4.1. Individual stars
4.1.1. Gl 212
Alonso (1996) used the Infrared Flux Method and ATLAS9 (Kurucz 1990) models to measure and found 3832 K. He adopted values of and equal to 5.0 and 0.0, respectively. Spectral types of dM1, dM2.5, and dM2 have been reported by Poveda et al. (1994), Eggen (1996), and Rutten et al. (1989), respectively. Measured emission fluxes in well known chromospheric lines such as Ca II HK and Mg II hk can be found, variously, in Panagi & Mathioudakis (1993), Rutten et al. (1989), and Giampapa et al. (1989). In their large catalogue of measured (H) values, Stauffer & Hartmann (1986) give (the negative sign indicates that H is in absorption).
We measure to be -0.40, which is stronger by than the value of Stauffer & Hartmann (1986). From Table 4 we see that the predicted value of of H in absorption first grows, then declines with increasing . Therefore, the closest fit to both of these measured values of could be provided by a model with a value of between -5.6 and -5.2 or between -4.8 and -4.6. The comparison of the observed and computed H line profiles is shown in Fig. 5. We note that while the region to the blue of H is generally well fit by the synthetic spectrum, the region to the red is very poorly fit. The line list that was used to calculate the synthetic spectrum was constructed with the goal of accurately reproducing broad and intermediate band photometric diagnostics (Kurucz 1990), rather than for the detailed fitting of high resolution spectra. Therefore, the line list is pervaded by inaccuracies in the oscillator strengths and transition wavelengths that cause local discrepancies in high resolution fit. In Fig. 5 we see that for a given value of , Series models produce line profiles that are deeper and wider than those of Series . A close fit to the shape of the observed line profile is provided by a model of either series with between -5.6 and -5.2 or a model of Series with , in agreement with the closest fit indicated by the value of . The measured of Gl 212 is hotter than that of our model by K Alonso (1996). On the basis of the perturbation analysis shown in Fig. 3, we expect the line profile to be minimally affected by variation in in the low pressure, absorption line regime.
From Fig. 6 we see that models of Series give rise to line profiles in which the core is wider than those of Series . Also, within each Series, the three models of lowest chromospheric pressure all give rise to line profiles that are almost identical. Therefore, we cannot expect the Na I D line to be an effective discriminator of chromospheric structure among inactive dM stars. The core of the observed line profile is in absorption, which corresponds to the behavior of the models of low chromospheric pressure. However, the synthetic spectra are too depressed in the inner wings and core by as much as in relative flux, despite having been rectified to the observed pseudo-continuum. From the perturbation analysis of Fig. 4, we note that for a model with a value of that is too low we expect the model to predict too much flux in the inner wings. Therefore, the discrepancy cannot be explained by the inaccurate value of in the model. Nevertheless, the shape of the inner wings is approximately fit by models with between -6.0 and -5.2. However, the contrast between the inner wings and core are fit most closely by a model with between -4.8 and -4.6. A model with too low a value of will predict too large an inner wing-to-core contrast, which mimics the effect of lower . Therefore, the value of derived from a fit to the contrast is an upper limit. In any case, models of Series fit the width of the observed line core better than those of Series .
Both lines can be fit approximately with a model of Series with a value of between -5.2 and -5.6. Series models are ruled out by the width of the Na I D line core, and models of Series with between -4.8 and -4.6 are ruled out by the narrowness of the observed H core.
4.1.2. Gl 382
Mathioudakis & Doyle (1991) give K and a spectral type of dM2. Eggen (1996) also reports a spectral type of dM2. Panagi & Mathioudakis (1993), Giampapa et al. (1989), and Mathioudakis & Doyle (1991) report integrated flux values for various chromospheric emission lines. Stauffer & Hartmann (1986) and Mathioudakis & Doyle (1991) give (H) equal to -0.37 and , respectively.
