4. The results
In Fig. 1 we show the computed models for both stars. For each star, we show the temperature and the electron density as a function of column mass. The atmospheric parameters for the models are given in the Appendix (Tables 4 and 5 are only available in electronic form). For comparison, we also show the model for AD Leo from Paper I, slightly modified due to the use of a new set of atomic parameters for Hydrogen. The new model has a slightly hotter chromosphere, and a chromospheric rise located at a slightly larger column mass than the model in Paper I. In any case, the differences between these two models are much smaller than the differences between AD Leo and the inactive stars.
As can be seen, the photospheric structure of the models for the three stars is very similar, differing only in the upper photosphere, just below the temperature minimum. Given the fact that the three stars have very similar colours (see Table 1), and that the observed continua coincide within the observation and calibration errors, we believe that there is no reason to assume a different structure in the region where the continuum emission is originated (below ). The difference just below is due to the fact that the Ca II minimum is narrower for Gl 628, as will be discussed below.
In Fig. 2 we show our computed continuum, together with the low resolution observations for the three stars between 3500 and 9000 Å. The heads of the most important molecular bands present in our spectra are indicated in Fig. 2. As can be seen, the three stars have a very similar behaviour within the estimated uncertainty of about 30% on our calibrated spectra. Although the general behaviour of the observed spectrum is reproduced within this uncertainty, and many observed spectral features are matched, there are still some molecular bands clearly absent from our calculations, notably CaOH and H2 O. It can also be noted that the opacity due to TiO seems to be overestimated. Kurucz (1991) cautions against using these opacities to compute the atmospheres of M stars, because they do not include triatomic molecules. However, based on the present results, we believe Kurucz's compilation yields results which are satisfactory enough for our purposes, since we are not interested in fitting any particular feature. In the present calculations, the continuum intensity is needed to account for the photospheric irradiation of the chromosphere, to properly compute the transition rates between levels. We point out that for each line we study in detail, the background opacities around the line were modified in order to match the observed continuum intensity.
In Table 2 we list the observed continuum fluxes for different visible filters (in the Cousin system), together with the values obtained by integrating the computed fluxes with the corresponding filter profile (Schaifers & Voigt 1982). Note the large differences for the visible filters between the fluxes computed with and without Kurucz's opacities, due to the much larger opacity when line blanketing is included. This implies that models based on a fit to filter values computed without taking into account these opacities can be strongly in error. Also shown in Table 2 are the infrared fluxes observed by IRAS, and the corresponding computed values. The computed values lie within the estimated error of the observations.
Table 2. Observed and computed fluxes at the star surface (105 erg cm-2 s-1 Å-1). For the continuum filters, the first value was computed considering line blanketing, and the second one without it
The profile of the Ca II K line, in particular the K1 minimum, is a very good indicator of the structure of the temperature minimum (Ayres and Linsky, 1976; Avrett, 1985). In Fig. 3-a we show the profiles of the K line for our models. The computed profiles have been convoluted with the instrumental response (FWHM = 0.18 Å).
In Fig. 3-b we show a detail of the observed profiles. As can be seen, the K1 minimum is narrower for Gl 628, and thus we had to modify the structure of the upper photosphere, where the inner wings of the line are formed. Also, the fact that the K1 minimum itself is more intense for Gl 628 implies that the chromospheric rise has to start at a larger column mass for this star than for Gl 588.
In Fig. 4 we show the four lower Balmer lines for both stars: it can be seen that and have a very low central depth, hidden in the noise. The Balmer lines determine the temperature of the chromosphere. It can be seen in Fig. 4 that the match for these lines is very good, well whithin the calibration error of the observations. In particular, we were able to reproduce the almost flat profile for and .
It should be noted that, although in Figs. 3 and 4 we show the relative intensity for clarity, we computed the models to match the absolute intensity at all wavelengths in the line, and we did not need to add any constant to the computed values, contrary to what was done by Houdebine & Doyle (1994b). The relative intensities can match the observations, but both the continuum and line center absolute intensities can differ from the observations.
To check the quality of our models, we also compare the observed profiles of the Na D lines with our computations. The results are shown in Fig. 5. As can be seen, the results are quite good for Gl 588 while for Gl 628 the computed profiles are too narrow, which is a consequence of the larger temperatures in the high photosphere that are needed to reproduce the narrower Ca II line.
In Table 3 we list the computed fluxes for the Mg II h and k lines, and for both stars. For the Mg II resonance lines, the values obtained can be compared with the observations listed in Table 1. Considering the errors in the observed values, and the fact that, without observed line profiles, it is very difficult to study these lines, we believe that the agreement within a factor of 3 is good enough. For , the agreement with the observed values of Table 1 is very good, as would be expected since the profiles agree very well with the observations. The largest difference is for Gl 628, for which the computed profile is too low. Unfortunately, no observed flux is available for the comparison.
Table 3. Computed line fluxes at the stellar surface (erg cm-2 s-1). Negative fluxes represent absorption lines.
© European Southern Observatory (ESO) 1997
Online publication: April 20, 1998