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

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1. Introduction

The observation of magnetically-induced activity of late-type stars is surely among the most important discoveries of modern astrophysics. Empirical studies which relate the characteristics of stellar activity (photospheric spots, chromospheric plages, and a hot X-ray emitting corona) with basic stellar parameters such as mass, age, and rotation are a natural way to test dynamo theories. Many studies have demonstrated the fundamental importance of stellar rotation for generating the mean activity level whereas the relation between turbulence and stellar activity has not yet been well determined observationally.

Stellar rotation and turbulent motions in the stellar interior produce not only a magnetic field, but also lead to differential rotation. The observed ([FORMULA]) pattern of the solar rotation (Libbrecht 1988; see also the review by Kennedy (1996) concerning the GONG experiment) is nearly explained by theory (Küker et al. 1993, Rüdiger & Kitchatinov 1994, Kitchatinov & Rüdiger 1995). However, any successful model must also predict the differential rotation of stars other than the Sun.

The observation of surface differential rotation (SDR) of stars is possible via three methods: i) stellar seismology; ii) analysis of the rotational broadening of spectral lines; and iii) observation of stellar `butterfly diagrams'. The two latter methods are restricted to delivering only the [FORMULA] dependence of rotation on the stellar surface. Slowly rotating stars like the Sun (2 km/s at the equator) produce a rotational line-broadening which is smaller than the broadening caused by turbulent motions in the stellar photosphere (the radial-tangential macroturbulence ([FORMULA]); values of [FORMULA] are given by Gray 1988). Although it may be possible to derive [FORMULA] even if its value is smaller than [FORMULA], the detection and measurement of surface differential rotation of slowly rotating stars from spectral line profiles is an unsolved question at least in practice (e.g., Gray & Baliunas 1997). The detection and measurement of differential rotation from observation of non-radial stellar pulsations like the 5 min oscillations of the Sun is an outstanding task and requires an observational accuracy which has not yet been reached from terrestrial sites.

The only method which is practical at present is the observation of stellar butterfly diagrams: i.e., monitoring the rotation period over an activity cycle. The basic idea is that (by analogy with the Sun) active regions (ARs) appear in two belts on the stellar surface at mid-latitudes (one in each hemisphere) at the beginning of an activity cycle, and shift monotonically towards the equator as the cycle progresses.

Stellar butterfly diagrams have been observed using survey data from Mount Wilson Observatory's HK Project (Baliunas et al. 1985; Donahue 1993). The presence of SDR has been detected for the Sun and 36 lower main-sequence stars with various spectral types, ages and rotation periods. As many as six different patterns have been identified showing changes in rotation period with time, or where possible, phase of the activity cycle (Donahue 1993; Donahue & Baliunas 1994). In some cases, stars with nearly identical parameters (i.e., mass and age) show completely different butterfly diagrams. This raises some questions, in particular, whether artifacts in the data can produce errors or inaccuracies in the determination of rotation period. Donahue (1993) notes that an important influence upon the determination of rotation period is active region growth and decay (ARGD).

ARGD appearing simultaneously at different stellar longitudes, will produce variability of not only the observed amplitude, but also the phase of rotational modulation, possibly including variations of the period. In those cases a deviation from sinusoidal modulation is also expected. Further, SDR can produce a multi-periodic signal caused by ARs positioned at different latitudes; the well-known butterfly-diagram of the Sun (Maunder 1913; Yallop & Hohenkerk 1980) shows that spots appear at a variety of latitudes at any given time. Hence, measurements of the gross rotation period will be variable on time-scales which are shorter than the activity cycle. Despite the possibility of strong scatter of the measured rotation period at shorter time-scales, we can hope that a general trend will be visible provided the length of the time series observed is sufficient to see this trend and, if the number of observations per time interval is large enough to determine the parameters of rotational modulation with sufficient precision.

This expectation must first be tested with the Sun; it is the only star for which we know both the differential surface rotation and can trace the appearance and disappearance of specific AR's on the stellar surface in detail. Although the pattern of surface differential rotation derived from different observational tracers differs slightly, the solar surface differential rotation can be reasonably approximated by (cf., Lang 1992)


where [FORMULA] is the solar latitude and [FORMULA] is the synodic rotation period. Hence, assuming a possible AR latitude range [FORMULA] a band-width [FORMULA]  d results. Donahue & Keil (1995) analysed a time series of disk-integrated Ca II K emission (Keil & Worden 1984) finding the correct qualitative behaviour: a jump from faster rotation to slower values at activity cycle minimum followed by a trend of increasing rotational velocity with cycle phase. Quantitatively, they derive a sidereal rotation period of 28.5 d at the beginning of Cycle 22, corresponding a synodic period of [FORMULA]  31 days that is too large as Schrijver (1996) has recently criticized. It also seems that the decline of rotation period observed is too steep, reaching [FORMULA]  d only five years later.

Because all of the parameters involved in detecting SDR are likely to also vary themselves (largely due to ARGD), a two-dimensional time-frequency analysis is preferred. In this paper we present the results of a wavelet analysis of disk-integrated solar Ca II K-line fluxes, including those analyzed by Donahue & Keil (1995).

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© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998