We measure to be , in close agreement with the value of of -0.37 measured by Stauffer & Hartmann (1986), but higher than the value of -0.23 measured by Mathioudakis & Doyle (1991). From Table 4 we see that, like Gl 212, all three of these measured values of indicate either between -5.6 and -5.2, or between -4.8 and -4.6. From Fig. 7, we see that the observed line profile is similar to that of Gl 212, but slightly weaker. Again, the closest fit is provided by a model of either series with between -5.6 and -5.2. However, in this case, models of both series with greater than -5.2 are ruled out by the narrowness of the observed line core. The measured temperature of Gl 382 is almost 300 K lower than that of our model (Mathioudakis & Doyle 1991). From the perturbation analysis of Fig. 3, we expect a model that is too hot to overpredict the strength of H in the low chromospheric pressure regime. Therefore, the values of derived here are lower limits.
From Fig. 8 we see that, for the low pressure models, the model that is fit most closely by the shape of the inner wings and that which is fit most closely by the contrast of the inner wing to the core are even more discrepant than was the case for Gl 212. However, a higher pressure model of Series with between -4.0 and -4.2 provides a close fit throughout the entire inner line profile. We see from Fig. 4 that, among models of lower chromospheric pressure, a model with too large a value of will underpredict the contrast of the inner wing to core, mimicking the effect of higher . Therefore, the value of is a lower limit. However, this dependency is slight and is not able to fully explain how the observed line profile, which has almost no detectable absorption core, could arise from a low pressure model such as that required to fit H.
The closest fit value of found from the Na I D line is over an order of magnitude greater in column mass density than that required to fit H. Indeed, the value of required to fit Na I D gives rise to an H profile that is strongly in emission , whereas the observed H profile is clearly in absorption. Furthermore, the dependency of both line profiles on the value of the stellar parameters cannot account for the entire discrepancy.
4.1.3. Gl 388
Gl 388 (AD Leo) is a well known flare star (Pettersen 1991). Giampapa et al. (1989) and Malyuto et al. (1997) report spectral types of dM4Ve and dM4.5Ve, respectively. Fleming et al. (1995) report solar metallicity, while Naftilan et al. (1992) found from spectrum synthesis. Caillault & Patterson (1990) found . Panagi & Mathioudakis (1993), Rutten et al. (1989), Giampapa et al. (1989), Doyle et al. (1990), Pettersen & Hawley (1989), Herbst & Miller (1989), and Elgaroy et al. (1990) give measured flux values in a variety of chromospheric emission lines including, variously, H, Ly, Ca II HK and Mg II hk. Doyle (1996), Mathioudakis et al. (1995), and Byrne & Doyle (1989) report fluxes in a variety of FUV and EUV lines that form in the TR. Stauffer & Hartmann (1986) and Pettersen & Hawley (1989) give (H) values of and , respectively. The positive value indicates net emission. Stauffer & Hartmann (1986) also give FWHM(H). Young et al. (1989) find a range in the value of excess (H) of 2.57 to . MacMillan & Herbst (1991) found the value of (H) to range from to with . The minimum value of (H) is interpreted as corresponding to a "quiet" photosphere, analogous to the quiet Sun, and the variation in (H) is interpreted to be due to a combination of spots and flares. Dempsey et al. (1993) in their data compilation give km s-1. Marcy & Chen (1992), on the basis of four lines in high resolution spectra derive km s-1.
The most extensive atmospheric modelling effort to date is that of Mauas & Falchi (1996), who performed two component, chromospheric/TR modelling in the approximation of the flaring atmosphere. They fit simultaneously the H, H, Ca I 4227, and Ca II K lines. This followed an earlier modelling effort by Mauas & Falchi (1994) in which they constructed chromospheric/TR models to fit simultaneously the H, H, H, H, Ca I 4227, Ca II K, Ca II 8498, Mg I b, Na I D, and Na I 8183/95 lines, and the optical continuum. Fig. 15 shows the temperature structure of the quiescent model.
