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Astron. Astrophys. 322, 266-279 (1997)

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3. Results and discussion

In the following discussion about the effect of a chromosphere on the D lines, we will often use the line profile produced by the photosphere alone as a reference. Fig. 4 shows such an emerging profile as computed with the adopted model photosphere.

[FIGURE] Fig. 4. The D lines computed from the model photosphere of Allard & Hauschildt (1995b). The dotted histogram represent the "background" photospheric flux (see text).

For the sake of consistency, the electron density has been recomputed following the same procedure, outlined in Sect.  2, as used for the model chromospheres. The resulting electron densities differ from the tabulated values by at most a few percent. The Na I and H I line profiles calculated from that purely photospheric model, will hereafter be referred to as the "photospheric" profiles.

The same figure also shows the "background" photospheric spectrum, at the resolution dictated by the input opacity tables (2 Å). Such a background spectrum represents the emerging spectrum obtained ignoring line absorption by sodium atoms; the variations in the background opacity affecting the wings of the D lines are readily apparent. Note also the severe blending between [FORMULA] and [FORMULA], which justifies our treatment of line overlapping (Sect.  2.1).

3.1. Dependence on atmospheric density and structure

We now proceed by first examining the effect of the structure of the chromosphere on the H I and Na I line spectrum. In the following sections we will deal with the other aspects we anticipated in Sect.  2 to be relevant for our discussion, namely the presence of an active corona (Sect.  3.2) and the importance of a correct treatment of collisions with hydrogen (Sect.  3.3) and of background opacities (Sect.  3.4).

3.1.1. Dependence on atmospheric density

As discussed in Sect.  2.2, one of the main parameters characterising our model chromospheres is the pressure atop the chromosphere or, equivalently, the column mass at the onset of the transition region, [FORMULA]. Fig. 5 shows the dependence of Na I D lines and of some H I spectral features (Ly [FORMULA], H [FORMULA], Pa [FORMULA]) upon this parameter, for a particular series of models (see figure caption). Along with H [FORMULA], we have chosen Pa [FORMULA] as a representative subordinate H I line instead of Pa [FORMULA] because, due to strong telluric absorption, the latter is practically unobservable from Earth.

[FIGURE] Fig. 5. Dependence upon the parameter [FORMULA] of some features of the Na I and H I spectra: the core of Na I [FORMULA] and [FORMULA] (upper-left panel), H [FORMULA] (upper-right panel), Ly [FORMULA] (lower-left panel), Pa [FORMULA] (lower-right panel). The line profiles correspond to model chromospheres type 1A (see Fig. 1). Note the logarithmic scale for the Ly [FORMULA] flux. For each profile, a dashed line indicates the corresponding value on the [FORMULA] axis.

The approximation of CRD, used in our calculations, is poor in the wings of Ly [FORMULA]. Therefore, the Ly [FORMULA] profiles shown here are purely indicative. However, the total flux in the line, dominated by the line core, can be regarded as more reliable. As is clear from Fig. 5, the latter quantity increases monotonically with [FORMULA]: in fact, it increases approximately as its square. This scaling law can be understood considering that Ly [FORMULA] forms mainly in the transition region within a limited temperature range whose lower and upper boundaries are defined, respectively, by the rapid decrease of the collisional excitation rate of level [FORMULA] and by the complete ionisation of hydrogen. Moreover, in all but the most active models the Ly [FORMULA] line is effectively optically thin: in other words, practically all the photons created in the line will eventually escape, possibly after multiple scatterings. The peak value of the source function (and of the emerging intensity, from the Eddington-Barbier relation) of a Doppler broadened line forming in an effectively thin, isothermal slab scales as [FORMULA] (Athay 1972, p. 64), where [FORMULA] is the collisional coupling parameter (see Eq.  1) and [FORMULA] is the slab total optical thickness at the line center. Both [FORMULA] and [FORMULA] depend linearly on density: hence the quadratic dependence on [FORMULA]. The optical depth [FORMULA] depends also on the gradient of the transition region, but we recall that within each series the models are scaled so that [FORMULA] in the transition region is kept constant (see Sect.  2.2).

Similarly, starting from an almost zero core flux in the photospheric profile, the core of the D lines shows a monotonic increase with chromospheric pressure, with self-reversed emission in the most active models. By contrast, both H [FORMULA] and Pa [FORMULA], practically absent from the photospheric spectrum, show at first a deeper absorption against the photospheric background, and are eventually quickly driven into emission, at [FORMULA]. The precise value of this "critical" pressure depends somewhat on the particular series of models and on the line: in our series of models, H [FORMULA] tends to be driven into emission at [FORMULA] between -4.4 and -4.2, while Pa [FORMULA] is driven into emission at [FORMULA]. Such a "non-LTE curve of growth" has been noticed by CM for H [FORMULA], but is common to other H I subordinate lines.

