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Astron. Astrophys. 326, 287-299 (1997)

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3. Results and discussion

3.1. Chromospheric line blanketing

Figs. 2 and 3 show the ratio of [FORMULA] to the total continuous mass absorption, [FORMULA], in the 500 to [FORMULA] range for the lowest and highest pressure models in the [FORMULA] series, respectively. The dashed lines indicate the location of [FORMULA]. The value of [FORMULA] reaches a local minimum near [FORMULA], then begins to rise again with decreasing m in the lower chromosphere. This mirrors the rise of [FORMULA] with increasing m in the upper photosphere just below [FORMULA]. In the upper chromosphere the higher temperatures dissociate molecules and ionize many metals leading to a decrease in [FORMULA] in this wavelength regime, and finally an abrupt decline in the transition region. The initial rise in [FORMULA] above [FORMULA] differs from the ad hoc gradual ramping down of [FORMULA] above [FORMULA] in the calculations of ADB. Therefore, their treatment of transitions that form in the upper chromosphere and transition region are based on background opacities that are significantly under-estimated.

[FIGURE] Fig. 2. Ratio of line to continuous absorption opacity, [FORMULA], computed by PHOENIX for the lowest pressure model in [FORMULA] series. The dashed lines show the position of [FORMULA].

[FIGURE] Fig. 3. Same as Fig. 2 for the highest pressure model in [FORMULA] series.

3.2. Radiative transfer in hydrogen

Fig. 4 shows the emergent flux, [FORMULA], in the Lyman and Balmer continua as computed by MULTI for the lowest and highest pressure models in the [FORMULA] series, with [FORMULA] only, with [FORMULA], and with [FORMULA]. The Lyman jump is strongly in emission in the highest pressure model. The H I lines have been left out of [FORMULA] because they are treated in detail in the MULTI calculation. All the important b-f continua of metals that are treated by the Uppsala Opacity package that accompanies MULTI have been included in all calculations. However, the presence of [FORMULA] obscures the corresponding jumps in the [FORMULA] distribution. In the region just to the blue of the Balmer jump, the inclusion of [FORMULA] lowers [FORMULA], which is consistent with the expected behavior of a blanketed radiation field. However, at still shorter wavelengths in the Balmer continuum, [FORMULA] has the opposite effect and causes [FORMULA] to be larger. For the lowest pressure model, this behavior extends to the blue side of the Lyman jump, where [FORMULA] is larger by [FORMULA] dex in the case of [FORMULA]. The main qualitative differences between the cases of [FORMULA] and [FORMULA] are that the [FORMULA] predicts larger [FORMULA] in the 1500 to [FORMULA] range of the Balmer continuum, and lower [FORMULA] in the Lyman continuum, compared to [FORMULA] values.

[FIGURE] Fig. 4. The [FORMULA] distribution for the radiative equilibrium model (thick lines), and chromospheric models of lowest (top panel) and highest (bottom panel) pressure in the [FORMULA] series. Solid line: [FORMULA] only; dashed line: [FORMULA] ; dotted line: [FORMULA].

Fig. 5 shows the mean intensity, [FORMULA], the monochromatic background intensity source function, [FORMULA], and the intensity contribution function, [FORMULA], for an angle near disk center ([FORMULA]), at two wavelengths on the Balmer continuum, for the cases of [FORMULA] only, and [FORMULA] values. The Planck function, [FORMULA], is also shown. For [FORMULA], in both models, the inclusion of [FORMULA] causes [FORMULA] to be reduced throughout most of the atmosphere, as expected. However, the effect of [FORMULA] on [FORMULA] is controlled by the condition that [FORMULA] at the depths where [FORMULA] is maximal, and the inclusion of [FORMULA] causes the peak of [FORMULA] to move outwards. Because [FORMULA] peaks well below [FORMULA], the inclusion of [FORMULA] causes [FORMULA] to form at depths where [FORMULA] is lower, and, hence, [FORMULA] is reduced.

[FIGURE] Fig. 5. Various radiative transfer quantities for [FORMULA] (top panels) and [FORMULA] (lower panels) for the lowest (left panels) and highest (right panels) pressure models in the [FORMULA] series. The thin solid line, thick solid line, and thin dotted line are [FORMULA], [FORMULA], and [FORMULA], respectively, for [FORMULA], for the case of [FORMULA] only. The thin short-dashed line, thick short-dashed line, and thick dotted line are the same quantities for the case of [FORMULA]. The long-dashed line is [FORMULA].