We measure to be , which falls between the values of 2.7 and 4.0 found by MacMillan & Herbst (1991) and Pettersen & Hawley (1989), respectively. From Table 4 we see that this range in can be accommodated by a very narrow range of models in the grid as result of the large sensitivity of (H) to changes in . Our observed value and that of MacMillan & Herbst (1991) may be fit by a model of either series with between -4.2 and -4.0. We measure the FWHM of the emission core to be , which is somewhat greater than the value of found by Stauffer & Hartmann (1986). The latter value is narrower than that prediced by any model in either series. However, our value may be approximately fit by a model of Series with equal to -3.8.
The theoretical profiles have been rotationally broadened with a value of equal to 5.6 km s-1 (Marcy & Chen 1992). In Fig. 9 we see that the closest fit model provided by either series has approximately equal to -4.2. The emission profiles of Series are narrower and have less pronounced central self-absorption reversals that those of Series and provide a better fit to the observed profile. The measured value of for Gl 388 is over 400 K less than that of our model (Caillault & Patterson 1990). From Fig. 3 we see that high pressure chromospheric models that are too hot will predict emission lines that are too weak. Therefore the values of that are found by our H fitting are upper limits.
In Fig. 10, for models that have emission cores, the Na I D core shows a much greater discrimination between the two model series than does H, with Series yielding profiles that have a clear central double reversal, and Series yielding profiles with a single reversal. As a result, the emission core very clearly distinguishes the highest pressure model from Series as the closest fit. In particular, a Series model with equal to -3.8 provides an approximate fit. As in the case of H, the high value of in the model yields line profiles that are too weak for a given value of . Therefore, the value of derived here is an upper limit.
Both lines have emission cores and, therefore, require high pressure chromospheres. However, the value of required by the H line is 0.4 dex lower in column mass density than that required to fit Na I . The discrepancy may in part be due to the high sensitivity of the emission strength to small changes in , especially in the case of H, and in part due to the inaccuracy of the value of used in the model, which limits the fit to a derivation of an upper limit only on the value of . The greater sensitivity of Na I D to chromospheric thickness and steepness allows us to identify Series as the closest fit.
Comparison with previous models
Mauas & Falchi (1994) derived a semi-empirical atmospheric model of the quiescent state of AD Leo (henceforth the MF model) by fitting several spectral features observed by Pettersen & Hawley (1989) with synthetic spectra calculated with the PANDORA (Avrett & Loeser 1992) model atmosphere code: the over all continuum, the profiles of the first four members of the Balmer series, Na I D, Mg I b, and Ca II K and lines, and the total fluxes of the Ly and Mg II h and k lines. They include the effect of line blanketing in their calculation by incorporating the line lists of Kurucz (1990), but it is unclear whether the line blanketing is calculated self-consistently for the entire model with the chromospheric/TR temperature rise, or if it is included for the photosphere only. Their model is shown in Fig. 15 along with models from both of our series with between -3.8 and -4.2, which spans the range in that was fit separately by either of our diagnostics.
The chromospheric structure of their model is more general than ours in that it deviates from a straight line. However, the mean slope is close to that of our Series models. The location of in the MF model is close to that of our highest pressure model. The value of falls within the range spanned by our closest fit models, but the TR temperature rise of the MF model is much more gradual than that of ours.
One possible reason for the discrepancies between the MF model and ours is the difference in the spectral resolution of the data being fit. We fit fully resolved profiles of emission cores, whereas the data of Pettersen & Hawley (1989) have a resolution, , of , which is not sufficient to resolve the detailed shape of the line profile. For example, the central absorption reversal of the H emission core in dMe stars is a diagnostic of the TR slope (Houdebine et al. 1995), and our Series model fits the reversal approximately. By contrast, the observed H spectrum used by Mauas & Falchi does not resolve the central reversal. Similarly, the slope, of the chromospheric temperature rise is distinguished in our fitting by the presence or absence of a central absorption reversal in the Na I D core, whereas the emission cores are not resolved at all in the data of Pettersen & Hawley (1989).
