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Astron. Astrophys. 336, 613-625 (1998)

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3. Modelling

ADB 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 ([FORMULA]K, [FORMULA], [FORMULA]). 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 [FORMULA] and [FORMULA] (chromospheric [FORMULA] constant) are shown in Fig. 1. ADB computed non-LTE H[FORMULA] and Na I D profiles for this grid. Their calculations included line blanketing opacity, computed with PHOENIX , due to [FORMULA] spectral lines of atoms, diatomic molecules, and [FORMULA]. However, this line blanketing opacity was only computed for the photospheric temperature structure below [FORMULA]. The line opacity was assumed ad hoc to approach zero within a decade of logarithmic column mass density above [FORMULA].

[FIGURE] Fig. 1. Temperature structure of models. Upper panel: [FORMULA] series, lower panel: [FORMULA] series. Solid line: [FORMULA], dotted line: [FORMULA], dashed line: [FORMULA], dot-dashed line [FORMULA], dot-dot-dot-dashed line: [FORMULA], long dashed line: [FORMULA], diamonds: [FORMULA], triangles: [FORMULA], squares: [FORMULA]

SD recomputed the PHOENIX line opacity spectrum for a small sub-grid of six models within Series [FORMULA] and [FORMULA] 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 increases with declining column mass density in the low chromosphere above [FORMULA], in sharp contrast to the assumption of monotonically decreasing chromospheric line opacity made by ADB. The complete, self-consistent line blanketing opacity has now been computed for the eighteen models presented here using the procedure described in SD.

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[FORMULA] and H[FORMULA] lines. Figs. 5 through 13 show the computed H[FORMULA] profiles, and Table 4 gives the [FORMULA] values of the predicted H[FORMULA] lines and the FWHM values for those models that have H[FORMULA] in emission. A comparison of the computed [FORMULA] values with those for the sub-grid treated by SD shows discrepancies, with the [FORMULA] values given here being variously smaller or larger than the previous value by as muxh as [FORMULA]. The reason for the discrepancy is an improved treatment of the optical depth grid spacing in the calculation of the statistical equilibrium and radiative transfer problem for Hydrogen. Negative [FORMULA] values indicate net absorption.


[TABLE]

Table 4. Computed H[FORMULA] line: [FORMULA] and FWHM (emission) in [FORMULA]


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 [FORMULA] 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 parameters

The reported values of [FORMULA] found in the literature for our program stars span the range from 3270 K (Gl 388) to 3870 K (Gl 494). The value of [FORMULA] 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 [FORMULA] dex if these stars have a luminosity class of type V . Our photospheric base model has [FORMULA] and [FORMULA] 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 [FORMULA] equal to 4.5 and [FORMULA] equal to 3400 and 3900 K, and for models with [FORMULA] equal to 3700 K and [FORMULA] 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 [FORMULA] 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 [FORMULA] fixed. Therefore, because the temperature structure below [FORMULA] is different for models with different values of the stellar parameters, the value of [FORMULA] is necessarily different in each of these models. The total range in [FORMULA] throughout the grid is about 400 K. It is not possible to hold both [FORMULA] and [FORMULA] 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 [FORMULA] rather than [FORMULA] is held constant, and some of the variation in the predicted H I and Na I spectra will be due to the variation is [FORMULA] as well as the variation of [FORMULA] and [FORMULA].

[FIGURE] Fig. 2. Grid of models with low ([FORMULA]) and high ([FORMULA]) chromospheric pressure that explores a range of photospheric stellar parameters. Solid line: [FORMULA] K, [FORMULA] (model closest to that used for entire grid), dotted line: [FORMULA] K, [FORMULA], dashed line: [FORMULA] K, [FORMULA], dot-dashed line [FORMULA] K, [FORMULA], dot-dot-dot-dashed line: [FORMULA] K, [FORMULA].

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[FORMULA]. 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.

[FIGURE] Fig. 3. Synthetic H[FORMULA] profile for models of fixed chromospheric/TR structure and varying stellar parameter. Left panel: models with chromosphere/TR structure of low pressure ([FORMULA]), right panel: models with chromosphere/TR structure of high pressure ([FORMULA]). Vertical lines in left panel indicate the position of H[FORMULA]. Solid line: [FORMULA] K, [FORMULA] (model closest to that used for entire grid), dotted line: [FORMULA] K, [FORMULA], dashed line: [FORMULA] K, [FORMULA], dot-dashed line [FORMULA] K, [FORMULA], dot-dot-dot-dashed line: [FORMULA] K, [FORMULA].

[FIGURE] Fig. 4. Synthetic Na I D profile for models of fixed chromospheric/TR structure and varying stellar parameter. See Fig. 3 caption.

