Issue 
A&A
Volume 569, September 2014



Article Number  A38  
Number of page(s)  18  
Section  Stellar atmospheres  
DOI  https://doi.org/10.1051/00046361/201323086  
Published online  16 September 2014 
Online material
Appendix A: deviations in the diskarea coverages by active regions
Fig. A.1
Photometric variability as a function of mean chromospheric activity calculated for α = 0 (original A_{S} coverages given by Eq. (1), solid curve), α = 0.5 (increased A_{S} coverages, dotted curve), α = − 0.5 (decreased A_{S} coverages, dashed curve). The calculations are performed for the homogeneous distribution of active regions. 

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Fig. A.2
Regression slope of the dependence of photometric brightness variation on HK emission variation, plotted vs. mean chromospheric activity calculated for α = 0 (original A_{S} coverages given by Eq. (1), solid curve), α = 0.5 (increased A_{S} coverages, dotted curve), α = − 0.5 (decreased A_{S} coverages, dashed curve). The calculations are performed for homogeneous distribution of active regions. 

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The relationships between diskarea coverages and chromospheric activity employed in the present study were established on the basis of solar data and then extrapolated to higher activity levels. While one might expect that the extrapolation works well for stars with activities similar to that of the Sun, the diskarea coverages of more active stars may deviate from the values given by Eqs. (1)–(2). To estimate the impact of such deviations on our results we recalculated the log (rms(b + y)/2) and Δ [ (b + y)/2 ]/ΔS values plotted in Figs. 10 and 11, first assuming that the coverage of the most active stars from the sample of Lockwood et al. (2007) is 50% larger then we expect from the extrapolation from the Sun and than that it is 50% smaller than we expect from solar extrapolation.
Namely, we applied following correction to the spot diskarea coverage: (A.1)where is the new spot diskarea coverage, A_{S}(S) is the spot diskarea coverage given by Eq. (1), α is the coefficient that determines the amplitude of the correction, S_{⊙} is the mean solar level of chromospheric activity, and S_{max} was chosen to be equal to 0.5, which is the highest mean chromospheric activity considered in the present study (see the description of the algorithm employed to produce Figs. 10 and 11 in Sect. 6).
The resulting dependences of log (rms(b + y)/2) and Δ [ (b + y)/2 ]/ΔS on S are plotted in Figs. A.1 and A.2 for three values of α. One can see that the scatter in the surface coverages may lead to significant deviations in the theoretical curves plotted in Figs. A.1 and A.2. At the same time, the general success of our approach in modeling the stellar data implies that the simple extrapolation of solar diskarea coverages works remarkably well even for stars significantly more active than the Sun.
Appendix B: comparison with the SATIRES results
Our stellar variability model is based on the representation of the facular and spot diskarea coverages as functions of the Sindex measured from a vantage point in the stellar equatorial plane (see Eqs. (1), (2)). For fixed inclination and distribution of active regions the change of the stellar brightness due to magnetic activity, δ(b + y)/2, is a singlevalue function of the observed Sindex.
In reality, Eqs. (1), (2) are only approximate. For every particular observational season the diskarea coverages may differ from the values given by these equations, because they define the relation averaged over the longest time interval for which solar data are available (see Sect. 3). For example, a transit of a large spot may cause the stars deemed faculaedominated by our analysis to be temporarily spotdominated.
To estimate the importance of this effect we considered the solar Sindex and photometric brightness (i.e. the Strömgren (b + y)/2 flux) timeseries. The Sindex was calculated from the Sac Peak Kindex K_{SP} (see Sect. 3). Because there are no longterm solar irradiance measurements equivalent to the Strömgren (b + y)/2 flux, we employed the SATIRES spectral irradiance timeseries (see description in Ball et al. 2012, 2013, and references therein) convolved with the Strömgren (b + y)/2 spectral filter profile. From these data we calculated the slope of the photometric brightness regression on the observed Sindex, Δ [ (b + y)/2 ]/ΔS, for 11 year time intervals (the value for the year X is the regression slope calculated for the [X–5, X + 5] dataset), offset by one year each.
Fig. B.1
Annual values of the solar spectral flux in the Strömgren (b + y)/2 filters according to the SATIRES model (upper panel) and the Sindex calculated from the Sac Peak measurements (middle panel) as well as the slopes of the activitybrightness correlation (lower panel). The solid line in the lower panel represents the slope calculated with the simplified model used in this paper, while the dashed line corresponds to the slope calculated with SATIRES data using the entire time series (1978–2010). 

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Fig. B.2
The same as Fig. B.1, but for threemonth averages. 

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The Sindex, photometric timeseries, and slopes are plotted in Fig. B.1. One can see that the photometric flux is not a singlevalue function of the Sindex and the regression slope varies with time. For example, the increase of the slope after 2003 might be explained by the decrease of the ratio between spot and facular diskarea coverages.
The variations of the slope may explain some of the scatter in the observed stellar slopes (see Fig. 11). The effect increases if instead of the annual data we consider threemonth averages (see Fig. B.2). The transit of large spots affects the photometric flux, but leaves the Sindex unchanged (because for a star with a solar activity level the contribution of spots to the Sindex is negligibly small). We note that the slopes calculated with our model using the entire time series (1978–2010) agree well with the more sophisticated and accurate SATIRES calculations.
Figures B.1 and B.2 reveal that according to the SATIRES model the variability of the solar Strömgren (b + y)/2 flux is always faculaedominated if the Sun is observed for at least 11 years. This does not contradict with its position in the shaded band in Fig. 11 because if the Sun is observed for a shorter period of time around its activity maximum it can be erroneously identified as spotdominated (see Fig. 7).
Fig. B.3
The same as Fig. 3, but for the adopted polar distribution of spots (two caps with latitudes between ± 45° and ± 90°). 

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© ESO, 2014
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