## 3. ModellingADB have presented a grid of 72 chromospheric and transition region
(TR) models in which the photospheric base was computed with
PHOENIX (Allard & Hauschildt 1995) and is
representative of a dM0 star (K,
, ). The grid explores a
wide range in chromospheric pressure from low to high, which
corresponds to the range in observed chromospheric activity level from
low (dM stars) to high (dMe stars). The grid also explores two values
of the chromospheric thickness (or, equivalently, the mean
chromospheric temperature gradient), two different functional forms of
the chromospheric temperature variation with column mass density, and
two different values of the TR thickness. The models of their Series
and (chromospheric
constant) are shown in Fig. 1. ADB
computed non-LTE H and Na I
SD recomputed the PHOENIX line opacity spectrum for
a small sub-grid of six models within Series
and of the grid of ADB. This opacity
calculation differed from that of ADB in that the line opacity was
calculated throughout the entire atmospheric model using the
chromospheric and TR temperature structure as input to the opacity
calculation. As a result, the background line blanketing opacity is
consistent with the chromospheric/TR temperature structure of the
outer atmosphere. SD found that in a chromospheric model the line
blanketing opacity initially SD used the MULTI non-LTE radiative transfer code
(Carlsson 1986) to recompute the non-LTE H I spectrum
with their complete line blanketing opacity included in the
calculation and compared the resulting line profiles with those
resulting from the adoption of the line blanketing of ADB. They found
that for the most active (dMe) models, the complete treatment of line
blanketing opacity is necessary to correctly model the
Ly and H lines. Figs. 5
through 13 show the computed H profiles, and
Table 4 gives the values of the predicted
H lines and the
We have also used MULTI to recompute the non-LTE Na I spectrum using the model atom that was described by ADB, but with our complete line blanketing included in the background opacity. Figs. 6 through 14 show the computed Na I profiles. ADB and SD contain extensive detailed discussions of the effect of different line blanketing treatments on the calculated non-LTE H I and Na I spectra. Here we will confine our discussion to the comparison with the observed spectra. ## 3.1. Dependence on stellar parametersThe reported values of found in the literature for our program stars span the range from 3270 K (Gl 388) to 3870 K (Gl 494). The value of for these stars has not been measured and previous investigators normally adopt values around 5.0 on the basis of spectral type when modelling the atmosphere. The actual value is not likely to differ from this by more than dex if these stars have a luminosity class of type V . Our photospheric base model has and values of 3700 K and 4.7, respectively. Therefore, we investigate the sensitivity of the predicted line profiles to variation in the stellar parameters by computing the H I and Na I spectra for models with equal to 4.5 and equal to 3400 and 3900 K, and for models with equal to 3700 K and equal to 4.0 and 5.0. The grid of models used in the perturbation analysis is shown in Fig. 2. For each set of stellar parameters we have attached the chromospheric/TR structure of Series with the lowest and highest value of the chromospheric pressure. We then compute line profiles for the grid. In this grid of models we have held the value of fixed. Therefore, because the temperature structure below is different for models with different values of the stellar parameters, the value of is necessarily different in each of these models. The total range in throughout the grid is about 400 K. It is not possible to hold both and fixed when attaching chromospheric structures to radiative equilibrium photospheric models with different temperature structures. We have chosen to construct a perturbation analysis grid in which rather than is held constant, and some of the variation in the predicted H I and Na I spectra will be due to the variation is as well as the variation of and .
The results of this perturbation analysis can be seen in Figs. 3 and 4. The blanketed line profile have been approximately normalized in relative flux by a single point division by the calculated value of the continuum flux at the wavelength of H. With the expanded relative flux scale of the left panel in Fig. 3 we can see the difference in the shape of the pseudo-continuum due to the different line opacity distributions for models of different parameters.
## 3.1.1. Low pressure chromosphere
For the low pressure chromospheric models of the left panel of Fig. 3, H shows a significant dependence: the absorption line is almost undetectable in the K model and is broader and stronger by about 0.05 in relative flux in K model. We note that the computed H line is blended with the background line blanketing opacity. Therefore, some of the dependence may be due to differences in the background line opacity at the wavelength of H, rather than to changes inherent in the H transition itself. The relative line strength is almost identical in the models with equal to 4.5 and 5.0, but is noticeable weaker in the model with equal to 4.0. However, comparing the difference between the profiles for the models of varying and of low chromospheric pressure in Fig. 3 to the difference between models of varying chromospheric pressure in Fig. 5, we can see that in the regime of low chromospheric pressure the change in the line profile due to variation in the stellar parameters is significantly less than the change due to a step in chromospheric pressure in the chromospheric/TR grid.
The left panel of Fig. 4 shows that the model with equal to 3400 K and that with equal to 4.0 have inner wings that are about 0.025 brighter, and the model with equal to 3900 K, and that with equal to 5.0, have inner wings that are about 0.025 darker, than the fiducial model. At the same time, all models have almost identical central core profiles. Therefore, there is a slight dependency of the inner wing-to-core contrast on the stellar parameters. From comparison with the low pressure synthetic line profiles in Fig. 6 (those with ), we note that the dependency of the inner wing-to-core contrast on stellar parameters is of the same size as the dependency on the location of .
## 3.1.2. High chromospheric pressure
The right panel of Fig. 3 shows that varying stellar parameters have a large effect on the predicted strength of H when it is in emission. Changing from 3700 K to 3400 K, or increasing from 4.5 to 5.0 approximately doubles the flux at line center, , and the equivalent width, . This change is equivalent to increasing the value of in the chromospheric grid by 0.3 dex in the high pressure regime where H is in emission. The dependency, in which emission strength relative to the local continuum varies inversely with may be partially understood as a contrast effect in which an emission line that forms in a fixed chromospheric/TR structure is being seen against a photospheric background of varying brightness temperature. However, a proper understanding of the dependency would require a detailed analysis of radiative transfer quantities such as intensity contribution functions and monochromatic source function throughout the line profile and adjacent continuum, as has been done in the case of chromospheric H I line formation by Short & Doyle (1997). The results of the perturbation study in the regime of high chromospheric pressure place severe limitations on the accuracy of chromospheric modelling of dMe stars with the H line.
The right panel of Fig. 4 shows that, as in the case of H, modest variation in and changes and by approximately a factor of two or more. As with H, a reduction of or an enhancement of causes a dramatic increase in the emission line contrast with the local continuum. The general observation made above for H I holds for Na I ; a proper understanding of the difference in line profile in different models requires an in depth radiative transfer analysis such as that provided for the chromospheric Na I spectrum by ADB.
The results of the perturbation study in the regime of high
chromospheric pressure place severe limitations on the accuracy of
chromospheric modelling of dMe stars with the H
and Na I If we attempt to fit either H or the
Na I If we attempt to fit the absorption core of the Na I
© European Southern Observatory (ESO) 1998 Online publication: July 20, 1998 |