3.1.2. Dependence on temperature structure

Fig. 5, while informative about the general trends as chromospheric activity increases, refers only to a specific temperature stratification in the chromosphere. Fig. 6 shows instead the consequences of differences in temperature structure on the same Na I and H I lines as the previous figure. For clarity, only the results for the last four models of each series are shown. As for the two D lines, only the core of [FORMULA] is displayed in Fig. 6 ; the behaviour of [FORMULA] is similar. Moreover, the photospheric profile is also plotted for comparison.

[FIGURE] Fig. 6. Line profiles for the most active models of the four series of models with [FORMULA]. From top to bottom: Na I [FORMULA], Ly [FORMULA], H [FORMULA], Pa [FORMULA]. For the Na I [FORMULA] line the purely photospheric profile is also shown (solid line).

The Ly [FORMULA] line seems insensitive to the chromospheric structure, a fact consistent with the picture of a collisionally dominated line forming in the transition region. To a large extent the same can be said of the subordinate H I lines when in emission. In this case, density (or pressure, parametrised, as usual, by [FORMULA]) seems to be the most important parameter (CM, Cram & Mullan 1985, have investigated such a dependence for H [FORMULA]).

Far more pronounced is the effect of the chromospheric structure on the [FORMULA] line. In particular, it is interesting how the position of temperature minimum leaves its signature on the profile, mainly on the dip between the emission core and the photospheric wings (compare, for example, the profiles of models [FORMULA] with those of models [FORMULA]). This is reminiscent of the effect on the Ca II H & K lines. From a computational point of view, however, the D doublet has the advantage over the Ca II lines that its profile is not affected by PRD problems (a fact already mentioned in Sect.  2.1).

For the most active models, at a given value of [FORMULA] the extension of the chromosphere seems to be the dominant effect on the profile of the emission cores, as hinted by the systematic changes in the inner-core profiles from models [FORMULA] (the thinnest chromospheres) to models [FORMULA] (the thickest chromospheres). In fact, it is easy to see the progression from the round-top, gaussian-like profiles of models [FORMULA], through the flat-top profiles of models [FORMULA], to the self reversed profiles of models [FORMULA] and [FORMULA].

3.1.3. Dependence on the gradient of the lower transition region

To complete our investigation on the effect of the temperature structure, it remains to be examined how the lines we have considered respond to changes in the gradient (i.e. thickness) of the transition region. For this purpose, Fig. 7 displays the line profiles for the most active models (the last three models of each series, in this case), both for [FORMULA], as in Fig. 6, and for [FORMULA].

[FIGURE] Fig. 7. Dependence on the gradient of the transition region. The thicker lines correspond, as in Fig. 6, to [FORMULA] ; the thinner lines to [FORMULA]. Only the last three models in each series are shown in this figure.

In the case of Ly [FORMULA], it is interesting to note how a temperature gradient change in the transition region appears to be equivalent to a pressure change. For example, the profiles with [FORMULA] and [FORMULA] nearly overlap with the profiles with [FORMULA] and [FORMULA]. This fact fits in the picture sketched above of a collisionally controlled Ly [FORMULA] whose main dependence is upon the temperature gradient (through the total optical depth of the emitting slab, [FORMULA]), as well as on the square of pressure. Thus, increasing the pressure in the transition region by 0.2 dex (the step between the models shown here) produces an increase of 0.4 dex in Ly [FORMULA] emission, almost equivalent to the change produced by the increase of the temperature gradient (0.5 dex).

The subordinate H I lines are less sensitive to changes in the transition region. This is consistent with the findings of Houdebine & Doyle (1994), but the effect of the gradient in the transition region cannot be neglected. In contrast, the [FORMULA] line (and [FORMULA], as well) is almost insensitive to the structure of the transition region. This implies that the core of the Na I lines forms at lower heights than the H I spectrum.

3.1.4. The region of formation of the D lines

The statement made at the end of the previous section, can be expressed in pictorial terms by the use of contribution functions. One possible definition of contribution function can be derived from the formal solution of the transfer equation for intensity in plane parallel, semi-infinite atmospheres. In this case, the contribution function, [FORMULA], at each wavelength, [FORMULA], and cosine of the angle with the normal to the surface, µ, is defined so that:


where x is a depth coordinate. With the choice [FORMULA], the contribution function becomes:


where [FORMULA] is the density, [FORMULA] is the emissivity and [FORMULA] is the optical distance to the surface. While alternative, more sophisticate definitions can be adopted (e.g. Magain 1986), Eq.  2 is sufficient for our purposes.