Another reason for discrepancies in the chromospheric/TR structure between our model and the MF model is the difference in the photospheric structure of the two models. The photospheric temperature structure of the MF model is flatter than ours, and we have already seen from the perturbation analysis presented in Figs. 3 and 4 that the calculated profiles of chromospheric features depend sensitively on the structure of the underlying photosphere. Because Mauas & Falchi derived the structure of the photosphere , as well as the chromosphere, semi-empirically, we cannot assign the usual stellar parameters to their model for the sake of comparison with our photospheric model. However, from Fig. 15, we also see that the MF model corresponds to a lower value of than ours, and is, therefore, in better agreement with the measured value of Gl 388. From the perturbation analysis given above, we expect that a model with a lower photospheric would give rise to a closest fit chromospheric structure with lower values of the column mass density, as does the MF model.
Fig. 15 also shows the model of AD Leo of Hawley & Fisher (1992) (henceforth the HF model). The photospheric base is provided by a model of Mould (1976) that corresponds to equal to 3500 K, equal to 4.75, and equal to 0.0. The temperature structure of the chromosphere and upper photosphere of this model was computed theoretically by adopting a model for the overlying corona and assuming that coronal X-ray illumination is the source of excess heating in the outer atmosphere during quiescence. The energy equilibrium structure is computed by balancing the coronal X-ray heating against the radiative losses in the H I spectrum and the Mg II hk and Ca II HK lines. The procedure was iterated with re-integration of the hydrostatic equilibrium equation.
The value of in the HF model is 0.2 dex less than that of the lowest pressure model that approximately fits either of our diagnostics. Also, the location of is almost 0.3 dex deeper than our well fitting model with the deepest . Furthermore, the slope, , of the chromospheric temperature rise is similar to that of our Series models, which are clearly ruled out by the lack of a central absorption reversal in the observed Na I D lines. However, the HF model is purely theoretical and has not been fit to any observed chromospheric diagnostics. Also, as with the MF model, the underlying photosphere has a flatter temperature structure, and, indeed, corresponds to a value that is 200 K lower than that of our model. On the basis of the perturbation analysis shown in Fig. 3, the amount of the radiation loss in the H I spectrum, which is important in determining the structure of the HF model, will be affected by the photospheric structure.
4.1.4. Gl 494
Alonso (1996) finds K by the Infrared Flux Method, having adopted values of and equal to 5.0 and 0.0, respectively. Rutten et al. (1989) give a spectral type of dM2e and Henry et al. (1994) find a type of dM3V. Panagi & Mathioudakis (1993), Rutten et al. (1989), Doyle et al. (1990), Pettersen & Hawley (1989), Herbst & Miller (1989), and Panagi et al. (1991) give measured flux values in various chromospheric emission lines, including the Na I D lines in the case of the latter. Stauffer & Hartmann (1986) give (H) and FWHM(H), while Panagi et al. (1991) give FWHM(H) and the ratio of line centre to continuum flux, , equal to 2.01. Pettersen & Hawley (1989) give (H) and Young et al. (1989) find the range in excess (H) to be 1.80 to . Panagi et al. (1991) report self-reversals in the cores of the Na I D doublet in WHT spectra. Stauffer & Hartmann (1986) find km s-1.
We measure to be , which is just below the range of the previous measurements of 2.12, 1.48, and (Stauffer & Hartmann 1986, Panagi et al. 1991, Pettersen & Hawley 1989). From Table 4, a model with between -4.4 and -4.2 provides a fit to our measured value and those of Stauffer & Hartmann (1986) and Panagi et al. (1991). We measure the FWHM of the emission core to be , which is slightly larger than the value of found by Panagi et al. (1991) and significantly smaller than the value of found by Stauffer & Hartmann (1986). Our measured value may be fit by a model of Series with between -3.8 and -4.0, whereas the value of Panagi et al. (1991) and Stauffer & Hartmann (1986) lie outside the range predicted by either series.
The synthetic profiles have been rotationally broadened with a value of of 10.0 km s-1 (Stauffer & Hartmann 1986). We see from Fig. 11 that, in keeping with the closest fit to the value of , models with of either series provide close fits to the observed profile, although the slightly closer spacing of the emission peaks of the Series model provides a slightly better fit. Because the value of this star is over 150 K hotter than that of the model, we expect, according to Fig. 3, that the model will predict emission lines that are too strong for a given value of . Therefore, the value of fit here is a lower limit.
