect to simultaneous exchange [FORMULA] and [FORMULA], allows us to limit the discussion to the case [FORMULA] It is expedient to distinguish among others the situation in which the structural elements radiate an equal amount of energy (i.e., when [FORMULA] [FORMULA], or [FORMULA]). This situation is unreachable if the condition [FORMULA] is satisfied together with inequality [FORMULA] In this case [FORMULA] is a monotonically decreasing function of N and goes to a nonzero limit as [FORMULA].

Fig. 1 shows that for ([FORMULA] [FORMULA]) fixed the values of [FORMULA] are smaller when the bright component is more probable. With increasing [FORMULA] this function becomes steeper, while an increase in the values of both of [FORMULA] not violating the inequality [FORMULA] leads to a smaller limit of [FORMULA] as N [FORMULA] [FORMULA]. As might be expected, with an increase of [FORMULA] from 0 to 1, (i.e., with decreasing contrast between [FORMULA]), the RelMSD becomes smaller (see first three graphs of Fig. 3a). The behaviour of [FORMULA] with respect to N is altered essentially when one of inequalities, [FORMULA] or [FORMULA], changes its sign (see Fig. 2). This corresponds to the case when the brighter component is less opaque than the fainter. We also observe that now the values of [FORMULA] for [FORMULA] are the largest. We see from Fig. 3a that for [FORMULA], close to [FORMULA], [FORMULA] becomes smaller for any value of N, and alters its behaviour by turning into a monotonically increasing function of N. When [FORMULA], [FORMULA] changes from an increasing function of N for [FORMULA] to a monotonically decreasing function for [FORMULA] (Fig. 3b).

[FIGURE] Fig. 2. The function [FORMULA] of N for various [FORMULA] for an atmosphere with the brighter component less opaque than the fainter ([FORMULA] and [FORMULA]). Now the values of [FORMULA] for [FORMULA] are largest.

[FIGURE] Fig. 3a and b. The function [FORMULA] of N for various values of the ratio [FORMULA]: a  [FORMULA], b  [FORMULA]. Depending on whether [FORMULA] or [FORMULA] [FORMULA] the behaviour of the function [FORMULA] is different.

Figs. 4a and b illustrate the relationship between [FORMULA] and [FORMULA] for [FORMULA] and [FORMULA], respectively. The minimum attained by [FORMULA] at [FORMULA] is clearly seen. It is noteworthy that depending on whether the inequality [FORMULA] or [FORMULA] holds, the behaviour of [FORMULA] as [FORMULA] is different for large values of N. The quantitative analysis of numerical results described above as well as their application to prominences will be given in Sect. 5.

[FIGURE] Fig. 4a and b. The function [FORMULA] of [FORMULA] for various N: a  [FORMULA], b  [FORMULA]. The minimum attained by [FORMULA] at [FORMULA] is discernible. The different behaviour of [FORMULA] for large [FORMULA] in the case a as compared to that in b is noteworthy.

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Online publication: February 23, 1999
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