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

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4. The NLTE atmosphere

As it was pointed out in the outset of the paper, the Non-LTE atmosphere differs significantly from that in LTE due to multiple scatterings which establish coupling between various volumes of the medium. Now the averaging can no longer be performed by parts as for LTE, since the mean intensity of radiation emerging from any part of an atmosphere is altered when adding a new layer (element) to it. As in Paper II, we confine ourselves by considering pure scattering (destruction coefficient [FORMULA]), for which the theory is simplified at great extent. We assume again that the time scale of variations in the structure of the atmosphere is much greater than the average time of the photon's travel in it. In determining the statistical characteristics of radiation emerging from an inhomogeneous Non-LTE atmosphere, in Paper II we used Ambartsumian's method of "addition of layers". We obtained explicit expressions for the RelMSD relevant to several particular small numbers, N, of structural elements. With increasing N this method becomes, however, too cumbersome to be applied, which stems from the high non-linearity of equations derived for outgoing intensity (see Eqs. 3 and 4 of Paper II). This disadvantage prevents the complete comparison with the LTE case.

To give more insight into the problem, we shall adopt in this section a new approach based on the concept of the photon escape probability. This enables one to obtain a closed-form analytical expression for the RelMSD for arbitrary N. We introduce the quantity [FORMULA] defined as the probability that a photon, randomly scattered at the optical depth t of a medium of optical thickness [FORMULA], will eventually escape from it through the boundary [FORMULA]. As is known (see, e.g., Sobolev 1963, Chapt. 6), in the case of conservative scattering,

[EQUATION]

Particularly,

[EQUATION]

by virtue of which Eq. (18) may be recast as

[EQUATION]

If the atmosphere contains energy sources radiating [FORMULA] in all directions, the intensities of outgoing radiation integrated over the line are given by

[EQUATION]

Consider now an inhomogeneous scattering atmosphere consisting of N structural elements. As above, the elements will be regarded as layers, the physical properties of which are described by two parameters: the optical thickness [FORMULA], and the power, [FORMULA], released by embedded energy sources. The latter is assumed constant within each individual element. The physical conditions in the layers undergo random variations in such a way that the pair of parameters [FORMULA] takes only one of two possible values, [FORMULA] [FORMULA], each occurring with probability [FORMULA]. We shall treat the idealized one-dimensional problem of the transfer of radiation through such atmosphere and focus our attention on the statistical characteristics of the emerging radiation, [FORMULA] and [FORMULA].

Let the atmosphere be composed of k layers of the first kind (i.e., described by parameters [FORMULA]), and [FORMULA] layers of the second kind (with parameters [FORMULA]). There obviously exist [FORMULA] different configurations of a given composition, each occurring with probability [FORMULA]. For pure scattering all the radiation energy generated in the medium escapes it, so that the total amount of energy radiated by such an atmosphere is [FORMULA], where [FORMULA] have the same physical meaning as the quantities [FORMULA], for the LTE problem discussed above.

For odd N, some configurations are symmetrical about the middle of atmosphere, i.e., the distribution of layers of different kinds, being referenced from each boundary, is the same. The intensities outgoing from each boundary of such atmosphere are equal to [FORMULA]. For non-symmetrical configurations the intensity [FORMULA] emerging from the boundary [FORMULA] of atmosphere may be represented in the form

[EQUATION]

where the superscript m is introduced to indicate the dependence on the arrangement of layers of the different types within the atmosphere. The explicit expression for [FORMULA] found in Appendix A (see Eq. A8) is

[EQUATION]

where [FORMULA]; [FORMULA] is the optical thickness of the atmosphere, and [FORMULA] are certain integers in the range [FORMULA] with clear combinatorial meanings explained in Appendix A. When [FORMULA], i.e., [FORMULA], we return to the just-considered case of the symmetrical distribution. The other values of [FORMULA] are distributed in pairs about a in such a way that the quantity [FORMULA] takes values of the form [FORMULA] (where [FORMULA] as we shall see later, are integers from the interval [FORMULA]) with each of signs corresponding to the intensity emerging from one of two boundaries of the atmosphere with a non-symmetrical distribution of layers of different types. It is evident that for every non-symmetrical configuration there exists a certain companion configuration of the same composition but with an inverted distribution of layers for which [FORMULA] has the opposite sign.

In calculating the mean intensity [FORMULA], the terms [FORMULA] disappear to yield

[EQUATION]

This result has been already used in Paper II, and differs fundamentally from its counterpart equation for LTE (cf. Eq. 4).

In contrast to the mean intensity, it is somewhat more difficult and lengthy to derive a closed-form expression for the RelMSD, since the integers [FORMULA] do not disappear. Even so, the needed summations may be performed, as is shown in Appendix B, to derive

[EQUATION]

where

[EQUATION]

Some special forms of Eq. (25) for small values of N were presented in Paper II. The direct calculation shows particularly that [FORMULA], and

[EQUATION]

We see that the RelMSD, as presented in Eq. (25), is a sum, in which only the second item depends on the relative proportion of layers of different types and their arrangement in the atmosphere. It is also seen that, depending on the individual properties of components, either of the two terms in brackets may become dominant. Examining Eq. (26), we observe that with increasing N the quantity [FORMULA] remains bounded from above by the limiting value [FORMULA] (where [FORMULA]). So we obtain an important assertion: the RelMSD of the intensity of radiation emerging from the multicomponent randomly inhomogeneous NLTE atmosphere decreases with an increase of the number Nof components, tending to 0as [FORMULA]. It should be noted that this assertion holds under general assumptions concerning the physical properties of structural elements, which was not the case in LTE.

