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Astron. Astrophys. 357, 767-776 (2000)

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2. Theory

The spectral emission from a plasma is governed by two sets of coupled equations, namely the radiative transfer equations and the equations of statistical balance. These may be written as (McWhirter 1965)

[EQUATION]

[EQUATION]

Here [FORMULA] is the photon intensity; dl is an element of distance along the line of sight; [FORMULA] is the line profile (emission and absorption profiles are assumed here to be equal); [FORMULA] is the photon emissivity - the number of photons emitted/unit time/unit volume/unit solid angle - which, if stimulated emission is ignored, is given by

[EQUATION]

where [FORMULA] is the Einstein A-coefficient. [FORMULA] is the absorption coefficient which is defined such that [FORMULA] is the number of photons absorbed/unit time/unit volume/unit solid angle. [FORMULA] is therefore given by

[EQUATION]

where [FORMULA] is the Einstein B-coefficient. Finally, [FORMULA] and [FORMULA] are the upper and lower level population densities respectively and [FORMULA] is the intensity averaged over direction. Note that Eq. 2 is equivalent to Eq. 1 of Bhatia & Kastner (1992).

Using the relation between the Einstein A and B-coefficients, Eqs. 1 and 2 become

[EQUATION]

[EQUATION]

where the spatial dependances have been put in explicitly.

The terms in the brackets are markedly similar, the only difference being in the specification of the intensity terms. In the radiative transfer case the intensity term [FORMULA] is the intensity at the point [FORMULA] due to the source [FORMULA] and so is related to the emissivity, [FORMULA], along the path [FORMULA]. In the population case, however, [FORMULA] is the radiation field at [FORMULA] due to the surrounding plasma and so is related to the integral of [FORMULA], and thus the integral of [FORMULA], over all points [FORMULA].

Eqs. 5 and 6 can be re-written as follows:

[EQUATION]

[EQUATION]

[FORMULA] is called the escape probability which represents the probability that a photon will propagate a distance [FORMULA] along the path [FORMULA] without being absorbed. [FORMULA] is the optical depth at line centre defined by

[EQUATION]

with

[EQUATION]

g is identical to what Irons (1979) calls the transmission coefficient but is called here the escape probability since the point r is outwith the emitting plasma. [FORMULA] is called the Biberman-Holstein coefficient (hereafter referred to as the absorption factor ) which represents 1 - the probability that a net absorption will occur at the point [FORMULA]. These quantities act to parametrically adjust the Einstien A-coefficients in the optically thin radiative transfer and statistical balance equations to account for the effects of self-absorption.

2.1. The escape probability

The escape probability, as introduced by Holstein (1947), has subsequently been considered by many authors (eg. McWhirter 1965; Irons 1979; Kastner & Kastner 1990). Holstein showed that assuming a constant source function and that spectral lines are purely doppler broadened, the escape probability may be written as

[EQUATION]

From Eq.  7 the solution of the equation of radiative transfer may be written as

[EQUATION]

where [FORMULA] corresponds to a position outwith the layer. Making the further assumption that [FORMULA] is constant we obtain

[EQUATION]

where [FORMULA] is the thickness of the emitting layer. [FORMULA] represents the mean probability that a line photon emitted somewhere along the line of sight in the direction of the observer will escape along the line of sight without being absorbed. [FORMULA] is the total optical depth of the layer along the line of sight. It was shown by Kastner & Kastner (1990) that this quantity, which they term [FORMULA], is given by

[EQUATION]

From this follows a simple idea, which was introduced by Jordan (1967), for extracting optical depth from observations of intensity ratios of spectral lines arising from a common upper level. Such ratios may be written as

[EQUATION]

In optically thin conditions this ratio reduces to that of the Einstein A-coefficients and opacity is the only mechanism that can modify it from this value. The transitions that are most significantly affected are those with large and long lived lower level populations - i.e. metastables. However, such populations are themselves not significantly modified due to their large value and are collisionally controlled. Thus the ratio of optical depths of two optically thick lines arising from a common upper level is constant - i.e.