Further along the Balmer continuum, at [FORMULA], the depth distribution of [FORMULA] is weighted more toward chromospheric depths, where [FORMULA]. The inclusion of [FORMULA] raises the chromospheric part of [FORMULA] and shifts [FORMULA] to shallower depths above [FORMULA] where [FORMULA], and hence [FORMULA], is larger. Furthermore, the inclusion of [FORMULA] raises [FORMULA] in the upper chromosphere. Both of these effects cause [FORMULA] to be increased. The increase in [FORMULA] in the upper chromosphere in the case of [FORMULA] is partly due to a very small increase in [FORMULA] at these depths. Normally one expects line blanketing to decrease the value of [FORMULA]. However, [FORMULA] will generally include many lines that form at depths above [FORMULA], and that have line source functions that are dominated by the thermal contribution ([FORMULA]). At [FORMULA] and [FORMULA] K, [FORMULA] is very sensitive to temperature, and at the chromospheric depths where these lines form, we may have the condition: [FORMULA]. In other words, the chromospheric temperature inversion may drive many of these lines in to net emission, and [FORMULA] may become a net contributor to [FORMULA] in the UV. This is consistent with the greater size of the [FORMULA] increase in the highest pressure model. The resulting increase in [FORMULA] is slight and can only account for a small fraction of the increase in [FORMULA]. The rest of the increase in [FORMULA] in the case of line blanketing is the result of [FORMULA] being treated as a purely thermal source of opacity which, therefore, increases the relative contribution of [FORMULA] to the value of [FORMULA].

Fig. 6 shows the same quantities as Fig. 5, but for [FORMULA]. The situation differs from that of the Balmer continuum in that [FORMULA] is sharply peaked at depths in the uppermost chromosphere and transition region. Because of the much larger value of the continuous opacity in the Lyman continuum, [FORMULA] throughout almost the entire atmosphere. However, in the lowest pressure model, [FORMULA] in the uppermost part of the chromosphere where [FORMULA] is maximal. This slight departure from LTE allows the value of [FORMULA] to influence [FORMULA], and furthermore, this figure shows that [FORMULA] in the upper chromosphere is increased slightly by the inclusion of [FORMULA], as was the case at [FORMULA]. Therefore, [FORMULA], and consequently [FORMULA], are increased by [FORMULA]. For the highest pressure model, [FORMULA] throughout the entire atmosphere, and [FORMULA] is so sharply peaked in the narrow transition region, that the inclusion of [FORMULA] does not affect its location or value. Therefore, [FORMULA] has negligible effect on [FORMULA].

[FIGURE] Fig. 6. Same as Fig. 5, but for [FORMULA]. The lower panels show the same quantities as the upper panels, but on an expanded scale in column mass density.

3.3. The [FORMULA] and [FORMULA] density structure

The left panel of Fig. 7 shows the radiative rate per atom from the [FORMULA] and [FORMULA] states of H I throughout the chromosphere in the same two models. Because the hydrostatic equilibrium population of H I and the statistical equilibrium population of the level populations may both be different for the cases with and without line blanketing, we show in the right panel the radiative rate per volume element. Figs. 5 and 6 show that for the lowest pressure model, in the case where [FORMULA] is included, [FORMULA] at 911 and [FORMULA] is lower throughout the [FORMULA] region and most of the chromosphere. Correspondingly, the radiative ionization rates from [FORMULA] and [FORMULA] are lower throughout the chromosphere.

[FIGURE] Fig. 7. Radiative ionization rates for H I from the [FORMULA] (thin lines) and [FORMULA] (thick lines) states for the lowest (top panel) and highest (bottom panel) pressure models in the [FORMULA] series. Solid line: [FORMULA] only; dashed line: [FORMULA]. Left panel: rates per atom; right panel: rates per unit volume.