We see that [FORMULA] depends merely on the ratio of [FORMULA] and [FORMULA]. In the special cases, (i)-(iii) discussed in Sect. 2 for LTE, the expression for [FORMULA] simplifies to a great extent.

(i) Let [FORMULA], then Eq. (25) takes the form

[EQUATION]

Now [FORMULA] is a function of [FORMULA], [FORMULA], and [FORMULA] For optical thickness [FORMULA] sufficiently small, the effect of scatterings is negligible, and [FORMULA]. In this special case there is no difference between LTE and NLTE (cf., Eq. 15) since in both cases all the quanta radiated in the atmosphere escape it. The dependence of [FORMULA] on [FORMULA] is not important for large N, considering that the term in brackets only varies from unity to [FORMULA] as [FORMULA] changes from 0 to [FORMULA].

(ii) If [FORMULA], Eq. (25) simplifies to result [FORMULA], where [FORMULA].

(iii) Suppose that all the structural elements radiate equal amounts of energy, i.e., [FORMULA]. Then we have [FORMULA], and the range of variation in the RelMSD is as much as [FORMULA] times greater than that in the case (i).

(iv) It is important for the further discussion concerning the H Ly-[FORMULA] line to consider the special case in which the components of a medium are supposed to be optically thick ([FORMULA]). Letting [FORMULA], for simplicity, Eqs. (25) and (26) take a much more simple form. Now [FORMULA], and for relatively large N, from Eq. (26), we find that [FORMULA] Excluding from the treatment the less interesting case of the quasi-homogeneous atmosphere when [FORMULA], and [FORMULA], in place of Eq. (25), one may write

[EQUATION]

where [FORMULA] is a certain constant from the interval [FORMULA]. We shall use this result in Sect. 5 below.

The numerical results based on Eq. (25) for [FORMULA] are given in Figs. 8-10, which are the Non-LTE analogues of Figs. 2, 3b and 4b, respectively. For convenience of comparison, the values of parameters in both cases are chosen the same. The conclusions we draw in comparing the theoretical values of [FORMULA] for the LTE and Non-LTE multicomponent atmospheres may be summarized as follows: in general, the RelMSD for a Non-LTE atmosphere can be less as well as greater than that for one in LTE. The greater [FORMULA] for Non-LTE are observed only for relatively small N when [FORMULA] [FORMULA] (or [FORMULA] [FORMULA], in view of the symmetry of the problem at hand). In this case the difference between [FORMULA] for LTE and Non-LTE is only quantitative, so we limit ourselves by plotting in Figs. 8, 9 the graphs that illustrate the case [FORMULA] [FORMULA], with the only exception being the last curve in Fig. 9. Fig. 8 illustrates the situation when the RelMSD for Non-LTE is much smaller as compared to that for LTE, even for small N. In addition, as N increases, [FORMULA] for the conservatively scattering atmosphere, which is not the case in LTE. This fact is important for any values of parameters involved, and leads to smaller [FORMULA] for Non-LTE as long as the number N of components is sufficiently large (cf. Figs. 10 and 4b). Note also that normally [FORMULA] is a monotonically decreasing function of N (if [FORMULA] is not too close to 1 (see Fig. 9)), while it is not the case in LTE for rather wide range of values of [FORMULA].

[FIGURE] Fig. 8. The function [FORMULA] of N for various [FORMULA] and indicated values of other parameters (analogue of Fig. 2 for the Non-LTE atmosphere).

[FIGURE] Fig. 9. The relationsip between [FORMULA] and N for various values of the ratio [FORMULA]. In contrast to the LTE atmosphere, the RelMSD is a monotonically decreasing function of N as long as [FORMULA] [FORMULA] is not too close to 1 (case [FORMULA]).

[FIGURE] Fig. 10. The same as in Fig. 4b for the Non-LTE atmosphere.

The final comment to be made in connection of the model problem considered in this section is that we did not specify the origin of the initial energy sources. In fact, these sources may be partially due to the external radiation incident on the atmosphere from outside. This is the case when considering the effect of the photospheric radiation incident onto prominence, which is of particular importance for the H Ly-[FORMULA] line. In the case of pure scattering adopted in our model problem, the contribution of the external radiation will be the same for each individual structural element, i.e., [FORMULA] will increase by the same value, so that [FORMULA]. As it follows from Eq. (25), this, in turn, leads to a decrease of the RelMSD, that is the external radiation tends to smooth the actual fluctuations in brightness. The quantitative estimation of the contribution of the incident intensity is afield of the theory we developed and must be derived from other reasonings as an input parameter.

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© European Southern Observatory (ESO) 1999

Online publication: February 23, 1999
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