## 5. Analysis of the observational dataThe spatial variations in the EUV line intensities have been analysed in Paper I. We attempted to explain some specific features of these by randomness in the number of threads along a line-of-sight. The predictions of the theory we developed are essentially in agreement with the observational data. It is only natural to surmise that the brightness fluctuations exhibited by prominences are due, at least partially, to physical inhomogeneities in the emitting medium. In this section we shall avail ourselves of the results obtained above to elucidate the role of this latter effect. The prominences, raster-images of which we use, are listed in Paper I. To begin, we briefly recall several conclusions of Paper I that are needed for the further discussion. The statistical analysis shows that, for a given region, the brightness fluctuations are usually the least for the H Ly- line. Among different lines formed by a similar mechanism, the more opaque lines are characterized by smaller fluctuations. This relationship particularly applies to the lines of the Lyman series. For the same line the smallest variations are usually observed in the bright regions of a prominence. This correlation is especially well pronounced for the H Ly- line. We note also that the largest RelMSD are specific to the lines Mg x 625 Å and O iv 554 Å, which exhibit strong absorption in the Lyman continuum. Let us examine each of the investigated lines separately.
The high opacity of prominences in
Ly- suggests that the brightness
fluctuations in this line must be smaller on the average than those in
the other EUV lines. The RelMSD for the
Ly- line is of the order of
for the bright regions, and rises
in passing to the fainter regions up to values of order 0.2 and even
greater. A typical example of such behaviour is presented below in
Table 1. In Paper I, our study of the brightness
fluctuations was based on the assumption that they are due only to
randomness in the line-of-sight number of threads. Adopting Poisson's
law for the variation of
Suppose that the two particular sources of fluctuation act jointly. Then the standard procedure of averaging over the Poisson distribution being applied to the case (iv) considered in the preceding section yields where is Poisson's mean of the
line-of-sight number of threads. For a homogeneous atmosphere
and we obtain
, thus returning to the case
discussed in Paper I. Eq. (30) shows to what extent the random
inhomogeneities in an atmosphere affect the observed value of
. The actual scale of variations in
the power of the internal energy sources and optical thickness of
threads is unknown a priori, though it is reasonable to expect that
this must be relatively small in the bright regions, and larger for
the faint top parts of prominences. A typical example of the latter is
region V in Table 1. For the bright regions the spread in
It is of particular interest to compare our results with those derived by others. Schmahl and Orrall (1979) inferred by analysing the Lyman continuum absorption that there must be at least cool threads along a typical line-of-sight. Fontenla and Rovira (1985) have constructed Non-LTE models of prominence threads and computed profiles and absolute intensities of the lines of the Lyman series and the Lyman continuum from an ensemble of threads. They found that the minimum number of threads along a line-of-sight ranges from 10 to 100. We see that their estimates agree with those stated in this paper. It is easily seen that the treated problem may be inverted in the sense that having reliable estimates for the number of threads, one can evaluate the scale of variations in the local physical characteristics of structural elements of prominences.
