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Astron. Astrophys. 343, 222-228 (1999)

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3. Results

Fig. 2 shows the width-luminosity relation of the MgII k line for our sample of 59 stars of spectral classes G, K and M. Six stars of spectral class F have been left out because they seem not to follow a Wilson-Bappu relation, as will be demonstrated in Sect. 3.1.

[FIGURE] Fig. 2. Width-luminosity relation of the Mg II k line for all sample stars of spectral classes G, K and M using both log W (above) and log W[FORMULA] (below) as parameters of the line width

When the diagrams in Fig. 2 are compared with previous results in Elgaroy et al. (1997), it is seen that the difference is extremely small. The scatter of the data around the regression line is about the same in the two cases. But whereas one previously had to take into account that an unknown part of the scatter might be due to erroneous absolute magnitudes, the new results are free of this deficiency. It is now possible to look for physical mechanisms causing deviations from the average behavior.

The following Wilson-Bappu relations are obtained for the 59 G, K and M stars in the sample:

[EQUATION]

(r is the correlation coefficient).

The uncertainties are roughly the same in the expressions (1) and (2).

Almost the same results are obtained using classical, earth-based absolute magnitudes.

3.1. Possible dependence on stellar temperature

The accurate Hipparcos absolute magnitudes permit a further differentiation of the material according to temperature, as given by stellar spectral classes. Width-luminosity relations for the MgII k line for stars according to spectral class are shown in Fig. 3.

[FIGURE] Fig. 3. Width-luminosity relations for stars of different spectral classes. The regression lines are given as solid lines

From the data the following Wilson-Bappu relations are derived:
F stars, n= 6 stars.

[EQUATION]

G stars, n= 22 stars.

[EQUATION]

K stars, n = 26 stars.

[EQUATION]

M stars, n = 11 stars.

[EQUATION]

From Fig. 3 and from the regression line derived for the F stars it is seen that our data reveal no relation between the luminosity and the width of the MgII k line for the F stars. But one must take into account that there are 6 F stars only in the sample and that they cover a relatively small range in absolute magnitudes (Mv = 2.2 to 4.0). In addition the MgII lines become weak and noisy for F stars and show strong central reversals, making good determinations of their widths difficult. We conclude that a weak relation between luminosity and width for F stars can not be completely excluded, although our present data reveal no connection.

For the stars of spectral classes G, K and M, there is a very good correlation between luminosity and line width. The respective Wilson-Bappu relations are well defined. It is seen that the slope of the regression line for the G stars is less steep than for the K and M stars. Between K and M stars there is no significant difference.

Since main sequence stars dominate in the lower left of the diagrams whereas giants and supergiants occupy the upper right the difference in slope of the G stars and the K-M stars suggest that there is a larger difference in line width between G dwarfs and giants than is the case for K-M dwarfs and giants.

3.2. Deviations from the Wilson-Bappu relation

In Elgaroy et al. (1997) it was found that active stars had broader lines and showed larger variations in line widths than quiet stars. In particular observations of the active RS CVn binary [FORMULA]Gem taken at epochs of different levels of activity, clearly demonstrated line broadening accompanying increased activity.

The sample of K stars may be used for a study of the dispersion of the data points around the regression line. [FORMULA]log W[FORMULA] is defined as the difference between log W[FORMULA] (measured) and log W[FORMULA] (average). The latter is obtained from the W-B relation using the absolute magnitudes of the chosen stars. The MgII k-line surface flux (Table 4) may be used as a parameter of activity. According to Robinson & Carpenter (1995) the strength of the emission is a direct measure of chromospheric activity. In Fig. 4 [FORMULA]log W[FORMULA] has been plotted against surface flux as given by log F[FORMULA]. It was found useful to make a further division of the material into dwarfs (KV) and giants (KIV, III, II, I).


[TABLE]

Table 4. Surface fluxes of the Mg II k line as determined by Elgaroy et al. (1997) updated with new parallaxes from the Hipparchos data together with deviations from the regression line.


[FIGURE] Fig. 4a and b. Deviation from the Wilson-Bappu relation ([FORMULA]log W[FORMULA]) plotted against surface flux ([FORMULA]) for K giants and supergiants (above) and K dwarfs (below)

We estimate the error in the derived surface fluxes to amount to approximately 50%, corresponding to an uncertainty of 0.3 in the logarithm of the surface flux. The errors in [FORMULA]log W[FORMULA] are equal to the errors in log W[FORMULA] (see Table 1).

Fig. 4 reveals an interesting difference between the behavior of K dwarfs and giants. Dwarfs show increasing line width with increasing surface flux. But in the case of giants, the line widths decrease with increasing surface flux for [FORMULA]0.02. Taking the limited material and the errors in the data into account, this suggests that there is a characteristic difference between dwarfs and giants in the case of the broadest line profiles.

The following regression line has been derived:
K dwarfs, n = 11

[EQUATION]

For the K giants a linear regression line is clearly not appropriate (cf Fig. 4).