3.1.1. Low pressure chromosphere

H[FORMULA]

For the low pressure chromospheric models of the left panel of Fig. 3, H[FORMULA] shows a significant [FORMULA] dependence: the absorption line is almost undetectable in the [FORMULA] K model and is broader and stronger by about 0.05 in relative flux in [FORMULA] K model. We note that the computed H[FORMULA] line is blended with the background line blanketing opacity. Therefore, some of the [FORMULA] dependence may be due to differences in the background line opacity at the wavelength of H[FORMULA], rather than to changes inherent in the H[FORMULA] transition itself. The relative line strength is almost identical in the models with [FORMULA] equal to 4.5 and 5.0, but is noticeable weaker in the model with [FORMULA] equal to 4.0. However, comparing the difference between the profiles for the models of varying [FORMULA] and [FORMULA] 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.

[FIGURE] Fig. 5. Gl 212, H[FORMULA]. Left panel: models of [FORMULA] series, right panel: models of [FORMULA] series. Crosses: observed spectrum. Models in order of increasing chromospheric pressure as measured by [FORMULA]: solid line: [FORMULA], dotted line: [FORMULA], dashed line: [FORMULA], dot-dashed line [FORMULA], dot-dot-dot-dashed line: [FORMULA], long dashed line: [FORMULA], diamond line: [FORMULA], triangle line: [FORMULA], square line: [FORMULA]

Na ID

The left panel of Fig. 4 shows that the model with [FORMULA] equal to 3400 K and that with [FORMULA] equal to 4.0 have inner wings that are about 0.025 brighter, and the model with [FORMULA] equal to 3900 K, and that with [FORMULA] 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 [FORMULA]), 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 [FORMULA].

[FIGURE] Fig. 6. Gl 212, Na I [FORMULA]. See Fig. 5.

3.1.2. High chromospheric pressure

H[FORMULA]

The right panel of Fig. 3 shows that varying stellar parameters have a large effect on the predicted strength of H[FORMULA] when it is in emission. Changing [FORMULA] from 3700 K to 3400 K, or increasing [FORMULA] from 4.5 to 5.0 approximately doubles the flux at line center, [FORMULA], and the equivalent width, [FORMULA]. This change is equivalent to increasing the value of [FORMULA] in the chromospheric grid by 0.3 dex in the high pressure regime where H[FORMULA] is in emission. The [FORMULA] dependency, in which emission strength relative to the local continuum varies inversely with [FORMULA] 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[FORMULA] line.

Na ID

The right panel of Fig. 4 shows that, as in the case of H[FORMULA], modest variation in [FORMULA] and [FORMULA] changes [FORMULA] and [FORMULA] by approximately a factor of two or more. As with H[FORMULA], a reduction of [FORMULA] or an enhancement of [FORMULA] 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.

Application to this study

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[FORMULA] and Na I D line. Unless the fundamental stellar parameters are known accurately, the closest fit chromospheric/TR structure is not well constrained. A proper analysis should employ a grid that spans a range of [FORMULA] and [FORMULA] values as well as a range of chromospheric/TR parameters. However, due to the enormous computational effort required to compute complete line blanketing opacity for a unified chromospheric/TR structure, we hold [FORMULA] and [FORMULA] fixed at 3700 K and 4.7 throughout the chromospheric modelling of individual stars and use the results of the perturbation analysis as a guide to the limitations on the accuracy of the modelling. Most of the stars in our sample do not have reported abundance measurements. Therefore, for the same reason of computational expediency, we have held [FORMULA] fixed at 0.0.

If we attempt to fit either H[FORMULA] or the Na I D core of an active (dMe) star that has a value of [FORMULA] that is higher than that of our model (3700 K), then the model will predict emission cores that are too bright with respect to the adjacent continuum (too contrasty) for a given value of [FORMULA]. Noting the dependence of the predicted H[FORMULA] profile on [FORMULA] shown in Fig. 9, we see that the inaccuracy in [FORMULA] will mimic the effect of a larger value of [FORMULA], which will be compensated for by reducing the value of [FORMULA] to achieve a close fit. As a result, the inferred value of [FORMULA] will be too small, and will be a lower limit to the actual value. Similarly, the closest fit value of [FORMULA] will be an upper limit in the case where we fit a star with a value of [FORMULA] lower than that of our model.

If we attempt to fit the absorption core of the Na I D line in the spectrum of a low activity (dM) star that has a value of [FORMULA] that is higher than 3700 K, then the model will predict a line profile in which the contrast between the inner wing and Doppler core is too large. By noting the dependence of the predicted line profiles in Fig. 6, we see that the inaccuracy in [FORMULA] will mimicking the effect of lower [FORMULA]. This will be compensated for by raising the value of [FORMULA] in the model to achieve a good fit. Therefore, the value of [FORMULA] derived from the fit will be too large,and will, thus, be an upper limit.

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© European Southern Observatory (ESO) 1998

Online publication: July 20, 1998
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