Unlike the synthetic H profiles, the synthetic Na I D profiles distinguish very clearly between the two model series. The presence of a central self-absorption reversal in the Series models rules them out as good fits. A model of Series with provides a close fit to the shape of the line core, although the predicted profile is lower in flux than the observed one by in relative flux throughout the entire inner line profile. This discrepancy may be due to inaccuracy of the rectification of the observed to the computed pseudo-continuum, or due to inaccuracy in the stellar parameters of the photospheric model. However, from Fig. 4, we expect that a model with too low a value of will predict too much flux in the inner wings rather than too little. As with H, the inaccuracy in the model value of leads us to take the value of derived here as a lower limit.
The value of that is found from a fit to the H line is 0.4 dex smaller in column mass density than that found from a fit to the Na I core. Indeed, the value found from the H profile corresponds to a model in which the Na I D core is in absorption , whereas the observed core is clearly in emission. The discrepancy may in part be due to the inaccuracy of the model value, which makes our derived values of from each line lower limits only.
4.1.5. Gl 900
Gl 900 is unique in our sample because it is the only "zero H" (dM(e)) star. There are no measured values of reported in the literature. Byrne & Doyle (1989) report fluxes in a variety of FUV lines that form in the TR and find that they are intermediate between those of quiescent dM and active dMe stars. From high resolution spectroscopy, Robinson & Cram (1989) find that the Ca II core flux is intermediate between that of dM and dMe stars. They also find that the H profile has emission wings and an absorption core.
Rutten et al. (1989) give a spectral type of dM1. Panagi & Mathioudakis (1993), Rutten et al. (1989), Giampapa et al. (1989), Herbst & Miller (1989), and Robinson et al. (1990) give measured flux values in a a variety of chromospheric emission lines. Stauffer & Hartmann (1986) give (H). Fleming et al. (1989), from moderate resolution spectroscopy, measure km s-1.
Although Gl 900 has been found to be a zero H star, we detect a distinct absorption line, although it is much weaker than that of Gl 212 or Gl 382. We measure to be , which is smaller than the values of 0.01 and 0.08 found by (Stauffer & Hartmann 1986, Robinson et al. 1990). The latter two values correspond to slight emission, rather than absorption. From Table 4 we see that in Series all three measured values may be fit with a model with between -4.8 and -4.6. Among Series models, our value corresponds to a model with between -4.8 and -4.6 and those of Stauffer & Hartmann (1986) and Robinson et al. (1990) indicate a model with between -4.6 and -4.4. In Fig. 13 the depth and width of the observed line profile is approximately fit by a model of Series with equal to -4.6. However, this model has small but significant emission wings that are not present in the observed line profile. A model with a value between -4.6 and -4.8, which would be deeper but have weaker or absent emission wings may provide a better fit, in agreement with the results of the fit. The Series profiles of the same range are slightly narrower and, therefore, provide a slightly worse fit.
In Fig. 14 we can see that a model of Series with equal to -4.2 provides a very close fit to the shape of the entire inner line profile. None of the models of Series is able to provide even an approximate fit.
The Na I line profile is almost perfectly fit by a model in which the value of is larger by at least 0.4 dex in column mass density than that needed to fit H. The usual caveats about inaccuracies in the stellar parameters of the model apply, although we cannot provide a quantitative assessment due to the lack of measured values of the parameters in the literature. In the case of Gl 900, the ability to simultaneously fit both lines is complicated by another consideration: the "zero-H" stars of intermediate chromospheric pressure exist in a regime where both H and the Na I D line cores are making a very rapid transition from being in absorption to being in emission with increasing . Therefore, the strength and shape of each line is more sensitive to slight changes in as compared to the low chromospheric pressure regime. As a result, unless the detailed modelling of the line formation and detailed structure of the models is very accurate, we may not expect to be able to fit both lines even approximately with a single model.
© European Southern Observatory (ESO) 1998
Online publication: July 20, 1998