SpringerLink
Forum Springer Astron. Astrophys.
Forum Whats New Search Orders


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

Previous Section Next Section Title Page Table of Contents

Appendix A: the derivation of Eq. (23)

For fixed N and k, and an arbitrary arrangement of layers (marked by the superscript m), the outgoing intensity may be expressed in terms of [FORMULA] as follows

[EQUATION]

wherein [FORMULA] is the optical thickness of the atmosphere. The limits of integrations represent the optical depths that correspond to boundary planes of a given layer. These are random quantities and depend on the realized particular ordering of components in the atmosphere. It should be seen that [FORMULA], and [FORMULA].

By making use of Eq. (20), the integrals in Eq. (A1) are evaluated immediately to give

[EQUATION]

[EQUATION]

where [FORMULA] [FORMULA] are the optical depths which correspond to the middles of the jth layer (counting from the boundary [FORMULA] of the atmosphere) of the first and second type, respectively. Now Eq. (A1) yields

[EQUATION]

where for brevity one of the arguments in [FORMULA] is omitted. Taking into account the second of Eqs. (19) one may write

[EQUATION]

The coordinates [FORMULA] and [FORMULA] can be represented in the form

[EQUATION]

[EQUATION]

wherein [FORMULA] and [FORMULA] are the numbers of the components of the opposite type preceding a given layer. Thus we have

[EQUATION]

[EQUATION]

with the quantities

[EQUATION]

representing the total numbers of layers of the opposite type preceding all the layers of a given type. These numbers admit an alternative interpretation that is convenient for further discussion. This rests on the concept of `transposition' widely used in the combinatorial analysis. By transposition we mean here any exchange, or swap, of two adjacent layers of different kinds. Let us agree to call `direct' the arrangement consisting of two series such as all the layers of the first structure type precede those of the second type. The opposite arrangement, obtained by a simple inversion of two series, will be referred to as the `inverted' order. It is easily seen that, for a certain random realization, the quantity [FORMULA] is nothing more than the total number of transpositions needed to establish the direct ordering of layers, while [FORMULA] is the total number of transpositions performed in obtaining the inverted order. On the other hand, the number of transpositions transforming the direct order into the inverted one is [FORMULA] so that we may write

[EQUATION]

Now utilizing Eqs. (A6), (A8) in Eq. (A4), we find

[EQUATION]

Thus, we obtained the requisite explicit expression for [FORMULA] (cf., Eq. 22) that allows us to elucidate the characteristic features of the emerging intensity. The layers of the same sort are obviously indistinguishable, so that for fixed N and k there exist [FORMULA] different configurations. The values of [FORMULA] (and then of [FORMULA] are determined completely by the discrete quantity [FORMULA] (or [FORMULA]), and obey the Fermi-Dirac statistics (see, e.g., Feller 1957). It follows from Eq. (A8), that the values taken by [FORMULA] must lie between 0 and [FORMULA]. The configurations with [FORMULA] (i.e., [FORMULA]) are symmetrical. The other values of [FORMULA] are grouped in pairs in such a way that the quantity [FORMULA] is of the form [FORMULA]where [FORMULA] are integers from the interval [FORMULA]. These correspond to the non-symmetrical configurations which are grouped in pairs (see Sect. 4).

Appendix B: the RelMSD N for the NLTE athmosphere

To proceed to the derivation of an explicit expression for [FORMULA], we employ Eqs. (24) and (A1) to write

[EQUATION]

By virtue of Eq. (A8) the internal sum in the right-hand side of Eq. (B1) takes the form

[EQUATION]

where

[EQUATION]

The bracketed term in Eq. (B3) may be rewritten as follows

[EQUATION]

wherein the coefficients [FORMULA] have a simple probabilistic meaning: [FORMULA] is the probability that the particular distribution of layers may be converted into the direct distribution as a result of exactly j transpositions. The rigorous evaluation of the sum Eq. (B4) is of interest from the point of view of combinatorial analysis and may become the subject of a separate treatment. Nevertheless, the direct calculation of this sum for successively large values of N allows us to establish that

[EQUATION]

Thus, in place of Eq. (B3) we find that

[EQUATION]

Substituting Eq. (B2) into Eq. (B1), we shall use the following easily checked identity

[EQUATION]

to obtain

[EQUATION]

where

[EQUATION]

Previous Section Next Section Title Page Table of Contents

© European Southern Observatory (ESO) 1999

Online publication: February 23, 1999
helpdesk.link@springer.de