[EQUATION]

where [FORMULA] and [FORMULA] are oscillator strengths. Since c is a constant, Eq.  15 may be written as

[EQUATION]

This equation provides the route for extracting optical depth from observations. Once the optical depth is known for one line, those of all the other lines of the same ion whose lower levels are metastable can be obtained via Eq.  16. Assuming the other lines of the same ion (those whose lower levels are not metastable) are optically thin, the opacities of all the lines of an ion can be obtained from the single extracted value. In Paper I this analysis was performed for lines of C II and C III using SUMER data to assess which lines should be rejected or intensity adjusted in a DEM analysis.

2.2. Absorption factors

The two principle effects of opacity are

  1. Loss of photons out of the line of sight

  2. Modification to the population structure due to photo-absorptions

The absorption of a photon by an atom or ion results in an electron excitation which can lead to either a re-emission of a photon in the reverse process or a collisional de-excitation. It is conventional within radiative transfer theory to distinguish between these two cases since the former (scattering) leads to no thermal coupling between the emitting and the absorbing plasma, whereas the latter case (pure absorption) does. Furthermore this distinction is useful in dealing with a third effect of opacity, namely partial frequency redistribution in which the emission profiles are modified due to scattering.

This distinction is not always so clear in escape probability literature. However, as described above, it influences the populations and emergent intensities via the effect of partial frequency redistribution and as yet this effect has not been included within an escape probability model.

The escape probability takes account of the loss of photons out of the line of sight due to both scattering and absorption. However, this quantity is not sufficient in describing the effects of opacity on spectral emission. Along the line of sight photons are absorbed from all directions leading to an enhanced upper level population and consequently an increase in emission in the line of sight. This is described by some authors as scattering into the line of sight (e.g. Kastner & Bhatia 1992).

In essence Kastner & Bhatia formulate the problem as follows:

[EQUATION]

where `[FORMULA]' denotes line-of-sight. Jordan (1967) wrote that the energy emissivity, E, of line i can be expressed as

[EQUATION]

where [FORMULA] the probability of escape (equivalent in principle to [FORMULA]) and [FORMULA] the probability that a photon will be emitted in line i. [FORMULA] was re-evaluated by Kastner & Bhatia (1992) as

[EQUATION]

where [FORMULA] is the photon loss probability, [FORMULA], and [FORMULA] 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, [FORMULA] (Irons 1979). [FORMULA] is given by [FORMULA] divided by the denominator of [FORMULA].

Jordan noted that in the case of intensity ratios of lines arising from a common upper level the denominator cancels and thus the population modification does not influence these ratios.

The problem with this approach is that it does not take full account of the indirect effects on the population structure. In the formulation of Eq.  19, Jordan considered emission in line i from a point to be characterised by an optically thin population structure. She then considered the ultimate fate of those emitted photons allowing them to be absorbed in line i , subsequently re-emitted in any line stemming from the same upper level, subsequently re-absorbed again, and so on and so forth. However, this does not take account of the contribution to line i of photons initially emitted in lines other than i , but from the same upper level, that are re-absorbed leading to emission in line i . This contribution is of particular importance for lines that are relatively thin if there exist thick lines stemming from the same upper level.

In addition if there is some line blending then photons emitted in one line in the blend may be absorbed by the other line. The indirect effects of this situation are complex and can effectively increase the optical depth of a line from a population modification point of view, whilst decreasing it from an emergent flux perspective and vice versa. If blending exists then the denominator of Eq. 20 will, in principle, not cancel. Furthermore it is not clear how the [FORMULA] expression should extend into a variable density model.

Following Bhatia & Kastner (1999), we choose to formulate the problem as follows:

[EQUATION]

where [FORMULA] is obtained via the solution of Eq.  6.

The use of absorption factors, or indeed escape factors, in the statistical balance equations to calculate an optically thick population structure has been considered by a number of authors (eg. McWhirter 1965; Kastner & Bhatia 1989; Bhatia & Kastner 1992, 1999; Brooks et al. 2000). The escape factor - the escape probability averaged over every line of sight through the layer - is often used in this context as it was shown by Irons (1979) that this quantity is equivalent to the absorption factor averaged throughout the layer. Bhatia & Kastner (1999) calculated optically thick populations iteratively starting from an optically thin solution and assuming that the source function ([FORMULA]) is constant throughout the emitting layer. Here we take a similar approach but for each iteration the populations are calculated at every point throughout the layer using the resolved absorption factor - the absorption factor as a function of space - to obtain an optically thick upper level population distribution (see Fig. 1). Thus here the spatial variation of the source function is included within the iterative process.