With the definition of contribution function given by Eq.  2, Fig. 8 summarises the main differences between the D lines and H [FORMULA] and, by extension, all H I subordinate lines. As an example, we consider an active model ([FORMULA]) with a chromosphere type 2A and [FORMULA]. In this case the contrast between H [FORMULA] and [FORMULA] is immediately apparent. The former is exclusively formed in the upper chromosphere, while the core of the latter forms throughout almost the entire chromosphere. Only a small contribution to the [FORMULA] emission, even for this very active model, comes from the lower transition region.

[FIGURE] Fig. 8. Contribution functions, [FORMULA], for intensity emerging at [FORMULA], as functions of wavelength and depth, for the core of [FORMULA] (left) and H [FORMULA] (right), in the model atmosphere with chromosphere type 2A, [FORMULA] and [FORMULA]. In each panel, the contribution function is also shown with a contour plot; on the same graph the locus [FORMULA] is drawn with a solid, thick line.

It should be remarked, however, that Eq.  2, as all the contribution functions proposed so far, is based on the formal solution of the transfer equation, which means that the source function and optical depth are considered as given quantities. It thus ignores the non-linear, non-local nature of radiative transfer in these strong lines, as well as the inter-dependence between different transitions. Therefore, while contribution functions can serve to illustrate some specific points, they cannot be a substitute of a more physically sound approach. Such an approach requires the study of the response of the emerging profile to variations of physical properties of the atmosphere, as we have done earlier in this section, and as we intend to do in the following discussion.

3.2. Dependence on coronal illumination

The response of the D lines to coronal illumination is one possible non-local effect that is of interest. Fig. 9 compares some "standard" results presented so far with results obtained considering an XUV flux incident atop the model atmosphere. Significant changes occur in the core of the D lines in the presence of a strong XUV flux illuminating the chromosphere.

[FIGURE] Fig. 9. [FORMULA] core profiles for [FORMULA] (thick lines) compared with the results obtained with the coronal illumination of Fig. 3 (thin lines). As in preceding figures, the photospheric line profile is drawn with a solid line.

A closer inspection of specific models can give further insights on the physical process behind the emerging profiles. The alterations in the Na ionic equilibrium, shown in Fig. 10, are of particular interest. Clearly, over-ionisation of [FORMULA] will propagate through the recombination chain to [FORMULA]. Fig. 10 also shows how the changes in the sodium ionisation equilibrium can reach the formation region of the [FORMULA] core, if marginally.

[FIGURE] Fig. 10. Fractional abundances of [FORMULA] (solid lines) and [FORMULA] (dashed lines), in the case of a model chromosphere type 2A, [FORMULA] and [FORMULA]. Thicker lines refer to a null coronal illumination, while thinner lines represent abundances calculated with the XUV flux of Fig. 3. All quantities are plotted versus the optical depth at the central wavelength of the [FORMULA] line.

As for the effect on the emerging profile of the D doublet, it is not easy to produce a simple quantitative model. Even limiting ourselves to the most active models, where the coupling of the line source function with the local Planck function is strongest, one should be aware that some of the relevant quantities vary considerably over the large region of line formation. However, in qualitative terms only, it is still possible to identify two possible effects.

The first possibility concerns the dependence on density of the emission cores we have noticed in Sect.  3.1.1 and Sect.  3.1.2. This density dependence is mainly due to the collisional coupling parameter [FORMULA] of Eq.  1. Even if [FORMULA] is in itself not constant throughout the line-forming region, the ratio between the source function and the Planck function is mainly determined by some "mean" value of [FORMULA] (Athay 1972, p. 48). Notably, the surface value of the source function (closely related to the line-center emerging intensity) will be proportional to [FORMULA]. The [FORMULA] ionisation front induced by coronal radiation will push down the upper boundary of the region where Na I chromospheric emission is produced, towards lower temperatures and electron densities. Consequently, [FORMULA] will decrease, causing a lower intensity in the line core.

The other possibility stems from the dependence of the emission cores on the thickness of the chromosphere (Sect.  3.1.2). In this case, an overionisation of [FORMULA] would in effect result in a reduction of the thickness of the chromosphere as seen by the D lines. Physically, the coronal photoionisation flux depletes the topmost layers of neutral sodium, thus exposing lower, cooler chromospheric layers. Again, the effect is a reduced core emission intensity.