Fig. 8 shows the [FORMULA] and [FORMULA] population densities normalized by the total H population density of the lowest and highest pressure models in the [FORMULA] series for the cases [FORMULA] and [FORMULA]. Fig. 9 show the corresponding [FORMULA] population density and also includes the case of [FORMULA]. The effect of including [FORMULA] is most pronounced in the lowest pressure model where it reduces [FORMULA] by as much as [FORMULA] dex at some depths in the chromosphere. The effect of [FORMULA] is less pronounced in the highest pressure model. As expected, the decrease in [FORMULA] in the chromosphere caused by [FORMULA] is mirrored by a corresponding decrease in [FORMULA]. However, the effect on the H I /H II balance due to [FORMULA] persists deeper into the atmosphere by over two decades in column mass density than the effect on [FORMULA]. Unlike solar type stars, hydrogen remains mostly neutral until near the top of the chromosphere, as can be seen from Fig. 8. Therefore, the details of the H I /H II balance only have a significant effect on the [FORMULA] density in the upper chromosphere. As a corollary, we conclude that it is necessary to determine the effect of [FORMULA] on the ionization equilibria of metals that are important electron donors in the lower chromosphere and [FORMULA] region, as well as on the H I /H II balance, in order to properly assess the sensitivity of the [FORMULA] density throughout the entire chromosphere to the opacity treatment.

[FIGURE] Fig. 8. The [FORMULA] (thick lines) and [FORMULA] number density (thin lines) resulting from non-LTE H I solution for the lowest (top panel) and highest (bottom panel) pressure models in the [FORMULA] series. Dotted line: [FORMULA] only; dashed line: [FORMULA].

[FIGURE] Fig. 9. The [FORMULA] (left panel) and [FORMULA] (right panel) population density resulting from the non-LTE H I solution for the lowest (top panel) and highest (bottom panel) pressure models in the [FORMULA] series. Solid line: [FORMULA] only; dashed line: [FORMULA], dotted line: [FORMULA].

The dotted line shows the [FORMULA] density resulting from a calculation with [FORMULA]. The neglect of the chromospheric component of the line blanketing causes [FORMULA] to be underestimated by [FORMULA] dex near the top of the chromosphere in the lowest pressure model. In the highest pressure model, the two treatments of [FORMULA] produce [FORMULA] densities that differ negligibly.

Any differences in the final [FORMULA] and [FORMULA] density that result from different treatments of the opacity may have two sources: 1) the H I /H II balance will differ as a result of different amounts of total opacity being included in the calculation of the radiative rates of the hydrogen transitions, 2) because the hydrostatic equilibrium equation was re-converged separately for each case, the equilibrium density structure differs for each case. In order to determine the relative importance of these two effects, we also show in Fig. 9 the [FORMULA] density normalized by the total H population ([FORMULA] /(H I [FORMULA] H II)). Noting that the [FORMULA] axis scale is necessarily more compressed in the [FORMULA] plot, Fig. 9 shows that the differences in [FORMULA] between the cases with and without [FORMULA] are almost entirely due to direct radiative transfer effects on the H I /H II balance. The reduction in [FORMULA] and [FORMULA] values due to [FORMULA] are consistent with the reduction in radiative photo-ionization rate seen in Fig. 7 and this is consistent with the changes being directly due to radiative transfer effects. For the highest pressure model, the effect of [FORMULA] on both the [FORMULA] values and the radiative rates is rather smaller, which corresponds to the smaller effect on the [FORMULA] and [FORMULA] densities.

3.4. The hydrogen spectrum

3.4.1. Ly [FORMULA]

Fig. 10 shows the Ly [FORMULA] flux profile for our entire grid with [FORMULA] only and with [FORMULA]. Also shown are line profiles for the [FORMULA] series with [FORMULA]. The computed flux level of the emission peaks and the central reversal for all the models is negligibly affected by the inclusion of [FORMULA] and by the particular treatment of [FORMULA]. Therefore, the absolute brightness of the line near [FORMULA] may be used as an accurate chromospheric and transition region diagnostic without the inclusion of [FORMULA]. However, the inclusion of [FORMULA] causes the computed flux level of the continuum in the region of Ly [FORMULA] to be increased by as much as [FORMULA] dex. This increase is consistent with the behavior of the computed broad-band continuum shown in Fig. 4.