[FIGURE] Fig. 1. a Optically thin (dashed line) and thick (solid line) population density distributions for model 3 (see Sect. 3) for the C II  2s22p 2P[FORMULA] - 2s2p2 2P[FORMULA] line at 904.143 Å for an optical depth of 4. b Optically thin (dashed line) and thick (solid line) population density distributions for model 3 for the C II  2s22p 2P[FORMULA] - 2s2p2 2S[FORMULA] line at 1037.020 Å for an optical depth of 1.46 which corresponds to the same circumstances as in a . In contrast to a the modified population distribution in b is closely approximated by a constant [FORMULA] the optically thin distribution.

[FIGURE] Fig. 2. a The population density ratio of C III (2p2 3P2)/(2p2 3P1) vs optical depth including blending, calculated using Eq.  31 (solid line), compared with the unblended calculation based on Eq.  28 (dashed line). Photons emitted in the 2-2 line may be absorbed by the 1-1 line and vice versa effectively increasing the optical depths in each line. Consequently the blended curve varies as the unblended one with a scaled optical depth. However, absorption of 2-2 photons by the 1-1 line enhances the 2p2 3P1 level, lowering the population ratio. b [FORMULA] unblended (dashed) and [FORMULA] blended (solid) vs optical depth. Since the optical depth ratio of the (1-2) line to the (2-1) line is [FORMULA] 1, their photon flux ratio is largely independent of the ratio of their [FORMULA]'s and thus is proportional to the curve in a . Photon flux ratios that do not have comparable optical depths and do not share a common upper level vary with optical depth in a manner that displays characteristics of both the curve in a and that in Fig. 8. Such behaviour is evident in Fig. 3 of Bhatia & Kastner (1999).

If we assume a stratified atmosphere where [FORMULA] and [FORMULA] are purely functions of height, h, above the solar surface then the absorption factor may be written as

[EQUATION]

where [FORMULA] is the first exponential integral.

From the lower level models described below, absorption factors were calculated throughout the layer. These were calculated iteratively assuming an initial optically thin solution since the evaluation of [FORMULA] requires knowledge of the upper level population distribution. The results of this calculation are illustrated in Fig. 1 for model 3 (see below) which comprises of a layer with a density which varies exponentially with height above the solar surface.

Interestingly, upon iteration the optically thick upper level distributions of the levels of interest here (see Fig. 1b) tend to [FORMULA]. This adds weight to the assumption that

[EQUATION]

If this assumption is made then Eq.  21 reduces to Eq.  13 as follows:

[EQUATION]

If we define [FORMULA] as

[EQUATION]

and make the substitution

[EQUATION]

then Eq.  24 becomes

[EQUATION]

A similar argument shows that in making the assumption of Eq.  23, [FORMULA] reduces to [FORMULA] where

[EQUATION]

This quantity was derived in Paper I.

2.3. Line blending

As discussed in Sect. 2.2, the overlap of spectral lines in frequency space influences both the emergent intensities and the population structure. Emitted photons may be absorbed by any line whose absorption profile overlaps at the nominal frequency. Furthermore the intensity term, [FORMULA], in Eq. 6 refers to the total intensity at frequency [FORMULA], not just that resulting from the line in question.

Blending may be included trivially within the expressions for g, [FORMULA], [FORMULA] and [FORMULA]. The results for [FORMULA] and [FORMULA] are shown below for line i .

[EQUATION]

where

[EQUATION]

[EQUATION]

Fig. 3a shows the C II  2s22p 2P - 2s2p2 2P multiplet for the case where the optical depth of the 3/2-1/2 component at 904.143 Å is 10, calculated using Eqs. 29 and 31, in comparison with the optically thin case. Fig. 3b shows the same in comparison with the thick multiplet calculated using Eqs. 14 and 28. These show that there is a clear difference in both intensity and line shape when blending is included. Note that the multiplet components are labelled by their J quantum numbers.

[FIGURE] Fig. 3. a The C II  2s22p 2P - 2s2p2 2S multiplet for the optically thin case (dotted and solid lines) and the case where the optical depth of the (3/2-3/2) component at 904.143 Å is 10 (dashed and chain lines) calculated using Eqs. 29 and 31. b The dotted and solid lines correspond to the optically thick multiplet calculated using Eqs. 14 and 28. The dashed and chain lines are as in a .

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Online publication: June 5, 2000
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