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Appendix A: deviations in the disk-area coverages by active regions

	Fig. A.1 Photometric variability as a function of mean chromospheric activity calculated for α = 0 (original AS coverages given by Eq. (1), solid curve), α = 0.5 (increased AS coverages, dotted curve), α = − 0.5 (decreased AS 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 AS coverages given by Eq. (1), solid curve), α = 0.5 (increased AS coverages, dotted curve), α = − 0.5 (decreased AS coverages, dashed curve). The calculations are performed for homogeneous distribution of active regions.
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The relationships between disk-area 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 disk-area 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 disk-area coverage: (A.1)where is the new spot disk-area coverage, A_S(S) is the spot disk-area 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 disk-area coverages works remarkably well even for stars significantly more active than the Sun.

Appendix B: comparison with the SATIRE-S results

Our stellar variability model is based on the representation of the facular and spot disk-area coverages as functions of the S-index 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 single-value function of the observed S-index.

In reality, Eqs. (1), (2) are only approximate. For every particular observational season the disk-area 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 faculae-dominated by our analysis to be temporarily spot-dominated.

To estimate the importance of this effect we considered the solar S-index and photometric brightness (i.e. the Strömgren (b + y)/2 flux) time-series. The S-index was calculated from the Sac Peak K-index K_SP (see Sect. 3). Because there are no long-term solar irradiance measurements equivalent to the Strömgren (b + y)/2 flux, we employed the SATIRE-S spectral irradiance time-series (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 S-index, 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 SATIRE-S model (upper panel) and the S-index calculated from the Sac Peak measurements (middle panel) as well as the slopes of the activity-brightness 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 SATIRE-S data using the entire time series (1978–2010).

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	Fig. B.2 The same as Fig. B.1, but for three-month averages.
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The S-index, photometric time-series, and slopes are plotted in Fig. B.1. One can see that the photometric flux is not a single-value function of the S-index 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 disk-area 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 three-month averages (see Fig. B.2). The transit of large spots affects the photometric flux, but leaves the S-index unchanged (because for a star with a solar activity level the contribution of spots to the S-index 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 SATIRE-S calculations.

Figures B.1 and B.2 reveal that according to the SATIRE-S model the variability of the solar Strömgren (b + y)/2 flux is always faculae-dominated 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 spot-dominated (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