[FIGURE] Fig. 10. Ly [FORMULA]   flux profiles. Left panel: [FORMULA] ; right panel: [FORMULA] only. Models in series [FORMULA]: thin dashed line; [FORMULA] series: thick dashed line; the [FORMULA] series only with [FORMULA]: dotted line. The lowest and highest pressure models are the ones that have the weakest and strongest line emission, respectively.

The increase in the local continuum level is reduced to the point of being negligible in the case of [FORMULA]. The effect of including [FORMULA] in the photosphere, but neglecting it in the chromosphere, can be understood from a consideration of Figs. 5 and 6. For [FORMULA] in the range 912 to [FORMULA], [FORMULA] arises largely at chromospheric depths. Therefore, [FORMULA] neglects line blanketing in the part of the atmosphere where the background [FORMULA] at [FORMULA] is forming. Therefore, if the relative brightness of the line core with respect to the local continuum, or the equivalent width, [FORMULA], is to be used as a diagnostic, then the accuracy will be affected by the treatment of background opacity. Table 2 gives the [FORMULA] values for the models of the [FORMULA] series with the various opacity treatments. The value of [FORMULA] is reduced by as much as [FORMULA] by [FORMULA] in the case of the highest pressure model. This large change in [FORMULA] is difficult to see by visual inspection of Fig. 10. However, a [FORMULA] dex increase in the background [FORMULA] corresponds to about a factor of five increase in linear flux units. Because the [FORMULA] level of the line core does not change significantly when [FORMULA] is added, the total area of the emission above continuum decreases by an amount that is approximately proportional to the increase in background [FORMULA].


[TABLE]

Table 2. [FORMULA] of Ly [FORMULA]


3.4.2. H [FORMULA]

Fig. 11 shows the computed H [FORMULA] profiles for the entire grid with [FORMULA] only, and with [FORMULA]. Also shown are line profiles for the [FORMULA] series only with [FORMULA]. Table 3 shows the [FORMULA] values for the models in the [FORMULA] series with the various opacity treatments. Comparison of the left and right panels of Fig. 11 shows that inclusion of [FORMULA] causes the "continuum" to be noticeably depressed and distorted. It is not straight-forward to rectify the computed line profiles to a continuum level of unity because the background radiation is not a true continuum due to the inclusion of line blanketing. Therefore, we have plotted absolute flux and the comparison of line profiles between the calculations with and without [FORMULA] is with respect to their respective continua.

[FIGURE] Fig. 11. H [FORMULA]   flux profiles. Left panel: [FORMULA] ; right panel: [FORMULA] only. Models in series [FORMULA]: thin dashed line; [FORMULA] series: thick dashed line; the [FORMULA] series with [FORMULA]: dotted line. The lowest pressure models are the ones that have the weakest line absorption.

[TABLE]

Table 3. [FORMULA] of H [FORMULA]


Historically, the morphology of H [FORMULA] has been the main diagnostic for classifying dM stars by activity level, and Fig. 11 shows that our grid spans the range of observed activity level from the least active dM(e) stars (profile with almost no absorption) to very active dMe stars (profile with the strongest emission). For the highest pressure model, the flux level of the emission peaks and central reversal are negligibly affected by [FORMULA], but the [FORMULA] value of the net emission above continuum is approximately doubled due to the depression of the continuum. For the models where the profile is in absorption, the effect on [FORMULA] is negligible. Therefore, H [FORMULA] modelling of dMe stars in particular must incorporate [FORMULA] in order to be accurate. The profiles of all the models are negligibly affected by the choice between [FORMULA] and [FORMULA].

The Ly [FORMULA] to H [FORMULA] flux ratio in dMe stars is a diagnostic of the thickness of the transition region (Houdebine & Doyle 1994 ). Table 4 shows the integrated line fluxes and the flux ratio in our most active models with the different treatments of line blanketing. The large dependence of [FORMULA] on background opacity treatment shown in Tables 2 and 3 is proportional to a corresponding change in the background continuum level. Therefore, the total flux in the emission lines has a much weaker dependence. However, the Ly [FORMULA] to H [FORMULA] flux ratio is reduced by [FORMULA] to [FORMULA] in the case of line blanketing. The flux ratio is only marginally affected by the choice between [FORMULA] and [FORMULA]. From Fig. 14 of Houdebine & Doyle (1994 ), we estimate that the neglect of line blanketing would cause an estimate of the transition region thickness based on a fit to Ly [FORMULA] H [FORMULA] flux to be too small by a factor of [FORMULA]. Doyle et al. (1997 ) present observed values of [FORMULA] that have been corrected for the interstellar attenuation of Ly [FORMULA] for a variety of M drawfs. For Gl278C (dM1.0e), they find a value of 3.5, which is in excellent agreement with value calculated here with line blanketing included.


