4.3 Activity cycles in the young Sun

Records of sunspot numbers back over several hundred years show a near-cyclic modulation that has turned out to be a central challenge for dynamo theories. The activity period between two successive spot maxima is approximately 11 years; because the magnetic polarity reverses after one period, the full magnetic cycle amounts to 22 yr. Activity cycles corresponding to the 11 yr solar cycle have been found on many cool stars, mostly as a result of the Mount Wilson HK project (Baliunas et al., 1995) that has collected a continuous data stream of the chromospheric Ca ii H & K line flux diagnostic for many stars over several decades. A subset of stars appear to lack such cycles, however, and very active stars tend to exhibit an irregular rather than a cyclic mode of variability (Hempelmann et al., 1996). An alternative method for finding activity cycles on magnetically active stars is the identification of cycles in the starspot coverage.

4.3.1 Starspot cycles of solar analogs

Messina and Guinan (2002Jump To The Next Citation Point) 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 7View Imagea). Among G-type stars, only EK Dra appears to be compatible with the third class (Messina and Guinan 2002Jump To The Next Citation Point).

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Figure 7: Left (a): Ratio between activity-cycle frequency and rotation frequency plotted vs. the inverse Rossby number, mostly for solar analogs. Three theoretical branches (“inactive”, “active”, and “super-active”) are shown by solid lines. Key to the labels: A = BE Cet, B = 1 κ Cet, C = 1 π UMa, D = EK Dra, E = HN Peg, F = DX Leo (K0 V), G = AB Dor (K0 V), H = LQ Hya (K2 V) (from Messina and Guinan 2002Jump To The Next Citation Point). Right (b): Amplitude of V-band variability as a function of the inverse Rossby number. The labels are as for the left panel (from Messina and Guinan 2002Jump To The Next Citation Point, reprinted with permission).

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 7View Imageb). 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, 2002Jump To The Next Citation Point).

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Figure 8: V band photometric time series and sinusoidal (plus long-term trend) fits for EK Dra (left, a) and κ1 Cet (right, b) (from Messina and Guinan, 2002Jump To The Next Citation Point, reprinted with permission).

The standard near-ZAMS solar analog, EK Dra, has been an important target for cycle studies. Dorren and Guinan (1994aJump To The Next Citation Point) 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, 1994aJump To The Next Citation Point), 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 8View Image). 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 2003Jump To The Next Citation Point and Järvinen et al. 2005Jump To The Next Citation Point). 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 (2003Jump To The Next Citation Point) 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, 2003Jump To The Next Citation Point), 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 (2003Jump To The Next Citation Point), 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 (2003Jump To The Next Citation Point) have found a power-law relation between the rotation-period variation, ΔP, and the average rotation period, P , of the form (see Figure 9View Imagea)

ΔP ∝ P 1.42±0.5. (2 )

Further, a tight correlation is found between differential rotation, parameterized by Δ Ω∕Ω, and the activity cycle frequency, ω cycl,

(−0.055±0.004)ΔΩ∕Ω ωcycl ∝ e , (3 )

suggesting that Δ Ω ∕Ω is a key parameter controlling the duration of the activity cycle (see Figure 9View Imageb).

Differential rotation has also been measured on other very active stars (e.g., Donati et al. 2003aJump To The Next Citation Point), among them a solar-like post-T Tauri star (Donati et al., 2000Jump To The Next Citation Point), 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).

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Figure 9: Left (a): Rotation period variation as a function of the mean rotation period, for a sample of young solar analogs. The solid line is a power-law fit to the entire sample also containing early K stars, while the dotted line is a fit to the G-star sample only. Key to the labels: A = BE Cet, B = κ1 Cet, C = π1 UMa, D = EK Dra, E = HN Peg, F = DX Leo (K0 V), G = AB Dor (K0 V), H = LQ Hya (K2 V), I = 107 Psc (K1 V), L = 61 UMa (G8 V), M = β Com, N = HD 160346 (K3 V), O = 15 Sge, S = Sun (from Messina and Guinan, 2003Jump To The Next Citation Point). Right (b): The activity cycle frequency is shown as a function of the relative surface differential rotation amplitude, Δ Ω ∕Ω. Vertical dotted lines connect multiple cycles. Three different branches are indicated by the solid curves (from Messina and Guinan, 2003, reprinted with permission).

4.3.2 X-ray cycles of solar analogs

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. (1995Jump To The Next Citation Point), a more complete time series has been published by Güdel (2004Jump To The Next Citation Point). 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, 1994Gagne et al., 1995aJump To The Next Citation Point,bStern et al., 19941995Jump To The Next Citation PointMicela et al., 1996Sciortino et al., 1998Grosso et al., 2000Marino et al., 20022003b).

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