Astron. Astrophys. 357, 697-715 (2000)

4. Escape probability analysis

Jordan (1967) introduced the simple idea of extracting optical depths from observations of intensity ratios of spectral lines arising from a common upper level. She wrote the ratio of the energy emissivities of lines 1 and 2 as

where and are the escape probabilities, and the probabilities that photons will be emitted in lines 1 and 2 respectively and the 's are the line wavelengths. Earlier, other authors had introduced ideas of escape probabilities for describing both emergent intensities (Holstein, 1947) and excited state population structures (McWhirter 1965) and since then escape probabilities and escape factors have been developed extensively by eg. Irons (1979) and more recently Kastner & Kastner (1990). Doyle & McWhirter (1980), using the same approach as Jordan, considered branching ratios to investigate opacity in spectral lines of C III at the limb of the sun utilising for this purpose Skylab cross-limb observations. They used a more sophisticated escape probability expression than Jordan which they called g, given by

This is identical to what Irons (1979) called the transmission factor, , and what Kastner & Kastner (1990) termed . represents the probability that a photon emitted at an optical depth in the direction of the line of sight will propagate along the line of sight without being absorbed. Kastner & Kastner describe this as the case of isolated emitters and absorbers (denoted by `0') as it does not include the effect of emission at other optical depths along the line of sight. Averaging this expression along the line of sight leads to what Kastner & Kastner describe as being the proper escape probability for emergent intensities, given by

This represents the mean probability that a photon emitted from some point in a layer of optical depth , in the direction of the line of sight, will escape along the line of sight.

The expression assumes that the only effect of opacity is to scatter photons out of the line of sight which is not true as photons may also be scattered into the line of sight. This was recognised by Jordan (1967) who wrote

where is the fraction of photons created, escaping in line i; is equivalent in principle to and the denominator accounts for the scattering into the line of sight. This was subsequently re-written by Kastner & Bhatia (1992) as the correct line of sight escape probability, , where

where is the photon loss probability, , and . is the mean probability that a photon emitted anywhere in the layer will travel to the surface and escape. This latter term is equivalent to Irons' escape factor, .

It was pointed out by Jordan, however, that for intensity ratios of lines arising from a common upper level the denominator of cancels out leaving just as the appropriate escape probability for this ratio analysis.

4.1. The escape probability

Since we are following the approach of Doyle & McWhirter it is instructive to consider the effects of using the less appropriate transmission factor, g (Eq. 2), compared with the escape probability (Eq. 3). may be obtained from g as follows: Following McWhirter (1965) the optical thickness of a spectral line at line centre is

where is the ion temperature (K), M is the atomic mass number, is the central wavelength (cm), is the number density of the lower level of the transition (), is the absorption oscillator strength and L (cm) is the physical thickness of the plasma along the line of sight. We introduce the absorption coefficient at the frequency in the dependence of the escape factor. Take the origin of coordinates at the outer intersection of the viewing line with the boundary of the emitting layer and as the intersection of the viewing line with the inner boundary. Assume that emission and absorption profiles are identical () and that the Einstein B-coefficient is defined in terms of energy density. Then the equation of radiative transfer, integrated over frequency gives

where , and is the probability of escape from the point x in the plasma towards the line of sight. is the absorption coefficient at line centre. For a Doppler broadened line, introducing the absorption profile of the line with the central frequency of the spectral line as , corresponding to the central wavelength , we have the following connections

with

The absorption coefficient , defined in terms of energy absorption per steradian is

Define also

and

Returning to the intensity calculation, assume that the ion populations are independent of position in the layer and correspond to those from an optically thin collisional-radiative model. Then the observed intensity is

where is the averaged escape factor over the layer and is the thickness of the layer.

Then

Reintroduce . For two lines with common upper level u and lower levels and , the observed intensity ratio is

From Eq. 6,

Provided the ratio of the two lower level population densities is not influenced by self absorption and can be estimated, then the optical depth of the lines may be inferred from the observed intensity ratio using Eq. 17. Doyle and McWhirter applied this analysis to pure branching ratios of the C III   - (1175 Å) multiplet for which the lower `metastable fine structure' levels have close to relative statistical populations at solar atmosphere densities.

To see how the use of g rather than compares, recall Eq. 13 and use the intermediate value theorem so that

where is some value of such that . Thus the Doyle and McWhirter analysis seems to require that g replaces and replaces . However, the question arises as to whether Eq. 18 is valid with replacing . We can write . For it can be seen from Fig. 11 that which is to say that the mean probability of escape from a layer of optical depth less than 1 is equivalent to the probability of escape from the centre of the layer. However, it can also be seen from Fig. 11 that for optical depths larger than 1, as expected since for layers of optical depth greater than 1 not all of the layer is `seen'. The implication for the opacity deduction from this is that

and the use of g instead of is inappropriate for optical depths greater than 1.

