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Appendix A: Differential rotation beyond α = 0.3

In Eq. (4) an upper limit of α_max = 0.30 was used while searching for a second period. In this section we demonstrate how different upper limits α_max change the total number of detections and the overall behavior of α with temperature and rotation period. We use ten equidistant values 0.05 ≤ α_max ≤ 0.50 (see Table A.1). With increasing α_max we found a larger total number

of periods. The case α_max = 0.5 is shown to demonstrate the limits of our method, which is evident in the next two figures. For all stars with two detected periods we plotted their density in the T_eff − α plane in Fig. A.1. The colors are the same as in Fig. 9 with bright regions representing high density. For each α_max value we found that α slightly increases towards cooler stars. The case with the lowest upper limit (α_max = 0.05) looks a bit scattered, but it is not very different from the general trend. The plot in the lower-right corner (α_max = 0.5) demonstrates the limits of our method. Many stars jumped to the upper limit α = 0.5 because an alias period was chosen by the algorithm. The T_eff values from the KIC are not very accurate and neither are the stellar radii. We found the same trend with respect to α however, i.e., the α value increases towards smaller radii (although this may not be an independent constraint).

	Fig. A.1 Density plot in the Teff − α plane for different values of αmax. For each αmax value we found that α slightly increases towards cooler stars. The plot in the lower-right corner (αmax = 0.5) demonstrates that our method is limited to αmax < 0.5. Colors are the same as in Fig. 9.
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Our previous result that α increases towards longer rotation periods holds for all α_max values. In Fig. A.2 we showed the density in the P_min − α plane. We see that the density is localized in a sharp strip that smears out for limits greater than α_max = 0.3. These large α values belong to periods P₂ longer than 45 days where instrumental effects play a dominant role. Again, the stars jumped to the upper limit in the α_max = 0.5 case.

	Fig. A.2 Density plot in the Pmin − α plane for different values of αmax. For each αmax value we found that α strongly increases with rotation period. The plot in the lower-right corner (αmax = 0.5) demonstrates that our method is limited to αmax < 0.5. Colors are the same as in Fig. 9.
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© ESO, 2013