## 3. Results and discussion## 3.1. Chromospheric line blanketingFigs. 2 and 3 show the ratio of to the
total
## 3.2. Radiative transfer in hydrogenFig. 4 shows the emergent flux, , in the
Lyman and Balmer continua as computed by MULTI for the lowest and
highest pressure models in the series, with
only, with , and with
. The Lyman jump is strongly in emission in the
highest pressure model. The H I lines have been left
out of 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
obscures the corresponding jumps in the
distribution. In the region just to the blue of
the Balmer jump, the inclusion of lowers
, which is consistent with the expected behavior
of a blanketed radiation field. However, at still shorter wavelengths
in the Balmer continuum, has the opposite
effect and causes to be
Fig. 5 shows the mean intensity, , the monochromatic background intensity source function, , and the intensity contribution function, , for an angle near disk center (), at two wavelengths on the Balmer continuum, for the cases of only, and values. The Planck function, , is also shown. For , in both models, the inclusion of causes to be reduced throughout most of the atmosphere, as expected. However, the effect of on is controlled by the condition that at the depths where is maximal, and the inclusion of causes the peak of to move outwards. Because peaks well below , the inclusion of causes to form at depths where is lower, and, hence, is reduced.
Further along the Balmer continuum, at , the
depth distribution of is weighted more toward
chromospheric depths, where . The inclusion of
raises the chromospheric part of
and shifts to shallower
depths above where , and
hence , is larger. Furthermore, the inclusion of
raises in the upper
chromosphere. Both of these effects cause to be
increased. The increase in in the upper
chromosphere in the case of is partly due to a
very small Fig. 6 shows the same quantities as Fig. 5, but for
. The situation differs from that of the Balmer
continuum in that 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,
throughout almost the entire atmosphere.
However, in the lowest pressure model, in the
uppermost part of the chromosphere where is
maximal. This slight departure from LTE allows the value of
to influence , and
furthermore, this figure shows that in the
upper chromosphere is
## 3.3. The and density structureThe left panel of Fig. 7 shows the radiative rate
Fig. 8 shows the and population densities normalized by the total H population density of the lowest and highest pressure models in the series for the cases and . Fig. 9 show the corresponding population density and also includes the case of . The effect of including is most pronounced in the lowest pressure model where it reduces by as much as dex at some depths in the chromosphere. The effect of is less pronounced in the highest pressure model. As expected, the decrease in in the chromosphere caused by is mirrored by a corresponding decrease in . However, the effect on the H I /H II balance due to persists deeper into the atmosphere by over two decades in column mass density than the effect on . 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 density in the upper chromosphere. As a corollary, we conclude that it is necessary to determine the effect of on the ionization equilibria of metals that are important electron donors in the lower chromosphere and region, as well as on the H I /H II balance, in order to properly assess the sensitivity of the density throughout the entire chromosphere to the opacity treatment.
The dotted line shows the density resulting from a calculation with . The neglect of the chromospheric component of the line blanketing causes to be underestimated by dex near the top of the chromosphere in the lowest pressure model. In the highest pressure model, the two treatments of produce densities that differ negligibly. Any differences in the final and 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 density normalized by the total H population ( /(H I H II)). Noting that the axis scale is necessarily more compressed in the plot, Fig. 9 shows that the differences in between the cases with and without are almost entirely due to direct radiative transfer effects on the H I /H II balance. The reduction in and values due to 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 on both the values and the radiative rates is rather smaller, which corresponds to the smaller effect on the and densities. ## 3.4. The hydrogen spectrum## 3.4.1. LyFig. 10 shows the Ly flux profile for
our entire grid with only and with
. Also shown are line profiles for the
series with . The
computed flux level of the emission peaks and the central reversal for
all the models is negligibly affected by the inclusion of
and by the particular treatment of
. Therefore, the absolute brightness of the line
near may be used as an accurate chromospheric
and transition region diagnostic without the inclusion of
. However, the inclusion of
causes the computed flux level of the continuum
in the region of Ly to be
The increase in the local continuum level is reduced to the point of being negligible in the case of . The effect of including in the photosphere, but neglecting it in the chromosphere, can be understood from a consideration of Figs. 5 and 6. For in the range 912 to , arises largely at chromospheric depths. Therefore, neglects line blanketing in the part of the atmosphere where the background at is forming. Therefore, if the relative brightness of the line core with respect to the local continuum, or the equivalent width, , is to be used as a diagnostic, then the accuracy will be affected by the treatment of background opacity. Table 2 gives the values for the models of the series with the various opacity treatments. The value of is reduced by as much as by in the case of the highest pressure model. This large change in is difficult to see by visual inspection of Fig. 10. However, a dex increase in the background corresponds to about a factor of five increase in linear flux units. Because the level of the line core does not change significantly when is added, the total area of the emission above continuum decreases by an amount that is approximately proportional to the increase in background .