[TABLE]

Table 4. Integrated line flux in Ly [FORMULA] and H [FORMULA] and Ly [FORMULA] H [FORMULA] in dMe (log [FORMULA]) models


3.4.3. Pa [FORMULA]

Recently, large gains have been made in detection technology in the near infrared spectral region. Therefore, the Paschen series of the H I spectrum has the potential to provide a useful diagnostic compliment to the Lyman and Balmer lines. Unfortunately, Pa [FORMULA] ([FORMULA]) lies in a region where telluric contamination is so large that the line is not a useful diagnostic. Therefore, we investigate the behavior of Pa [FORMULA] in our models. Fig. 12 shows the computed Pa [FORMULA] flux profiles with and without line blanketing. Also shown are line profiles for the [FORMULA] series with photospheric blanketing only. The lowest pressure models are the ones with almost non-existent absorption. Table 5 shows the [FORMULA] values for the various cases. Fig. 12 shows that, for the highest pressure models, Pa [FORMULA] is much more sensitive to change in the chromospheric [FORMULA] (or, equivalently, the location of [FORMULA]) than either Ly [FORMULA] or H [FORMULA]. Therefore, this line provides a valuable additional constraint for semi-empirical models of dMe stars in particular.

[FIGURE] Fig. 12. Pa [FORMULA]   flux profiles. Left panel: [FORMULA] ; right panel: [FORMULA] only. Models in series [FORMULA]: thin dashed line; [FORMULA] series: thick dashed line; the [FORMULA] series with [FORMULA]: dotted line. The lowest pressure models are the one with the weakest line absorption.

[TABLE]

Table 5. [FORMULA] of Pa [FORMULA]


The effect of background opacity treatment on the computed line profiles is marginal. The largest relative change is in the lowest pressure models where [FORMULA] reduces [FORMULA] by [FORMULA]. The decrease in sensitivity of the H I spectrum to the inclusion and treatment of [FORMULA] as [FORMULA] increases is consistent with the general decrease in spectral line blanketing as [FORMULA] increases.

3.5. Chromospheric energy budget

One of the important conclusions of Houdebine et al. (1996 ) is that, contrary to what has been assumed previously, chromospheric radiative cooling in the emission continuum is greater than that in the H I emission lines for many chromospheric models. For their series of models with the largest value of [FORMULA] (3000K), they find that the relative contribution of the total H I series to the energy loss is at most [FORMULA]. For their lower [FORMULA] (2660K) models, the continuum contribution is about [FORMULA]. For both the lines and continuum, they define cooling as the excess flux above that from a basal flux model with minimal emission. The model with the lowest total emission (continuum + H I lines) in our grid is the lowest pressure model of the [FORMULA] series. We take this model as a fiducial basal flux star and subtract its H I line spectrum and continuum emission from each of the other models in the grid to produce excess emission values.

Table 6 shows the relative contribution to the total excess of the total H I line series up to and including the series of [FORMULA], and of the total H I continuum emission up to and including the B-F continuum of the [FORMULA] level. We confirm, qualitatively, the result of Houdebine et al. (1996 ) for the higher [FORMULA] models, where we find that the continuum carries over [FORMULA] of the excess flux at high chromospheric pressure. We also confirm the trend shown in their Fig. 7 toward increasing dominance of the continuum as chromospheric pressure decreases. However, our results differ radically from those of Houdebine et al. (1996 ) for lower [FORMULA] models, where we find that the continuum is dominant as in the higher [FORMULA] models. Lines play a minor role in the chromospheric energy loss throughout our entire grid. At this time we are unsure of the reason for the difference in the results for the lower [FORMULA] models. Fig 13 shows the overall distribution of [FORMULA] for the entire grid. The strong response of the blue and UV continuum to increasing chromospheric pressure, combined with the large wavelength range over which the excess flux contributes, accounts for the dominant role of the continuum in the energy balance. Table 6 shows that the inclusion of [FORMULA] has a minor effect on the relative contributions, and shifts the balance even more toward the continuum.


