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Astron. Astrophys. 323, 151-157 (1997)

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2. Activity-cycle length, rotation period and color

The stars of the HK-project can be divided into two groups according to their rotation rate: rapidly rotating young stars with a high activity level and slowly rotating old stars with a low activity level (Baliunas et al.  1995, 1996b). Activity variations with well-defined periods are observed predominantly in older stars. Younger stars tend to display stronger, more irregular activity variations.

We focus on the slowly rotating stars, which we define as those stars having a Rossby number larger than 0.9, and we exclude the rapidly rotating stars from our analysis. The resulting subset is similar to the group of solar-type stars examined by Soon et al. (1994) and Baliunas & Soon (1995).

In Table 1 we summarize the relevant properties of all known lower main-sequence stars with well-defined activity cycles. The third column gives cycle periods, compiled from Baliunas et al. (1995). We have included only stars with well-defined cycles (those rated "good" or "excellent"), as well as four stars that may be in the equivalent of a Maunder minimum or, alternatively, have cycle periods longer than about 20 years (Baliunas & Soon  1995). There are no stars with well-defined periods shorter than 7 years. The intervals between consecutive maxima in the sunspot record, as measured since the beginning of the 18th century, have a mean length of 11 years, and a standard deviation of about 2 years, i.e. 18%. This is taken to be indicative for the variability of stellar cycles. We thus estimate the deviation of the measured cycle period from its mean value as

[EQUATION]

where [FORMULA] is the number of cycles covered by 25 years of observations.


[TABLE]

Table 1. Stars with periodic chromospheric activity


The fifth column contains the convective turnover times [FORMULA] near the bottom of the convection zone. These are based on calculations by Kim & Demarque (1996) of the local turnover time (in fact, half the global turnover time) for lower main-sequence stars in the mass range [FORMULA]. Their results indicate that [FORMULA] is roughly independent of age on the main sequence, depending only on stellar mass, i.e. on [FORMULA]. We chose a stellar age of 2 Gyr and we performed a cubic fit in terms of [FORMULA] to their data,

[EQUATION]

where [FORMULA] is in days. The resulting curve is shown in Fig. 1. We reiterate that, unlike [FORMULA], the increase of [FORMULA] with [FORMULA] does not come to a halt or slow down significantly beyond [FORMULA].

[FIGURE] Fig. 1. Logarithm of the convective turnover time [FORMULA] (in days) versus [FORMULA], for lower main-sequence stars. The data points represent model calculations by Kim & Demarque (1996); the dotted line is a cubic fit (Eq.  2).

In trying to identify trends in the cycle periods of slowly rotating stars we proceed along lines similar to those followed by Noyes et al. (1984b). As a first step we consider the dependence on the rotation period. In Fig. 2 we plot the cycle period [FORMULA] versus [FORMULA] - the slowly rotating stars are indicated by the full circles. For stars with similar values of [FORMULA] we observe an increase of [FORMULA] with increasing [FORMULA]. But stars of different spectral type seem to be located on different curves, that are shifted to the left by an amount that increases with decreasing [FORMULA].

[FIGURE] Fig. 2. [FORMULA] versus [FORMULA], for all the stars in Table 1. Open symbols denote the rapidly rotating stars; arrows denote the four 'Maunder-minimum stars'. The labels indicate [FORMULA].

Hence the cycle period cannot be parametrized by the rotation period alone and must also involve a color-dependent term, which may be provided by the convective turnover time. We assume that [FORMULA] depends on [FORMULA] and [FORMULA] through a powerlaw, i.e.

[EQUATION]

where [FORMULA] is measured in years, and [FORMULA] and [FORMULA] in days. We determine a, b and c by performing a least square fit of the form [FORMULA]. The uncertainty in the values of [FORMULA] is [FORMULA], and we assign a weight [FORMULA] to each data point.

The best fit has [FORMULA], [FORMULA] and [FORMULA], with correlation coefficients [FORMULA], [FORMULA] and [FORMULA]. This yields [FORMULA] (note the effect of the strong anticorrelation between b and c). Thus our result is in agreement with a parametrisation [FORMULA], where Ro is the Rossby number,

[EQUATION]

In Fig. 3 we plot [FORMULA] versus [FORMULA], so that all the data points are shifted along the horizontal axis by the amount [FORMULA], relative to their positions in Fig. 2.

[FIGURE] Fig. 3a and b. [FORMULA] versus [FORMULA], with [FORMULA] (the uncertainties in a, b and c are not indicated in this plot). The dotted line, defined by [FORMULA], results from a least square fit applied to the slowly rotating stars (excluding the four 'Maunder-minimum stars'). The symbols have the same meaning as in Fig. 2. Labels indicate the Rossby number. Top (a): slowly rotating stars. Bottom (b): the complete sample of stars.

Some of the measured cycle periods show significant deviations from the best fit, for which their may be two reasons. First, our estimates of [FORMULA] are based on the assumption that the level of variability in [FORMULA] (cf. Eq.  1) is the same for all the stars in our sample. Perhaps we underestimate the variability in some cases. Second, we have ignored the uncertainty in the rotation periods. No reliable estimates of this uncertainty are available, but it may contribute to the scatter.

Although the four "Maunder-minimum stars" were not included in the least square fit because their cycle periods are (as yet) unknown, the resulting shift puts three of them at a location in Fig. 3, that is roughly in agreement with a cycle period of about 20 years. This suggests that their chromospheric activity may prove to be periodic in the future.

The cycle periods of rapidly rotating stars, indicated in Fig. 3b by the open circles, are much longer than what would be expected on the basis of the relation for the cycle periods of slowly rotating stars, and the deviation appears to increase with decreasing Rossby number. Hence the powerlaw as derived for the slowly rotating (old) stars does not hold for rapidly rotating (young) stars.

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© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998

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