To ascertain to what extent these two processes are effective, we examine in one particular model, as an example, the behaviour of the source function of the [FORMULA] line (Fig. 11). An inspection of the left-hand panel of Fig. 11 reveals that in reality the ratio [FORMULA], as function of the optical depth, hardly changes in the presence of photoionising photons. The two curves in the left-hand panel of Fig. 11 are indeed practically indistinguishable. This rules out the first possible effect, the reduction of [FORMULA]. In fact, since the region of formation of the line, i.e. the region where [FORMULA] is set, lies quite deep in the atmosphere, far more substantial XUV fluxes would be required to have an effect this way.

[FIGURE] Fig. 11. Run of the source function and of the Planck function at the central wavelength of the [FORMULA] component of the sodium doublet. The model chromosphere is the same as in Fig. 10. As in that figure, thinner lines represents quantities calculated accounting for the XUV coronal flux.

The dominant effect seems to be instead the lowering of the upper boundary of the "sodium chromosphere". This fact is clearly evident in the right hand panel of Fig. 11: at a given optical depth, the value of the Planck function in the chromosphere (and therefore of the source function) is lower in the presence of coronal illumination.

3.3. The effect of collisions with hydrogen

Another important issue is whether collisions with hydrogen can have a significant effect on the Na I D line profile. Fig. 12 addresses this problem, showing the inner profile of the doublet obtained in the usual "reference" active model we have considered in the previous sections. It is clear that the inclusion of the Kaulakys rates does not appreciably change the cores, and only has a limited influence on the upper photospheric profile. Note, in particular, the change in the region where the wings of the two components overlap.

[FIGURE] Fig. 12. Effect of collisions with hydrogen on the D profiles obtained with model chromosphere type 2A, [FORMULA] and [FORMULA]. The solid line is the "standard" calculations, i.e. Na-H collisions computed with Kaulakys cross-sections. The dotted line is the profile obtained neglecting collisions with hydrogen. The dashed line refers to calculations performed using the Drawin formulae.

On the other hand, the Drawin rates do produce more significant changes, even in the cores. But, considering that those rates are as much as two or three orders of magnitude larger, the changes are surprisingly small. As for the emission core, it forms in the chromosphere, where hydrogen starts to release more electrons. Therefore it is reasonable to expect that electron collisional rates are not easily overwhelmed in that region. The relatively small effect on the deeper parts of the line profile can instead be justified by considering that in those regions departures from LTE are not very strong. Thus, since adding any further thermalisation process brings the source function closer to LTE, the changes are relatively modest.

Fig. 13, the analogue of Fig. 11, quite clearly illustrates this point. In fact the thermalisation depth (the depth where the source function approaches LTE) changes when using Drawin instead of Kaulakys rates. But over much of the chromosphere there is little or virtually no change. Nevertheless, the particular choice of the collisional rates does have consequences on the source function in the temperature minimum region and lower chromosphere.

[FIGURE] Fig. 13. Run of the source function and of the Planck function at the central wavelength of the [FORMULA] component of the sodium doublet, as in Fig. 11. The thinner line refers to quantities obtained with Drawin cross-sections for H-Na collisions, the thicker lines refer to Kaulakys rates.

3.4. Dependence on background opacity

Finally, it is interesting to examine the effect of the choice of background opacities on the calculated line profiles. Fig. 14 shows the effect of neglecting plasma background line opacities on the visible and infrared chromospheric diagnostics considered in this work.

[FIGURE] Fig. 14. The D doublet (upper panels), H [FORMULA] (lower-left panel) and Pa [FORMULA] (lower-right panel) computed taking into account background line opacities (solid lines) or continuum opacities only (dotted line). All the calculations were made for model chromosphere type 2A, [FORMULA] and [FORMULA].

It should once more be stressed that the low wavelength resolution (2 Å) of the original opacity table adopted here is not suitable for comparisons with high resolution data. However, the chromospheric cores of the strong lines we have considered are affected relatively little by these computational details. A major effect is instead seen in the wings of the D lines: it is clear that any comparison with observations of cool stars cannot exclude a realistic treatment of the background opacity. This is especially important if the sensitivity of the D lines to the lower chromosphere, demonstrated in Sect.  3.1, is to be exploited.

Similar considerations on the importance of a proper treatment of background opacities in theoretical modelling, can be extended to other chromospheric diagnostics as well. In general, the theoretical "continuum" fluxes computed neglecting atomic and molecular line absorption in the photosphere of cool stars are overestimates. This fact may impair the correct evaluation of the source function of the chromospheric diagnostics. But even for a collisionally dominated line, such as H [FORMULA] in the most active models, the level of the continuum is important for comparing synthetic spectra with observed profiles. Unless the line is optically thin, its chromospheric emission is not just superimposed on the photospheric background: the computed relative profile will then depend on the accuracy of the computed photospheric flux. A comparison between H [FORMULA] and Pa [FORMULA] in Fig. 14 is sufficient to illustrate this point.

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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998