Fig. 11. Comparison of escape factors versus optical depth . (a) the layer averaged emergent flux escape factor ; (b) ; (c) the emergent flux escape factor ; (d) the population escape factor .

Table 2. Summary of NIS and GIS B-like ion multiplets suitable for opacity analysis. Lines marked * are also detected by SUMER.

4.2. Opacity deduction from observed branching line ratios

Following the Doyle and McWhirter analysis, the relative optical depths of the two lines of the ratio are obtained from Eq. 17. The relative populations in the lower levels were examined in an optically thin population calculation (see Sect. 4.4) at , for C III and , for C II . The electron temperatures, , were taken from the peak of the functions (at fixed pressure) for the lines and the electron density, , from the Vernazza et al. (1981) model. For C III , the electron density from the model is significantly larger than that typically used for quiet sun (). At the model density, the fine structure levels of the lower metastable term are closely relatively statistically populated (ratios 1:3:5). This is not so at the lower density when the decay via the mixing reduces the relative population (ratios 1:2.8,5.0). If these fine structure population ratios were sensitive to opacity then it would violate the conditions for Eq. 18. We have investigated the effect of optical depth on these ratios using the ADAS code ADAS214 (see Sect. 5). The opacity modified ratios are 1:3.1:4.7 The self-absorption as expected drives the population closer to its statistical proportion. However the effect is weak and so, for the purposes of the present paper, little error is introduced by using the optically thin metastable fine structure population ratios. We have adopted the ratios at as representative. For the C II case the lower ground term relative populations are very close to statistical, insensitive to the plasma conditions and unchanged at finite optical depths. Then in both cases Eq. 17 is used with the theoretical escape probabilities and experimental intensity ratios to deduce the optical depth in one of the lines in each ratio. Tables 3 to 7 summarise the values of , , and g at different solar positions across the limb from the C III and C II multiplet branching ratios. An original objective was to choose multiplets for different members of the same isoelectronic sequences, for example C II , N III and O IV to show the progression of optical depth effects along sequences. The multiplets amenable to analysis provide only a direct comparison of C II and O IV which show a marked reduction of optical thickness along the sequence. Dissimilar multiplets in C II and C III again indicate a strong reduction of optical thickness with increasing ion charge. The results available here suggest that neutral to doubly ionised ions are the principal concerns for optical depth effects on differential emission measure analysis.

Table 3. Summary of the data for C III transition for each raster scan position.

Table 4. Summary of the data for C III transition for each raster scan position. . Since the opacity ratio is fairly close to unity, the absolute opacities inferred here are expected to have large errors and would be more reliably modelled from the 2-2 opacity of the previous table (see Sect. 4.4).

Table 5. Summary of the data for C II transition for each raster scan position. .

Table 6. Summary of the data for C II transition for each raster scan position. . The opacity ratio is again fairly close to unity and the comments on Table 5 apply.

Table 7. Summary of the data for C II transition for each raster scan position. .

4.3. Modelling opacity variation in branching ratios at the limb

For a spherically symmetric stratified atmosphere, the extension of the geometrical length of an emitting layer along the line of sight at longitudes approaching the limb from the disk is the determinate factor in optical thickness variation. For thin emitting layers therefore, the variation of optical depths and branching ratios are also geometrically determined (Doyle & McWhirter 1980). We propose a simple two layer model. Assume that the layer may be divided into two parts, namely a potentially optically thick primary layer where the ionisation stage has its peak abundance (called the `inner model') and an optically thinning outer layer where the population densities are rapidly decreasing (called the `outer model'). The peak emission of an emitting layer is in the line of sight for all raster positions across the disk up to the radial location of the emitting layer at the limb. In Fig. 12, the theoretical curve for C III of versus position relative to disc centre has been superimposed on the experimental intensity ratio data. Here with inside the visible limb and outside the visible limb. is the optical depth at disc centre. Note that the approximation of variation as (cf. Doyle and McWhirter) breaks down at angles near depending on emitting layer thickness. The position (in arc sec relative to the disc centre) of the C III layer and are treated as variables chosen to minimise the functional

Fig. 12a uses the nominal angular position of the visible limb from the observing sequence telemetry. Positional error in SUMER is 10 arc sec. We have therefore conducted a further fit, shown in Fig. 12b, in which the visible limb position was varied to optimise the match of theory and observation. The optimised position is well within acceptable limits. Note also that the theoretical inner model curve based on our preferred shows a significant improvement of fit compared with that using the Doyle and McWhirter g. We take this as support of the use of in basic escape factor analysis.