[TABLE]

Table 6. Relative contribution to radiative cooling


[FIGURE] Fig. 13. Overall [FORMULA] distribution for entire grid. Upper panel: [FORMULA] ; lower panel: [FORMULA] only. Models in series [FORMULA]: thin dashed line; [FORMULA] series: thick dashed line; radiative equilibrium model: solid line.

These results are only approximate because the emergent flux in a transition is only equal to the total cooling in that transition if the transition is optically thin. The central double reversal of the Ly [FORMULA] and H [FORMULA] line profiles indicate that there is some chromospheric self-absorption in these transitions and that, therefore, [FORMULA] at [FORMULA]. Therefore the cooling rate for a particular transition computed here and in Houdebine et al (1996 ), especially in the case of a line, is only a lower limit. Nevertheless, the dominance of the continuum in the total loss is so large that it is unlikely to be entirely due to an underestimate in the line cooling.

The amount of non-radiative heating required to energize the chromospheres of late-type stars is a fundamentally important constraint on theoretical heating mechanisms. Traditionally, the amount of heating has been measured by summing the excess flux in emission lines such as the H I Lyman and Balmer series and the Ca II and Mg II resonance doublets. However, the results shown here indicate that the energy loss in these lines may be only a small fraction of the excess flux in the continuum, and, therefore, the total cooling rates are much larger than previously thought.

3.6. Background non-LTE effects

3.6.1. Metallic [FORMULA] continua

There are at least two important limitations in the computation of the emergent UV flux in both this work and in that of Houdebine et al. (1996 ). The first is that the background continuous absorbers have been treated in LTE. In their detailed modelling of the solar atmosphere, Vernazza et al. (1981 ) (VAL) showed that the continuum between 1000 and 1200 A is reduced by about an order of magnitude as a result, largely, of the ground state C I [FORMULA] continuum being out of LTE. VAL tabulate departure co-efficients, [FORMULA], for the first eight levels of C I, Si I, Mg I, and Al I, and for eight combined levels of Fe I, all of which showed large non-LTE departures in their model. The lower excitation energy levels that are more heavily populated have [FORMULA] values that fall as low as [FORMULA] in the [FORMULA] region where the near UV continuum forms, and then rise to values of [FORMULA] in the upper chromosphere. To properly account for non-LTE metallic opacity in our models, we should solve the combined radiative transfer and statistical equilibrium equations for at least these five elements, then re-compute the non-LTE H I /II solution and hydrostatic equilibrium equation with the modified background opacity and metallic [FORMULA] contribution, and iterate this procedure until the [FORMULA] values converge at all depths. This is a massive computational undertaking and is beyond the scope of the current study. However, we intend to perform such a calculation once we have acquired all the necessary atomic data.

For now, we have made an initial estimate of the importance of metallic non-LTE departures with respect to the Sun in determining [FORMULA] in the UV by scaling the chromosphere and photosphere of the VAL solar model to one of our models (the lowest pressure model in the [FORMULA] series), and interpolating the VAL [FORMULA] values for the five metals listed above onto our model. We then recomputed [FORMULA] with the VAL [FORMULA] values incorporated into the calculation of the background opacity. There are at least two sources of error in this procedure: 1) The VAL [FORMULA] values are determined by the particular densities and radiative intensities in the VAL model; a full non-LTE treatment of the metals in our model will certainly yield different [FORMULA] values, and 2) we are only taking into account the non-LTE departures of the level populations; the [FORMULA] source function, [FORMULA], is still equal to the Planck function in our calculation. Therefore, this is not a consistent non-LTE treatment of the background continuous opacity. Nevertheless, it gives us an approximate indication of the extent to which [FORMULA] is sensitive to the metallic [FORMULA] values as compared to the solar case.