Rather than using [FORMULA]log W[FORMULA] (or [FORMULA]log W) one may use the relative deviation, i.e. [FORMULA] as a parameter, where [FORMULA] is determined from the regression line.

Let [FORMULA], then one finds: K dwarfs

[EQUATION]

The two main results from the analysis of the log F[FORMULA]W[FORMULA] relations are:

  1. For main sequence K stars [FORMULA]log W[FORMULA] increases with increasing activity (as given by log F[FORMULA]).

  2. K giants and supergiants show the same behavior for [FORMULA]log W[FORMULA]0.02. For broader profiles activity seems to decrease with increasing line widths. More stars in this range are needed since the current sample contains three stars only.

The different relations between activity and excess line width for dwarfs and giants may be associated with the fact that strong magnetic fields are present in active dwarfs, whereas activity in many giants and supergiants may be more non-magnetic in nature (Cuntz, 1996).

To explain the properties given in 1) and 2) one needs detailed calculations of atmospheric (chromospheric) models and calculations of the line profiles using hydrodynamic methods and theory of line formation. Before such work has been performed, which requires large resources, some effects which are important for the explanation of the observations may be pointed out.

3.2.1. Main sequence stars

For these stars the temperature gradient in the chromosphere increases in active regions and the fluxes and line widths both become larger. When a star is more active (increased heating) the Mg II lines are formed at a level where the column mass and therefore the opacity is larger than in the quiet case. Consequently the fluxes and the widths of the chromospheric lines increase.

3.2.2. Giants and supergiants

In this case the situation is more complex. These stars have a much smaller surface gravity in the chromosphere than main sequence stars. The surface gravity and the metal abundance, which are important for the chromospheric temperature structure, can show large variations from one star to another, and are also responsible for the difference between dwarfs and giants. Model calculations (Cuntz et al., 1994) suggest that the giants have a much more extended chromosphere than main sequence stars due to the smaller gravity. The chromospheres are quasi-isothermal, i.e. the temperature increases very slowly up to the transition region. In and above active regions giants show higher temperatures leading to more extended quasi-isothermal chromospheres. These calculations show that increased heating, i.e. increased activity, may lead to more narrow lines. Lower metal abundances also give narrow lines as a result. This occurs because the line opacity is proportional to the Mg II number density, and therefore decreases in proportion to the Mg II abundance (Cuntz et al., 1994). In short: individuality, effects of non-magnetic activity and variations in metal abundances may explain our results for the giant stars.

3.3. Metallicity

Since metallicity may be a factor influencing the Mg II k line width (Lutz & Pagel 1982), data on abundances have been compiled for the K stars in our sample. The abundances were derived from Cayrel de Strobel et al. (1992), Jones et al. (1992), Kovács (1983), Taylor (1991), McWilliam (1990), Fanelli et al. (1990), Pallavicini et al. (1992) and Tokovinin (1990).

In many cases several different results have been found for the same star. In such cases the average value was used. The data for giants and supergiants are given in Table 5 and plotted in Fig. 5. The regression line is given by:

[EQUATION]

Data for 15 stars are used.

Even though the uncertainties in the abundance values (about 0.2 dex) and the relative deviations (Table 5) are large, our result clearly reveal an important relation between metallicity and line width, in qualitative agreement with the theoretical results of Cuntz et al. (1994).


[TABLE]

Table 5. Abundances as given by log Fe/H = (log Fe/H)star - (log Fe/H)Sun and relative deviation.


[FIGURE] Fig. 5. Relative deviation from the Wilson-Bappu relation D plotted against metallicity A = log(Fe/H), for 15 K giants and supergiants

The variation of Mg II surface flux for the K giants in our sample is relatively small (cf Fig. 4). This is also in agreement with the theoretical results of Cuntz et al. (1994), who mention three factors contributing to this end: a) the mean chromospheric temperature is slightly higher in models with lower metallicity. This tends to increase the Mg II emission flux. b) In models with lower metallicity the optical depth in the Mg II k line is reduced, forcing the line to be formed at higher column mass densities, where larger total particle densities occur. As a consequence, the reduced metal abundance is compensated. c) The wave shape of the shock waves (assumed to heat the chromosphere (Cuntz & Ulmschneider, 1988) is the same in large portions of atmospheres of different metal abundances, allowing the Mg II k line to be formed under similar thermodynamic conditions. The fact that chromospheric Mg II h and k line fluxes are independent of metal abundances is also in accordance with the scaling laws of Ayres (1979), which were found to be valid for various semiempirical chromospheres based on IUE data.

As regards the K dwarfs, abundances were found for 7 of the 11 stars in the sample. No connection between Fe/H abundance and deviation from average line width could be detected. Since the dwarfs are subject to magnetic activity it is likely that a possible effect of metallicity is overshadowed by the variation of line width with activity.

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© European Southern Observatory (ESO) 1999

Online publication: March 1, 1999
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