## 3.4.2. HFig. 11 shows the computed H profiles for the entire grid with only, and with . Also shown are line profiles for the series only with . Table 3 shows the values for the models in the series with the various opacity treatments. Comparison of the left and right panels of Fig. 11 shows that inclusion of 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 is with respect to their respective continua.
Historically, the morphology of H 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 , but the 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 is negligible. Therefore, H modelling of dMe stars in particular must incorporate in order to be accurate. The profiles of all the models are negligibly affected by the choice between and . The Ly to H 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
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 to H
flux ratio is reduced by
to in the case of line blanketing. The flux
ratio is only marginally affected by the choice between
and . 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 H
flux to be too small by a factor of . Doyle
## 3.4.3. PaRecently, 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 () 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 in our models. Fig. 12 shows the computed Pa flux profiles with and without line blanketing. Also shown are line profiles for the series with photospheric blanketing only. The lowest pressure models are the ones with almost non-existent absorption. Table 5 shows the values for the various cases. Fig. 12 shows that, for the highest pressure models, Pa is much more sensitive to change in the chromospheric (or, equivalently, the location of ) than either Ly or H . Therefore, this line provides a valuable additional constraint for semi-empirical models of dMe stars in particular.
The effect of background opacity treatment on the computed line profiles is marginal. The largest relative change is in the lowest pressure models where reduces by . The decrease in sensitivity of the H I spectrum to the inclusion and treatment of as increases is consistent with the general decrease in spectral line blanketing as increases. ## 3.5. Chromospheric energy budgetOne of the important conclusions of Houdebine Table 6 shows the relative contribution to the total excess of the
total H I line series up to and including the series of
, and of the total H I continuum
emission up to and including the B-F continuum of the
level. We confirm, qualitatively, the result
of Houdebine
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 and H line profiles
indicate that there is some chromospheric self-absorption in these
transitions and that, therefore, at
. Therefore the cooling rate for a particular
transition computed here and in Houdebine 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 continuaThere are at least two important limitations in the computation of
the emergent UV flux in both this work and in that of Houdebine For now, we have made an initial estimate of the importance of metallic non-LTE departures with respect to the Sun in determining 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 series), and interpolating the VAL values for the five metals listed above onto our model. We then recomputed with the VAL values incorporated into the calculation of the background opacity. There are at least two sources of error in this procedure: 1) The VAL 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 values, and 2) we are only taking into account the non-LTE departures of the level populations; the source function, , 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 is sensitive to the metallic values as compared to the solar case. The left panels in Fig. 14 show with in LTE and with the VAL 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 blanketing is excluded and included, respectively. Careful examination of the upper panel shows that the 1000 to 1200 A region ( to 3.1 in the figure) is more affected by the metallic 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 reduction found for the VAL solar model. For both the VAL model and our dM star model, throughout most of the photosphere and lower chromosphere is dominated by C I around A and by C I and Si I around A, and in both cases the effect of non-LTE departures is to enhance the contribution of both metals so that they provide of the opacity. The difference in the behavior of 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 behavior between the two models is unwarranted at this time. Inspection of the right panel shows that the suppression due to non-LTE effects is reduced to negligibility by the inclusion of line opacity. Also, we find an enhancement in due to non-LTE departures in the to 3.2 region. At A the dominant contributor changes from Si I to a combination of Si I, Mg I, and Al I as increases. These metals contribute close to of the opacity in both the LTE and non-LTE case. Once again, an explanation of the excess 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.
## 3.6.2. Background line scatteringThe second limitation in this work is that the line blanketing opacity, , 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 artificially close in value to in the outer layers where the UV continuum is forming. In the case of coherent isotropic scattering the line source function, , is approximately given by and the extent to which a line is scattering rather than absorbing
is determined by the relative weights of where is in eV and for Fe I lines and for ionic lines. Furthermore, the line scattering albedo, , is equal to . We have modified our treatment of by adding
the quantity to the total scattering opacity
and to the total thermal opacity at each
depth. The purely thermal treatment of that
was described in the previous sections corresponds to the case of
. Our 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
For , , as computed
by the approximate formula, is very low ()
throughout most of the model above the deep photosphere. As a result,
is almost unity and
for 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
, is reduced by
dex with respect to the thermal case.
Höflich
(1995 ) has suggested that the value of 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 The 0.2 dex decrease in the UV flux due to line scattering with
either extremal value of The extent to which the non-LTE departures discussed above affect 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 coloursHoudebine
Amado & Byrne (1997 ) have analysed de-reddened two-colour diagrams in the Johnson system for a large sample of late-type stars with in the range 0.2 to 2.2. They have found that stars classified as "active" on the basis of their observed H profile, on average, have a colour that is 0.042 magnitudes bluer, with , 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 dependence on chromospheric activity at the centimagnitude level found in our calculations. © European Southern Observatory (ESO) 1997 Online publication: April 20, 1998 |