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Astron. Astrophys. 322, 835-840 (1997)

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5. Discussion and conclusion

Our time series analysis of solar disk-integrated Ca II K emission using the wavelet transform yields nearly the correct pattern of SDR. Maximum values of periods are found at the beginnings of Cycle 21 and Cycle 22. Over the course of both cycles, the period decreases systematically with almost the same rate of decline. There is also a clear transition from low to high values of rotation period between the end of Cycle 21 and the start of Cycle 22. Thus, the qualitatively correct behaviour is found. However, the quantitative behaviour is more complex. Several peaks are seen simultaneously and the slope of increasing rotation rate over the activity is not smooth, at least for Cycle 21.

It is known from monitoring of the sunspot cycle that spots appear at a variety of latitudes at any given time, and that the latitude spread in the solar butterfly diagram is large. Therefore, the large scatter in the 'stellar' butterfly diagram seen here should be expected. However, the scatter will be reduced by smoothing resulting both from the low time and frequency resolutions. A time-resolution of one year will smooth any internal pattern resulting from ARGD. Insofar our wavelet map will reflect an averaged behaviour, this is really what is of interest.

Compared with the observed disk-resolved differential rotation of the Sun, our result is reasonable. According to Eq. (1), and taking into account that at early phases of a solar cycle spots appear around [FORMULA] one expects a rotation period of [FORMULA]  d, and at the end of a cycle approximately 27 d. Indeed, we find the correct value at the beginnings of both cycles. However, there is a slight deviation of -0.5 d at the end of Cycle 21 in 1987. This deviation is small because one has to take into account the low frequency resolution which corresponds to a uncertainty of 0.5 d.

Cycle 22 shows two large mountain ranges with somewhat peculiar behaviour. While the range with the longer period dominates, the second range is also powerful and therefore cannot be neglected. The question is whether this period splitting is of intrinsic nature or if it is an artifact of some kind? Szatmáry et al. (1994, c.f. Sect. 3) list three possibilities: double mode oscillation, phase modulation on a time-scale smaller than our chosen time resolution or regular gaps in the data. Since there are no regular gaps, the remaining possibilities are double mode oscillation or phase modulation.

Double mode oscillation could result from the preferred appearence of plage regions at high and low latitudes, but with a lack of those regions at intermediate latitudes - possibly arising from an asymmetry between the northern and the southern hemispheres. Phase modulation must also be expected because of the appearence of active regions at different solar longitudes, which in connection with ARGD, will not only modulate the zero phase, but will also shorten or lengthen the period of rotational modulation. This is indeed the case demonstrated in Fig. 2. Because our time-resolution is much too small to follow the ARGD of the individual ARs, side-lobes can appear in the wavelet map as discussed in Sect. 3. This could also be the reason for the third but very weak mountain range found in the wavelet picture around [FORMULA]  d. This range cannot be explained by appearence of plages at high latitudes because it would involve the existence of plages latitude [FORMULA] and higher - not seen in the disk-resolved data.

We can conclude from the subsequent observation of stellar butterfly diagrams it should be possible to derive the sign and magnitude of the period change over the activity cycle. Therefore, wavelet analysis appears to be an effective tool to investigate stellar differential rotation by observation of stellar butterfly diagrams. However, the appearence of side-lobes in the wavelet picture must be handled with some caution and could possibly caused by active region growth and decay rather than by two independent active belts or by a north-south asymmetry in activity.

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© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998