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Astron. Astrophys. 351, 212-224 (1999)

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5. Conclusions

The seasonal (SET) light curves of V 1794 Cyg were modelled with a second order Fourier function (Eq. 1). The [FORMULA] were determined separately for each subset to enable untroubled modelling of two phenomena undoubtedly present in V 1794 Cyg: differential rotation and longitudinal shifts of activity centres, also referred to as longitudinal activity migration that is connected to the concept of active longitudes . Our rejection of the traditional constant period ephemeris approach eliminated unique phases for the whole data. Unique phase information could, however, be retrieved from the epochs of the primary and secondary minima in time ([FORMULA] and [FORMULA]) that are nearly independent of the seasonal P (see Jetsu et 1993: Fig. 8). Some outliers were removed as flares before normalizing the UBVRI magnitudes and modelling. If some of these outliers do represent photometric flares, they are of short duration, and temporally unconnected to the seasonal light curve maxima. Earlier studies have indicated that photometric flares in evolved stars tend to occur close to the light curve maximum, but are rare events of short duration (e.g. Jetsu et al. 1993, Henry & Newsom 1996).

Significant activity cycles were not detected in the mean (M) and the total amplitude (A) of seasonal UBVRI light curves. The previously reported M ([FORMULA]) and A ([FORMULA]) cycles in Jetsu (1990a, 1990b) were not significant, the former cycle essentially represented the time span of those earlier data. The difficulties of cycle detection in chromospherically active stars could be explained with a plausible solar-stellar analogy, i.e. by considerable changes in the total sunspot number from one solar cycle to another. Apart from not being strictly periodic (i.e. predictable), the solar cycle has even faded for prolonged intervals, like the Maunder Minimum.

The normalized magnitudes ([FORMULA]) utilize the combined information from all UBVRI magnitudes. The seasonal [FORMULA], [FORMULA] and P were modelled from [FORMULA] with the three stage period analysis method (Paper i: TSPA). The complementary methods identified spurious periodicities ([FORMULA]) and confirmed that the [FORMULA] of V 1794 Cyg is indeed [FORMULA] (Paper i: Sect. 6.3.). Bootstrap provided all model parameter estimates, and their errors. All models violating reliable bootstrap statistics were consistently discarded, i.e. significant deviations of any model parameter distribution from a gaussian. Three rejection rules excluded unreliable [FORMULA], [FORMULA] and P estimates from further analysis (see Table 3).

Active longitudes were searched for in the most reliable [FORMULA] and [FORMULA] epochs that are circular data when folded with an arbitrary P. The nonparametric methods from Paper iii detected no periodicities reaching a significance level of [FORMULA] for rejecting the "null hypothesis" that the [FORMULA] and [FORMULA] represent a random sample drawn from a uniform phase distribution. The critical levels for the best periods [FORMULA] and [FORMULA] were [FORMULA] and [FORMULA] (Table 2). The latter periodicity may represent an active longitude having rotated with a constant angular velocity for about two decades (Fig. 5).

If compared to the estimates in Hall & Busby (1990), the 7.5% variations of the most reliable seasonal P estimates would imply strong differential rotation in V 1794 Cyg (Fig. 6: closed squares). A lower 3.3% estimate was determined from the yearly [FORMULA] (Fig. 7). These regular yearly P changes did not correlate with those of M, and were therefore not interpreted as the "butterfly" diagram of V 1794 Cyg. As in the Sun, this absence of correlation between M and P might due to a phase shift of several years, the sunspot maximum coinciding with main activity on mid-latitudes, while at the sunspot minimum the activity shifts from solar equator (smallest P) to the highest latitudes (largest P).

Finally, the predictiveness of the yearly ephemerides was tested by checking whether the [FORMULA] changes in V 1794 Cyg qualitatively resemble the continuous O-C modulations of eclipsing binaries (e.g. Jetsu et al. 1997). Such continuity would indicate that the activity centres survive the disruptive influence of differential rotation and/or latitudinal migration. But the yearly predictions failed (Table 4). Nevertheless, only one abrupt [FORMULA] shift was observed on September 1990 , while the other [FORMULA] discontinuities coinciding with observational gaps could not be reliably established (Fig. 9).

We encountered several manifestations of the solar-stellar-connection in the long-term photometry of V 1794 Cyg, and our time series analysis leaves an impression that unpredictability is one of the foremost characteristics for magnetic activity in this object.

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© European Southern Observatory (ESO) 1999

Online publication: November 2, 1999
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