## 4. The NLTE atmosphereAs 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 ), 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, 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 by virtue of which Eq. (18) may be recast as If the atmosphere contains energy sources radiating in all directions, the intensities of outgoing radiation integrated over the line are given by Consider now an inhomogeneous scattering atmosphere consisting of
Let the atmosphere be composed of For odd where the superscript where ;
is the optical thickness of the
atmosphere, and are certain
integers in the range with clear
combinatorial meanings explained in Appendix A. When
, i.e.,
, we return to the just-considered
case of the symmetrical distribution. The other values of
are distributed in pairs about
In calculating the mean intensity , the terms disappear to yield 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 do not disappear. Even so, the needed summations may be performed, as is shown in Appendix B, to derive Some special forms of Eq. (25) for small values of 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 We see that depends merely on the ratio of and . In the special cases, (i)-(iii) discussed in Sect. 2 for LTE, the expression for simplifies to a great extent. (i) Let , then Eq. (25) takes the form Now is a function of
,
, and
For optical thickness
sufficiently small, the effect of
scatterings is negligible, and . 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 on
is not important for large (ii) If , Eq. (25) simplifies to result , where . (iii) Suppose that all the structural elements radiate equal amounts of energy, i.e., . Then we have , and the range of variation in the RelMSD is as much as times greater than that in the case (i). (iv) It is important for the further discussion concerning the H
Ly- line to consider the special
case in which the components of a medium are supposed to be optically
thick (). Letting
, for simplicity, Eqs. (25) and (26)
take a much more simple form. Now ,
and for relatively large where is a certain constant from the interval . We shall use this result in Sect. 5 below. The numerical results based on Eq. (25) for
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 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
for Non-LTE are observed only for
relatively small
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- 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., will increase by the same value, so that . 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. © European Southern Observatory (ESO) 1999 Online publication: February 23, 1999 |