## 2. The LTE atmosphereLet us treat the number Thus, suppose that we have a set of where and , or the more general result A more rigorous derivation of Eq. (1) might proceed along the
following lines. Let a certain configuration of
elements occur with probability
where and 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 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 where and . Eq. (5) allows us to find in terms of and write The summation in the right-hand side of Eq. (6) can be easily performed to yield where , and . which by virtue of Eqs. (4) leads to the requisite equation for the RelMSD: where . Note also that Eqs. (3) and (8) allow us to write two equivalent recurrence relations for determining , which we present here in a more visualizable form where 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 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 and
for such media exhibit the same
behaviour with varying It is of particular interest to study the asymptotic behaviour of fluctuations when . Eqs. (4) and (9) show that while , the RelMSD goes generally to the nonzero limit, viz., 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 be , common for all structural elements. Then , and we obtain from Eq. (9) where As might be expected, the
fluctuations stem from the differences in values of (ii) Now we suppose that all components of a medium are characterized by the same value of 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 and , we obtain It follows from Eq. (16) that for a homogeneous atmosphere the
RelMSD tends exponentially to when
. One may show that this is the only
case in which fluctuations vanish with increasing (iii) Also of interest is the situation in which all structural
elements are emitting equal amounts of energy
so that the fluctuations are due
only to the difference in absorption of the emerging radiation. Now
and
which represent the minimal values
of these quantities for a given set of
, 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
follows from the equality of
(and hence
), we are led back to the homogeneous
atmosphere. In this particular case, when
are also equal,
is obviously zero, otherwise
is zero only for
, and increases monotonically with an
increase of In order to make an impression on the run of
with
Depending on the values of the parameters given by Eq. (12), the
function can exhibit a broad variety
of different behaviours. It may decrease or increase monotonically
with or exhibit an initial decrease
followed by an increase for greater values of
To facilitate further discussion, we
note that the symmetry of the problem with respect to simultaneous
exchange and
, allows us to limit the discussion
to the case It is expedient to
distinguish among others the situation in which the structural
elements radiate an equal amount of energy (i.e., when
,
or ). This situation is unreachable
if the condition is satisfied
together with inequality In this
case is a monotonically decreasing
function of Fig. 1 shows that for (
) fixed the values of
are smaller when the bright
component is more probable. With increasing
this function becomes steeper, while
an increase in the values of both of
not violating the inequality leads
to a smaller limit of as
Figs. 4a and b illustrate the relationship between
and
for
and
, respectively. The minimum attained
by at
is clearly seen. It is noteworthy
that depending on whether the inequality
or
holds, the behaviour of
as
is different for large values of
© European Southern Observatory (ESO) 1999 Online publication: February 23, 1999 |