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

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4. Results

4.1. Fourier analysis

The first question as to whether the rotation period is visible at all can be answered by Fourier analysis of the time series as a whole, i.e., the solution without time-resolution. In Fig. 3 we show the power spectrum resulting from application of Breger's PERIOD code (Breger 1990) applied to the entire time series. Apart from the appearence of enhanced power at low-frequencies ([FORMULA] cycles/day) which arises from a non-perfect fit of the solar cycle, the spectrum is clearly dominated by the solar synodic rotation: maximum power appears at a period of 28.3 days. This would correspond to the solar rotation at latitude [FORMULA] where active regions are concentrated at the beginning of the solar cycle. A second (but less dominant) peak appears at a period of 26.8 days corresponding to the solar rotation rate at latitude [FORMULA], the latitude of the active zone near the end of the solar cycle. Surprisingly, there is also a third peak at a period of 30.2 days which would require the existence of ARs at a latitude of [FORMULA]. This peak is likely an artifact resulting either from the window function or from ARGD (cf., Sect. 3). The window function (Fig. 4) does not show any enhanced frequencies. Hence, the 30.2 d period cannot be the result of the data sampling: a time-resolved analysis is required to uncover its nature.

[FIGURE] Fig. 3. Fourier power spectrum of the solar data from 1967 until 1992
[FIGURE] Fig. 4. The window function.

Another surprising feature to Fig. 3 is that the 28.3 d and the 26.8 d periods are clearly separated from each other. If there were a continuous shift of AR's in latitude during a solar cycle one would expect a single but very broad peak to appear in the power spectrum. The existence of two separate peaks could possibly be explained by the uneven distribution of power at isolated times, for example, at the beginning and the end of a solar cycle. Power might be reduced at solar maximum because the number of ARs at different longitudes is large enough that rotational modulation is smeared out and power is reduced.

4.2. Wavelet analysis

In a first step we have applied the wavelet transform to the whole time series including cycles 20-22. However, the data sampling before 1977 is poor compared to the years afterwards. The wavelet map, a three-dimensional plot of power as a function of time and frequency, looks quite different for these two intervals. The wavelet map after 1977 is rich with ranges of peaks, while the map from the early years (cycle 20) is almost empty. Therefore, we have concentrated our subsequent analysis to Cycles 21 and 22, only.

The wavelet plot of Cycle 21 (Fig. 5) shows a broad mountain range with a tendency to fall from [FORMULA]  d (in 1978) to [FORMULA]  d near the end of that cycle. This range is structured as individual peaks, but there is a second (but much weaker) mountain range adjacent to the main range at longer ([FORMULA]  d) periods. This could be the wavelet answer to the appearence of the 30.2 d period found from Fourier analysis above. The time-resolution of [FORMULA]  y and the period resolution of [FORMULA]  d do not allow for a more-detailed examination. However, from slices made at a series of epochs it is clear that there are several maxima of power present at any given time even within the prominent mountain range (e.g., during 1983).

[FIGURE] Fig. 5. Wavelet map of cycle 21 from 1977 until 1987 shown as a contour plot above the time-period plane.

The wavelet plot of Cycle 22 (Fig. 6) is characterized by two rather straight mountain ranges where the more prominent range follows the same values as the primary mountain range of Cycle 21. As in case of Cycle 21, there is a spurious third range near 30 days. Somewhat surprising is the comparatively strong power of the second range. Obviously it has its analogy in the strong 26 d side-lobe in the Fourier spectrum (Fig. 3). It cannot be due to rotation alone because the period falls below the limit given by Eq. (1). We discuss its possible nature below.

[FIGURE] Fig. 6. Wavelet map of cycle 22 from 1987 until 1992.

We assume that the prominent mountain ranges of both Figs. 5 and 6 reflect the true rotation period from three reasons: first, from the fact that maximum power is concentrated there in both Cycle 21 and Cycle 22; and second, that the 26 d range is only visible during Cycle 22 whereas it is absent during Cycle 21. Finally, the values of [FORMULA] inside the dominant range are reasonable, lying within the interval given by Eq. (1) such that [FORMULA].

To record changes in rotation period we have made cuts along the period axis at specific intervals of time with a time resolution typically half of the intrinsic wavelet resolution of [FORMULA]  y. In Fig. 7 the periods of all maxima found along those cuts but inside the prominent mountain range are plotted versus time (dots). Then weighted least squares fits were made where the weights were set proportional to the power of a given maximum (the two straight lines in Fig. 7).

[FIGURE] Fig. 7. 'Stellar butterfly diagram' of the Sun.

Fig. 7 shows a `stellar butterfly diagram' close to what is expected for the Sun: the fits yield a synodic period of 28.6 d at the early phase of Cycle 21, and 26.5 d at the end of this cycle. However, the latter value is a bit too low, lying below the 26.75 d limit at the solar equator. Then, the new cycle starts again with a 28.6 d period and decreases with nearly the same slope as Cycle 21. If one extrapolates the line towards 1998 (the expected year of the end of this cycle), then a synodic period of 26.9 d is found, which is a reasonable value at solar activity minimum.

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© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998