The left panels in Fig. 14 show [FORMULA] with [FORMULA] in LTE and with the VAL [FORMULA] values. The upper panel shows the important UV region and the lower panel shows the overall distribution. The left and right panels show the cases where [FORMULA] blanketing is excluded and included, respectively. Careful examination of the upper panel shows that the 1000 to 1200 A region ([FORMULA] to 3.1 in the figure) is more affected by the metallic [FORMULA] values than the nearby regions, with the flux there being suppressed by non-LTE effects. This is in general agreement with the VAL results for the Sun, however, the effect is much less that the order of magnitude [FORMULA] reduction found for the VAL solar model. For both the VAL model and our dM star model, [FORMULA] throughout most of the photosphere and lower chromosphere is dominated by C I around [FORMULA] A and by C I and Si I around [FORMULA] A, and in both cases the effect of non-LTE departures is to enhance the contribution of both metals so that they provide [FORMULA] of the opacity. The difference in the behavior of [FORMULA] between the two models when departures are allowed for must depend on the exact location of the continuum formation in this region in each model. Due to the crudeness of our non-LTE treatment in the M star model, further analysis of the difference in [FORMULA] behavior between the two models is unwarranted at this time. Inspection of the right panel shows that the [FORMULA] suppression due to non-LTE [FORMULA] effects is reduced to negligibility by the inclusion of line opacity.

Also, we find an enhancement in [FORMULA] due to non-LTE departures in the [FORMULA] to 3.2 region. At [FORMULA] A the dominant contributor changes from Si I to a combination of Si I, Mg I, and Al I as [FORMULA] increases. These metals contribute close to [FORMULA] of the opacity in both the LTE and non-LTE case. Once again, an explanation of the excess [FORMULA] in the non-LTE case will require a careful study of where the continuum forms and should wait until the non-LTE treatment of metals in our model is treated properly.

We conclude tentatively that the impact of metallic non-LTE departures on the UV emission in our models is significantly less that found for the Sun by VAL and probably does not effect our conclusion that the continuum emission dominates the chromospheric energy budget. However, a self-consistent solution of the non-LTE problem for hydrogen and all five dominant metals must be undertaken for our models before we can reach a definite conclusion on this point.

[FIGURE] Fig. 14. The effect of various opacity treatments on [FORMULA]. Left panels: [FORMULA] only - LTE treatment (solid line), VAL [FORMULA] values for five metals (dotted line). Right panels: [FORMULA] blanketing - [FORMULA] in LTE (solid line), [FORMULA] with VAL [FORMULA] values (dotted line), [FORMULA] in LTE and a line scattering albedo in [FORMULA] for the case of minimal scattering (long dashed line) and Anderson's atomic scattering (short dashed line). Top panels: UV region. Bottom panels: overall distribution.

3.6.2. Background line scattering

The second limitation in this work is that the line blanketing opacity, [FORMULA], is assumed to be entirely thermal throughout the model. The extensive non-LTE line blanketing calculation of Anderson (1989 ) for the Sun demonstrates that many atomic and ionic lines become scatterers rather than thermal absorbers of radiation in the upper photosphere of a radiative equilibrium model where the gas density becomes relatively low. As a result, our thermal treatment has the effect of keeping [FORMULA] artificially close in value to [FORMULA] in the outer layers where the UV continuum is forming. In the case of coherent isotropic scattering the line source function, [FORMULA], is approximately given by

[EQUATION]

and the extent to which a line is scattering rather than absorbing is determined by the relative weights of J and B in this sum. The line thermalization parameter, [FORMULA], is equal to [FORMULA], where [FORMULA] is the rate of collisional transitions between the upper and lower states, and [FORMULA] is the rate of spontaneous radiative decay from the upper state. The dependence on [FORMULA] makes [FORMULA] depth dependent, and Anderson (1989 ) has suggested the following approximate formula for [FORMULA]:

[EQUATION]

where [FORMULA] is in eV and [FORMULA] for Fe I lines and [FORMULA] for ionic lines. Furthermore, the line scattering albedo, [FORMULA], is equal to [FORMULA].

We have modified our treatment of [FORMULA] by adding the quantity [FORMULA] to the total scattering opacity and [FORMULA] to the total thermal opacity at each depth. The purely thermal treatment of [FORMULA] that was described in the previous sections corresponds to the case of [FORMULA]. Our [FORMULA] tables include all lines from atoms, ions, and molecules added together. Therefore, we are not able to treat lines from different types of absorbers separately. However, we have tried two extreme values of q found in the literature: Anderson's atomic value of [FORMULA], and a value of [FORMULA] that that corresponds to mostly thermal lines with minimal scattering (Avrett et al. 1994 ). The right panels of Fig. 14 show the computed [FORMULA] with [FORMULA] included in the treatment of [FORMULA]. Fig. 15 shows the value of [FORMULA] and [FORMULA] throughout the model for both values of q.

