Messina and Guinan (2002) specifically studied starspot cycles of stars from the “Sun in Time” program. Activity cycles are found in all of them, with periods ranging from 2.1 yr to 13.1 yr. A comparison with the more comprehensive survey and the theoretical interpretations presented by Saar and Brandenburg (1999) confirms the presence of two or three branches: i) inactive solar analogs show cycles about 100 times longer than the rotation period, P ; ii) active stars reveal cycles 200 – 600 times longer than P ; iii) and “super-active” stars show cycles about 4 order of magnitude longer than P (Figure 7a). Among G-type stars, only EK Dra appears to be compatible with the third class (Messina and Guinan 2002).
The brightness amplitude increases with increasing inverse Rossby number (i.e., increasing rotation rate for constant turnover time), indicating that spots produce progressively more modulation toward higher activity levels (Figure 7b). A plateau is suggested for the most active stars, indicating a saturation effect when spots cover a large fraction of the stellar surface (Messina and Guinan, 2002).
The standard near-ZAMS solar analog, EK Dra, has been an important target for cycle studies. Dorren and Guinan (1994a) and Dorren and Guinan (1994b) showed that its long-term photometric variations by 0.07 mag are consistent with a period of about 12 yr. Variations compatible with this time scale are also found in Ca ii HK, Mg ii h and k, and the ultraviolet C iv, C ii, and He ii fluxes (Dorren and Guinan, 1994a), and in X-rays (see Section 4.3.2 below). The spot periodicity has been confirmed by Messina and Guinan (2002) although their best estimate for the period is 9.2 ± 0.4 yr (Figure 8). The star is optically faintest when chromospheric and transition-region activity is highest. On top of this cyclic behavior, there is a long-term trend in the optical light, namely a decline of the blue light by 0.0017 ± 0.0004 mag yr–1 for an investigated time span of 35 yr (Fröhlich et al. 2002; see also Messina and Guinan 2003 and Järvinen et al. 2005). Further, two active longitudes shift in phase in concert with the activity cycles (Järvinen et al., 2005). The dominant spot concentration switches between the two preferred longitudes with a “flip-flop” cycle of about 4 – 4.5 yr. These features suggest the coexistence of axisymmetric and non-axisymmetric dynamo modes (Berdyugina et al., 2002).
Messina and Guinan (2003) have studied photometric periods from spot modulations, arguing that – as in the solar case – the varying dominant latitudes of the spots should induce a periodic variation of the rotation period in phase with the activity cycle, because of differential rotation. These correlated period variations are indeed present in solar analogs (Messina and Guinan, 2003), although two patterns are seen: either, the period decreases as the cycle proceeds (solar behavior), or it increases (anti-solar behavior). The former effect is due to spot migration toward the equator where the surface rotation is faster. The second effect could be due to pole-ward acceleration of rotation at higher latitudes where active stars predominantly show spots, while the individual spots may still migrate toward the equator. Messina and Guinan (2003), however, suggest that high-latitude spots migrate toward the (slower-rotating) pole, which induces an anti-solar behavior in particular for stars with small inclination angles. This model is preferred because correlations involving the fractional variation in the rotation period show the same behavior for the two subclasses. Support for the model comes from simulations of magnetic-field migration toward the poles in very active stars, as performed by Schrijver and Title (2001) (Section 4.1.2). Specifically, Messina and Guinan (2003) have found a power-law relation between the rotation-period variation, P, and the average rotation period, P , of the form (see Figure 9a)
Further, a tight correlation is found between differential rotation, parameterized by , and the activity cycle frequency, ,
suggesting that is a key parameter controlling the duration of the activity cycle (see Figure 9b).
Differential rotation has also been measured on other very active stars (e.g., Donati et al. 2003a), among them a solar-like post-T Tauri star (Donati et al., 2000), in particular based on Doppler imaging techniques (Section 4.1.1). Differential rotation was found to be solar-like in these examples, although time-variable, which may hint at dynamo processes that periodically convert magnetic into kinetic energy and vice versa (Donati et al., 2003a).
Given the much stronger variability in the outer, coronal layers of a stellar atmosphere, an activity cycle may be more easily identified in the X-ray or radio domains. However, such cycles have eluded detection until recently because no appropriate program had been carried out for sufficiently long periods. A few notable examples have now been reported from X-ray monitoring.
First tentative evidence for an X-ray cycle came from the young solar analog EK Dra that was monitored between 1990 and 2000 using ROSAT, ASCA, and XMM-Newton. Initial results were presented in Dorren et al. (1995), a more complete time series has been published by Güdel (2004). There is a suggestive anti-correlation between X-ray flux and photospheric brightness (the star is optically brightest at its activity minimum), although the total X-ray luminosity varies by no more than a factor of 2 – 3.
Other reports refer to inactive solar analogs and K-type stars. Hempelmann et al. (2003) and Hempelmann et al. (2006) have reported a correlation between X-ray luminosity and the Ca H & K S index for the two K-type stars 61 Cyg A and B. Both show chromospheric modulations on time scales of about 10 years, one being regular and the other irregular. A gradual X-ray modulation was also seen during a time span 2.5 years in the G2 V star HD 81809, although there seems to be a phase shift by about 1 year with respect to the Ca cycle (Favata et al., 2004).
Additional information on potential coronal activity cycles has been collected from multiple observations of young open clusters and star-forming regions. Generally, such samples indicate that magnetically active, solar-like stars mostly lack well-expressed X-ray cycles unless their cycle-induced variability is no more than a factor of 2 (Gagné and Caillault, 1994; Gagne et al., 1995a,b; Stern et al., 1994, 1995; Micela et al., 1996; Sciortino et al., 1998; Grosso et al., 2000; Marino et al., 2002, 2003b).
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