Fig. 12. a Calculated intensity ratios for the C III  -  (1175 Å) multiplet component ratios I(2-2)/I(1-2) and I(0-1)/I(2-1) vs raster position in arc sec relative to the disc centre compared with the observed ratios. The calculated results use the position for the visible limb from the telemetry (959.60 arc sec) The solid curve uses and the dashed line, g in the calculation. b The same multiplet component ratios are shown, but the position of the visible limb is shifted to optimise the match between theory and experiment. The shifted limb position is at 956.20 arc sec.

For the region beyond the limb we consider three models. Firstly we consider an emitting atmosphere as predicted by the model of Vernazza et al. (1981) (henceforth referred to as the VAL model) which assumes that the temperature and density gradients in the transition region are determined by a constant conductive flux from the corona to the chromosphere. From this model we have and which we use in conjunction with the contribution function, , for a line with the assumption that

where . Thus the optical depth of a line is given by

where l.o.s. stands for line of sight and ds is an element of distance along the line of sight. Here a is a constant which is found by matching the inner and outer models at the layer edge. Fig. 13 shows the inner and outer models superposed on the experimental data for the C II  -  (1036 Å) multiplet component ratio I(3/2-1/2)/I(1/2-1/2) and the C III  -  (1175 Å) multiplet component ratio I(2-2)/I(1-2). The inner and outer models give slightly different positions for the C III layer.

Fig. 13. a Calculated intensity ratio for the C II  -  (1036 Å) multiplet component ratio I(3/2-1/2)/I(1/2-1/2) based on the VAL model vs raster position in arc sec relative to the disc centre compared with the observed ratios. The solid curve uses and the dashed line, g. The shifted limb position at 956.20 arc sec is used. b Calculated intensity ratios for the C III  -  (1175 Å) multiplet component ratios I(2-2)/I(1-2) and I(0-1)/I(2-1) vs raster position in arc sec relative to the disc centre compared with the observed ratios.

It is clear that in both cases the VAL model fails to describe the emission above the limb. The functions are sharply peaked in temperature and for C II and C III the peaks occur at temperatures corresponding to the very sharp decrease in temperature and density predicted by the hydrostatic atmosphere model. Consequently the predicted ratios move sharply to the optically thin value on reaching the layer edge.

It has been known for some time that the off-limb EUV emission is dominated, over the height range of interest here, by spicule-like structures. Several authors (eg. Mariska & Withbroe 1975; Mariska et al. 1978) have shown that models such as the VAL model are unable to predict the observed line intensities in this region and so it is no surprise that they also fail to predict the observed opacities.

Though the signal is dominated by inhomogeneities, our models are stratified as regards emergent intensitities and plane parallel as regards escape probabilities and so we envisage two alternative empirical models in an attempt to account for the spicular component to the observed signals. Firstly the notion of a spherical shell of constant density, as previously envisaged on-disk, was extended to beyond the limb. This follows the approach of Kastner & Bhatia (1992) who used such a model to capture the grosser characteristics of the spicular contribution to the signal. Consequently the chosen layer thickness was 5 arc sec.

Mariska, Feldman & Doschek considered models where the dominant contribution to the EUV signal was due to transition-zone sheaths around isolated cylindrical spicules. They noted that `above the emission peak the amount of emitting material in the line of sight for any spectral line must decrease exponentially with height with a scale height that depends on temperature'. In particular, they found that the density scale heights were around 1500km but decreased with decreasing electron temperature of line formation. Thus in our second empirical model we envisage a stratified emitting layer with a density variation versus height given by

where B is a constant, is the inner edge of the layer which is taken from the VAL model and H is the density scale height. These models are shown in Figs. 14 and 15 and they both represent a marked improvement on the VAL case with the exponential atmophere clearly the most effective. Upon optimising H in the latter model we get, when using , scale heights of 1.3 arc sec and 1.4 arc sec,equivalent to 942 km and 1015 km, for the C II and C III cases respectively. These are smaller than that suggested by Mariska, Feldman & Doschek but do decrease with decreasing temperature of line formation.

Fig. 14. a Calculated intensity ratio for the C II  -  (1036 Å) multiplet component ratio I(3/2-1/2)/I(1/2-1/2) for a spherical shell of constant density vs raster position in arc sec relative to the disc centre compared with the observed ratios. The solid curve uses and the dashed line, g. The shifted limb position at 956.20 arc sec is used. b Calculated intensity ratios for the C III  -  (1175 Å) multiplet component ratios I(2-2)/I(1-2) and I(0-1)/I(2-1) vs raster position in arc sec relative to the disc centre compared with the observed ratios.