Table 1 shows that the intensities of all the lines decrease on the average with height. Some deviations from this relationship, which is usually the case for relatively non-extended and inhomogeneous prominences, may result from a scarcity of the volume of statistics. Comparing the regions I and V, for instance, we find the largest drop in for Ly- (by a factor 10.2), while in the lines C ii 1336 Å, C iii 977 Å, and O vi 1032 Å the mean intensity decreases only by factors of 6.5, 3.6, and 2.9 times, respectively. It is notable that these factors decrease with an increase in the temperature of line formation. The data we present may be used to estimate the relative dimensions of regions radiating in a given PCI-line, which ultimately allows to get a rough picture of the temperature gradient in the prominence-corona transition zone. By assuming that the second energetic level of an atom is excited predominantly by electron collisions from the ground state, we may express the line intensity by (Dupree 1972) where is the total density of a given element, is the fractional ion concentration. The integration is performed along a line-of-sight over the path where a given spectral line is formed. Being written for the optically thin resonance lines, Eq. (31) remains valid in the conservative case for the opaque lines (like H Ly-) in so far as all the radiation energy released in the medium escapes it. Under the condition of pressure equilibrium, , the usual procedure of evaluating the integral in Eq. (31) (see e.g., Pottasch 1964, Dupree 1972) yields for the relative sizes of zones radiating in the proper lines: (O vi (C iii (C ii (H Ly-) for the bright region I, and for the top region V. The relatively large range of the O vi line formation zone deserves attention but might be overestimated due to difficulties in correcting the coronal background. Some interesting relationships are observed when treating the
behaviour of the brightness fluctuations. As has been already
mentioned, there exists a pronounced correlation between the mean
intensity of a region and the RelMSD for the H
Ly- line: the brighter the region
under consideration, the smaller the value of
. This correlation is valid for all
prominences. Table 1 shows a similar correlation between
and
for the C ii
1336 Å line, while for the other
lines this effect is less pronounced. On the other hand, the
variations in and
for the line C ii
1336 Å are definitely
correlated with those for Ly-. It is
noteworthy that this effect does not depend on whether or not the
changes are due to the difference in height. These two lines have
similar behaviour in many respects which may be explained by strongly
correlated variations in the emission measure. The similarity between
variations in suggests that these
variations may be a result (at least partially) of changes in the
line-of-sight number of threads, provided that the parts of the PCI
radiating in the line C ii
1336 Å are present around each
thread. This proposition is supported by the possibility of rendering
a good fit to the observed values of
for the line C ii
1336 Å by taking the same
numbers of threads along a line-of-sight as those for the line H
Ly- Indeed, let the number where , , . Now taking, for instance, , (again, we suppose the presence of a small portion of relatively bright elements in the radiating gas), , (), and , we find from Eq. (32) that , and . This allows one a rough idea of the total optical thickness in the line C ii 1336 Å: for faint and rarified regions, and , for bright and dense regions. Variations in and within reasonable limits lead only to minor changes in the estimated values of . To proceed to the lines O vi 1032 Å and C iii 977 Å, notice first that for an optically thin atmosphere when , Eq. (32) simplifies to where we took into account that , , . The correlation between and for the mentioned lines is not so pronounced as in the case of H Ly- and C ii 1336 Å. If we start from the assumption that the lines O vi 1032 Å and C iii 977 Å are optically thin, then we can use Eq. (33) to evaluate the number . The consistency with the results obtained above will be attained by admitting that inhomogeneities of various amplitudes are equally probable in the radiating area. If this is not the case, Eq. (33) leads, under otherwise identical conditions, to larger , which is not actually observed. The most probable explanation for this discrepancy is that the PCI regions radiating in these lines are present mostly around the bulk of threads. This concerns primarily to the line O vi 1032 Å. The line C iii 977 Å occupies, in all connections, a somewhat intermediate place between C ii 1336 Å and O vi 1032 Å.
Let be the intensity of the line
radiation incident from the region of its formation behind the
prominence and Knowledge of and appearing in Eq. (35) allows us to infer the values of . Let us go further into this point by considering, for instance, the line O iv 554 Å. Since the temperature of formation of this line () is close to that for the lines O vi 1032 Å () and C iii 977 Å (), it is reasonable expect that for the O iv line does not differ strongly from the observed values of for the mentioned lines. The typical values of for the lines O vi 1032 Å and C iii 977 Å are ranged from 0.01 to 0.1 (see Paper I). The similar values for the line O iv 554 Å vary within fairly wide limits between 0.1 and 0.6. In any case, estimates of based on Eq. (35), for the bright regions and for the faint regions, may be considered realistic. Adopting the multithread model, we can use this result to infer the
individual properties of structural elements. For simplicity, let us
suppose, as above, that the absorbing matter consists of only two
types of threads characterized by the optical thickness
and opacity
(), each occurring with the
probability . The total number
where and are the mean values of the optical thickness and the opacity of a single thread. where and It is to be noted that although Eqs. (36)-(39) are derived by
assuming the presence of only two types of structural elements
, the final results remain valid for
an arbitrary number, Eqs. (36)-(43) show that, for fixed physical parameters of
elements, the RelMSD of the total optical thickness decreases with
© European Southern Observatory (ESO) 1999 Online publication: February 23, 1999 |