[FIGURE] Fig. 15. Top panel: approximate line thermalization parameter, [FORMULA] at - 926 A (dotted line), 1026 A (dashed line), 1226 A (squares), 3836 A (solid line), 4863 A (thick dashed line). Bottom panels: line scattering albedo, [FORMULA]. Anderson's albedo parameter, q, equal to [FORMULA] (left panels), and [FORMULA] (right panels).

For [FORMULA], [FORMULA], as computed by the approximate formula, is very low ([FORMULA]) throughout most of the model above the deep photosphere. As a result, [FORMULA] is almost unity and [FORMULA] for [FORMULA] is an order of magnitude brighter than that of the thermal case and is indistinguishable from that of an unblanketed model because the lines are barely absorbent. However, for [FORMULA], [FORMULA] is reduced by [FORMULA] dex with respect to the thermal case. Höflich (1995 ) has suggested that the value of [FORMULA] based on the treatment of Anderson (1989 ) may be as much as two orders of magnitude too small, which may account for the underblanketing found with the atomic q parameter at [FORMULA]. At the other extreme, [FORMULA] gives an [FORMULA] distribution that differs negligibly from the purely thermal [FORMULA] for [FORMULA]. For [FORMULA], [FORMULA], is very close to that computed with the much higher q value. This relative insensitivity of the mid-UV flux to the albedo is due to the rapid increase of [FORMULA] with decreasing [FORMULA] and decreasing [FORMULA], which causes [FORMULA] to be [FORMULA] at the depths where radiation with [FORMULA] forms for both values of q.

The 0.2 dex decrease in the UV flux due to line scattering with either extremal value of q is much less than the four orders of magnitude difference between the UV [FORMULA] values for the chromospheric model and the radiative equilibrium model. We conclude, for now, that the chromosphere produces substantial excess continuum emission, even in the presence of blanketing by scattering lines.

The extent to which the non-LTE departures discussed above affect [FORMULA] may depend on the chromospheric pressure. Therefore, these effects will be investigated, more accurately, for the entire grid of models in a future study.

3.7. Photometric colours

Houdebine et al. (1996 ) have found that in the highest pressure chromospheric models, the structure of the outer atmosphere has a detectable effect on the [FORMULA] colour due to the increasing enhancement of [FORMULA] with decreasing [FORMULA] seen in Fig. 13. In Table 7, we show for our grid of models with and without line blanketing the excess flux with respect to the basal model normalized by the flux from the basal model in the Johnson UBV pass-bands. In Table 8, we show the computed [FORMULA] and [FORMULA] colours. For the unblanketed models, the highest pressure model has a value of [FORMULA] that is 0.015 magnitudes lower than that of the lowest pressure model. The changes in [FORMULA] due to the addition of [FORMULA] increase [FORMULA] by [FORMULA] magnitudes and [FORMULA] by [FORMULA] magnitudes. The sensitivity of [FORMULA] to chromospheric pressure is reduced somewhat in the presence of [FORMULA], but is still significant; the difference between the lowest and highest pressure models is reduced to [FORMULA] magnitudes. Therefore, we confirm the result of Houdebine et al. (1996 ) that the chromospheres of the most active stars should be detectable with centimagnitude precision broad band photometry in the violet and blue spectral regions.


[TABLE]

Table 7. Relative flux excess in UBV with respect to basal model (series [FORMULA], log [FORMULA])



[TABLE]

Table 8. Computed colours of grid models


Amado & Byrne (1997 ) have analysed de-reddened two-colour diagrams in the Johnson system for a large sample [FORMULA] of late-type stars with [FORMULA] in the range 0.2 to 2.2. They have found that stars classified as "active" on the basis of their observed H [FORMULA] profile, on average, have a [FORMULA] colour that is 0.042 magnitudes bluer, with [FORMULA], than inactive stars. They caution that the dependence of colour on metallicity has not been accounted for in their study. However, their results suggest that there is a [FORMULA] dependence on chromospheric activity at the centimagnitude level found in our calculations.

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Online publication: April 20, 1998
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