5. Analysis of the observational data
The 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 line H Ly-. The optical thickness of prominences in this line is, in general, very large (), and may in some cases reach values up to (see e.g., Morojenko 1984, Tandberg-Hanssen 1995). Under such conditions the effects of the multiple scattering in the strong resonance line become critical. For typical values of temperature ( K), and the electron density ( cm-3) we find that (for the rates of spontaneous and collisional de-excitation the standard notations are used), so that the photon destruction coefficient is of the order of . Thus the assumption of the conservative scattering made in this paper is believed to be a fair approximation in determining the integrated intensity of outgoing line-radiation.
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 N, we inferred that the mean number of threads must be in the bright regions of prominences, and of about , in the faint regions. Making allowance for an additional source of the brightness fluctuations namely, physical inhomogeneities, the aforementioned estimates will be obviously changed so that greater values of N are needed now to explain a given observed value of the RelMSD. We will proceed to discuss this point in more detail.
Table 1. Typical values for the mean intensity (erg cm-2 sec-1st-1) and the RelMSD for different regions of a prominence.
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 f within the order of magnitude leads to , and in place of the previous estimates () of , we obtain . Similar estimation for the faint and top parts of prominences yields . It must be noted however that the use of Eq. (30) in this latter case is not quite substantiated, as some of the assumptions underlying this formula (e.g., high opacity of threads) may be failed. Other plausible causes of large dispersion in the faint regions are a large spread in the physical characteristics of the threads and a non-uniform probability distribution of inhomogeneities of different scale. We recall that, according to our theory, the presence of a small portion of bright components in an ensemble of weakly radiating threads leads to a sharp increase in the observed value of the RelMSD. As was shown in Paper II, such a composition of differently radiating elements is characterized by a kind of asymmetrical Poisson-like profile for the probability distribution, which is actually observed. These arguments suggest that the above estimates for the faint parts of prominences may be considered as the lower limit to the real values of .
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.
The lines C ii 1336 Å, C iii 977 Å, O vi 1032 Å. These lines are formed in the prominence-corona interface (PCI) at K K, and are thought to be optically thin (with the only exception perhaps the line C ii 1336 Å) thus providing for a wealth of information on prominence structure. However, the observational data and the proper theory for these lines are characterized now by a wide variety of features, which offer difficulties in attempting to interpret unambiguously the information obtained. We shall consider here some properties of the brightness variations which are common, to a degree, for different prominences. Typical values of the spatially averaged intensity and RelMSD for the PCI-lines and H Ly- are presented in Table 1. The data refer to a prominence with a clear-cut height-variation. Sequences I, III and V, and likewise II, IV and V, indicate progressively higher regions above the limb. Each of these regions encompasses rows of pixels. In spite of the difference in brightness, regions I and II (as well as III and IV), refer on the average to the same height (the data concerning the line O vi 1032 Å are tabulated after subtracting the background from the corona).
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 N of structural elements to be a random quantity distributed according to the Poisson law. Averaging expressions for , (Eqs. 4 and 8) over this law yields
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 .
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.
The lines O iv 554 Å and Mg x 625 Å. Additional information on the structural pattern may be obtained by studying the brightness fluctuations of prominences in these lines, which as it has been said, are essentially larger than those for other lines. In some cases the RelMSD for the O iv and Mg x lines takes values of the order of and higher. This is easy to understand remembering that these lines are formed in the PC transition sheaths and corona, whereas, laying inside the Lyman continuum ( Å), they undergo absorption in a different medium namely, in the cool central parts of threads. Hence the observed fluctuations in brightness result from inhomogeneities in these two media.
Let be the intensity of the line radiation incident from the region of its formation behind the prominence and I be the observed intensity attenuated in the cores of threads. Then one can write , where is the opacity, and is the total optical thickness of the absorbing matter along a line-of-sight. All the three quantities introduced are obviously random. Since the emission and absorption of the line radiation occur in the different media, it is reasonable to suppose that these two processes (thereby the quantities and Q) are statistically independent. This allows us to express the RelMSD, of the observed intensity in terms of similar quantities for and Q. Indeed, it is easily seen that
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 N of the encountered threads will be assumed at this stage to be fixed, while the numbers k and of each kind of threads are allowed to be random. Then it is seen that
where and are the mean values of the optical thickness and the opacity of a single thread.
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, n, of element types. In this latter case the probability of a certain configuration of threads is subject to the polynomial law. Finally, if N is allowed to be random, being distributed according to Poisson's law Eqs. (36)-(39) yield
Eqs. (36)-(43) show that, for fixed physical parameters of elements, the RelMSD of the total optical thickness decreases with N (or ) whereas the quantities and increase (cf. Eqs. 38, 42 and 39, 43). This effect is in agreement with the observed increase of the normalized fluctuations in the brightness of the O iv 554 Å line when passing from bright regions to the faint ones. The compliance of the results predicted by Eqs. (36)-(43) with those obtained above in considering other lines may be checked by taking the obtained estimates of N to solve Eqs. (40)-(43) with respect to . Considering, for instance, the faint regions and letting for , and , we find that and if and The dispersion in agrees with that used above in treating the line Ly-. The estimate (O iv 554 Å) obtained by Schmahl et al. (see Schmahl et al. 1974) apparently applies to the dense, bright region of the prominence. In principle, agreement with observational values of the RelMSD can be accomplished in this case by taking . However, the dispersion in that results is unrealistically small. This can be explained by possible deviations from the Poisson law or by relinquishing certain assumptions underlying our model problem.
© European Southern Observatory (ESO) 1999
Online publication: February 23, 1999