Fig. 15. a Calculated intensity ratio for the C II  -  (1036 Å) multiplet component ratio I(3/2-1/2)/I(1/2-1/2) for the exponential atmosphere model vs raster position in arc sec relative to the disc centre compared with the observed ratios. The solid curve uses and the dashed line, g. The shifted limb position at 956.20 arc sec is used. b Calculated intensity ratios for the C III  -  (1175 Å) multiplet component ratios I(2-2)/I(1-2) and I(0-1)/I(2-1) vs raster position in arc sec relative to the disc centre compared with the observed ratios.

A feature of all three models considered is that the ratios all return to the optically thin limit whereas in the C III case the observed ratios do not. Referring to Fig. 6 it is clear to see that at large heights, beyond 965 arc sec say, the observed fluxes are very weak which is suggestive of low opacity yet the ratios indicate opacities at these heights similar to those on disk. It is our feeling, as suggested in Sect. 2.4, that the signal at these heights is dominated by instrumentally scattered light which comprises of light from the whole disk and so it is expected that such a signal would reflect similar opacities to those on disk. As discussed in Sect. 2.5, the C II ratios are ambiguous in this region and so no conclusions may be drawn about the success, or otherwise, of the models here.

4.4. Population modelling for optically thin to moderately optically thick plasma

We can compare ratios of intensities of component lines of a multiplet which do not have the same upper level. However, in this case does not cancel in the ratio although the physical layer thickness does. Thus the replacement of Eq. 17 for the intensity ratio is

The suffix on the upper level population density ratio indicates that it depends in general on self absorption. Consider now the determination of the local population structure in a layer which is optically thick in the direction normal to the layer. We assume that the populations are independent of position in the layer and may be represented by the populations at the layer centre; also that the layer may be treated as plane parallel from the point of view of evaluating the radiation field at the centre of the layer. Take the origin of coordinates at the centre of the layer and let the layer have thickness . For simplicity write the equation of radiative transfer as

Evaluating the average over solid angle of the radiation intensity at the origin integrated over the layer

Substituting and into the local population equations, we have

Thus

where we call the `population escape factor'. It represents 1- the probability that a re-absorption event will occur at layer centre. Thus is equivalent to what Irons (1979) calls the `Biberman-Holstein coefficient' evaluated at layer centre. We regard, however, as representing re-absorption characteristics of the emission layer as a whole so in this respect is equivalent to Irons' escape factor, . The graph of is also shown in Fig. 11. Provided the optical depth can be determined in one line then the optical depths and population escape factors of all other lines of the same ion connected to the same lower level may be calculated. For the ground term of ions of interest here, the levels are close to statistical relative populations in the optically thin plasma, thus the population escape factors of all lines connected to levels of the ground term may be determined. In the moderately thick case, we must also be concerned with lines connected to levels of the metastable term. It is a good approximation for ions relevant here that the metastable level populations are close to relative statistical populations and that the metastable term population remains close to its optically thin value relative to the ground term population (see the discussion in Sect. 4.2). Thus the population escape factors may be established for all relevant lines.

4.5. Classification of emergent spectral fluxes

We classify observed spectrum lines into five categories according to the influence of optical depth along the line of sight and of the influence of self absorption on the formation of the excited populations in the emitting layer. Thin lines (t) are those which are optically thin on the limb and whose emitting excited populations are not influenced (directly or indirectly) by self-absorption. Modified thin lines (mt) are optically thin on limb but their emitting populations are modified by the indirect effect of self absorption in other lines. Weakly thin lines (wt) are thin on the disk but thick on the limb, with their emitting populations unaffected by self absorption in other lines. Modified weakly thin lines (mwt) are optically thin on disk but their emitting populations are affected by self absorption in other lines and they are thick on limb. Modified lines (m) are lines whose emitting excited populations are influenced directly by self-absorption. We choose a 10% modification to the Einstein A coefficient as the criterion for assessing whether opacity affects either the emergent flux or the population structure. Indirect effects on populations are indicated by shifts at the 10% level at disk centre. Since for and for the classification gives (i) thin   at the limb, populations unmodified; (ii) modified thin   at the limb, populations indirectly modified; (iii) weakly thin   on disk, at the limb, populations unmodified; (iv) modified weakly thin   on disk, at the limb, populations indirectly modified; (v) modified   on disk. Final disk centre results and line classifications are illustrated for C III and C II in Tables 8 and 9.

Table 8. Characterisation and classification of some spectrum lines of C III at disk centre.

Table 9. Characterisation and classification of some spectrum lines of C II at disk centre. The extrapolation to disk centre is subject to significant error in this case because of the relatively flat slope of the intensity ratio curve evident in Fig. 13a. A more secure classification would use direct measurements at the disk centre.

© European Southern Observatory (ESO) 2000

Online publication: June 5, 2000
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