 |  |
Astron. Astrophys. 322, 266-279 (1997)
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]](img64.gif) |
Fig. 4. The D lines computed from the model photosphere of Allard & Hauschildt (1995b). The dotted histogram represent the "background" photospheric flux (see text).
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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
and , 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, . Fig. 5 shows the
dependence of Na I D lines and of some
H I spectral features (Ly , H
, Pa ) upon this parameter,
for a particular series of models (see figure caption). Along with H
, we have chosen Pa as a
representative subordinate H I line instead of Pa
because, due to strong telluric absorption, the
latter is practically unobservable from Earth.
![[FIGURE]](img69.gif) |
Fig. 5. Dependence upon the parameter of some features of the Na I and H I spectra: the core of Na I and (upper-left panel), H (upper-right panel), Ly (lower-left panel), Pa (lower-right panel). The line profiles correspond to model chromospheres type 1A (see Fig. 1). Note the logarithmic scale for the Ly flux. For each profile, a dashed line indicates the corresponding value on the axis.
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The approximation of CRD, used in our calculations, is poor in the
wings of Ly . Therefore, the Ly
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 : in fact,
it increases approximately as its square. This scaling law can be
understood considering that Ly 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 and
by the complete ionisation of hydrogen. Moreover, in all but the most
active models the Ly 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
(Athay 1972, p. 64), where
is the collisional coupling parameter (see
Eq. 1) and is the slab total optical
thickness at the line center. Both and
depend linearly on density: hence the quadratic
dependence on . The optical depth
depends also on the gradient of the transition
region, but we recall that within each series the models are scaled so
that 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 and
Pa , practically absent from the photospheric
spectrum, show at first a deeper absorption against the photospheric
background, and are eventually quickly driven into emission, at
. 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 tends to be driven
into emission at between -4.4 and -4.2, while
Pa is driven into emission at
. Such a "non-LTE curve of growth" has been
noticed by CM for H , 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 is displayed in Fig. 6 ; the
behaviour of is similar. Moreover, the
photospheric profile is also plotted for comparison.
![[FIGURE]](img76.gif) |
Fig. 6. Line profiles for the most active models of the four series of models with . From top to bottom: Na I , Ly , H , Pa . For the Na I line the purely photospheric profile is also shown (solid line).
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The Ly 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 ) seems to
be the most important parameter (CM, Cram & Mullan 1985, have
investigated such a dependence for H ).
Far more pronounced is the effect of the chromospheric structure on
the 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
with those of models ).
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
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
(the thinnest chromospheres) to models
(the thickest chromospheres). In fact, it is
easy to see the progression from the round-top, gaussian-like profiles
of models , through the flat-top profiles of
models , to the self reversed profiles of models
and .
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 , as in Fig. 6, and for
.
![[FIGURE]](img79.gif) |
Fig. 7. Dependence on the gradient of the transition region. The thicker lines correspond, as in Fig. 6, to ; the thinner lines to . Only the last three models in each series are shown in this figure.
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In the case of Ly , 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
and nearly overlap with
the profiles with and .
This fact fits in the picture sketched above of a collisionally
controlled Ly whose main dependence is upon the
temperature gradient (through the total optical depth of the emitting
slab, ), 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 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
line (and , 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, , at each wavelength,
, and cosine of the angle with the normal to the
surface, µ, is defined so that:
![[EQUATION]](img84.gif)
where x is a depth coordinate. With the choice
, the contribution function becomes:
![[EQUATION]](img86.gif)
where is the density,
is the emissivity and 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 and, by extension, all H I
subordinate lines. As an example, we consider an active model
( ) with a chromosphere type 2A and
. In this case the contrast between H
and 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
emission, even for this very active model, comes from the lower
transition region.
![[FIGURE]](img92.gif) |
Fig. 8. Contribution functions, , for intensity emerging at , as functions of wavelength and depth, for the core of (left) and H (right), in the model atmosphere with chromosphere type 2A, and . In each panel, the contribution function is also shown with a contour plot; on the same graph the locus is drawn with a solid, thick line.
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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]](img94.gif) |
Fig. 9. core profiles for (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.
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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 will
propagate through the recombination chain to .
Fig. 10 also shows how the changes in the sodium ionisation
equilibrium can reach the formation region of the
core, if marginally.
![[FIGURE]](img96.gif) |
Fig. 10. Fractional abundances of (solid lines) and (dashed lines), in the case of a model chromosphere type 2A, and . 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 line.
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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 of Eq. 1. Even if
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
(Athay 1972, p. 48). Notably, the surface value
of the source function (closely related to the line-center emerging
intensity) will be proportional to . The
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,
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 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 line (Fig. 11). An
inspection of the left-hand panel of Fig. 11 reveals that in
reality the ratio , 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 . In fact, since the
region of formation of the line, i.e. the region where
is set, lies quite deep in the atmosphere, far
more substantial XUV fluxes would be required to have an effect this
way.
![[FIGURE]](img101.gif) |
Fig. 11. Run of the source function and of the Planck function at the central wavelength of the 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.
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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]](img103.gif) |
Fig. 12. Effect of collisions with hydrogen on the D profiles obtained with model chromosphere type 2A, and . 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.
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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]](img105.gif) |
Fig. 13. Run of the source function and of the Planck function at the central wavelength of the 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.
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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]](img107.gif) |
Fig. 14. The D doublet (upper panels), H (lower-left panel) and Pa (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, and .
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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 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
and Pa in Fig. 14 is
sufficient to illustrate this point.
© European Southern Observatory (ESO) 1997
Online publication: June 30, 1998
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