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

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4. Normalized magnitudes

In this section the three stage period analysis (TSPA) from Paper i is applied to the [FORMULA] of V 1794 Cyg. The 21 subsets with less than 7 nights of observations were not normalized, and subsequently not analysed with the TSPA. Table 3 gives the P, [FORMULA] and [FORMULA] for the remaining 93 subsets with [FORMULA]. Paper i summarized three rules for rejecting some P, [FORMULA] or [FORMULA] estimates. The first rule rejected the P, [FORMULA] and [FORMULA] of a SET, if the bootstrap distribution of any model parameter was not gaussian (Paper i: [FORMULA]). Two detailed TSPA analysis examples for the subsets SET=42 and 114 were presented in Paper i. When we performed the same TSPA analysis procedures for each of the 93 subsets of normalized photometry, the [FORMULA] rejection rates were 5/12 and 8/81 for [FORMULA] and [FORMULA] subsets, i.e. more data increased modelling reliability. These thirteen [FORMULA] rejections are indicated by "r" in column [FORMULA] of Table 3. Thus our second rule (Paper i: [FORMULA]) rejected the P, [FORMULA] and [FORMULA] of every [FORMULA] subset (Table 3: 12/93 subsets). Finally, the third rule rejected nine "unreal" [FORMULA] in Table 3, which were those not present in [FORMULA] bootstrap samples out of 200 (Paper i: [FORMULA]). A detailed [FORMULA] and [FORMULA] rejection example for SET=114 was given in Paper i (Sect. 6.2). Our Table 3 also contains all rejected P, [FORMULA] and [FORMULA], because these undoubtedly contain some, although less reliable, TSPA modelling information.

4.1. Active longitudes: the [FORMULA] and [FORMULA] changes

Numerous studies have uncovered active longitudes in chromospherically active stars (e.g. Eaton & Hall 1979; Hall 1987, 1991; Zeilik 1991; Jetsu 1996). The starspots seem to concentrate on such long-lived regions rotating with a constant angular velocity. Here the active longitudes of V 1794 Cyg are searched for in the [FORMULA] and [FORMULA] epochs using the nonparametric methods from Jetsu & Pelt (1996: Paper iii). Active longitudes were detected in four RS CVn binaries with these methods in Jetsu (1996), where the data were the epochs when the starspots modelled by Henry et al. (1995) crossed the centre of the visible stellar disk. Here the data ([FORMULA]) are the [FORMULA] and [FORMULA] not rejected with [FORMULA], [FORMULA] and [FORMULA] (Table 3: [FORMULA]). These [FORMULA] are circular data when folded with an arbitrary P. We searched for periodicity in [FORMULA] with the nonparametric methods by Kuiper (1960: the K-method) and Swanepoel & DeBeer (1990: the SD-method), and their weighted versions that can also utilize [FORMULA] (Paper iii: the WK- and WSD-methods). These methods test the "null hypothesis" ([FORMULA]): "The [FORMULA] of [FORMULA] with an arbitraryPare randomly distributed between 0 and 1." We tested [FORMULA] between [FORMULA] and [FORMULA], i.e. within [FORMULA]% from [FORMULA]. As in Jetsu (1996), the significance level for rejecting [FORMULA] was [FORMULA]. The best [FORMULA] and [FORMULA] had critical levels [FORMULA] and [FORMULA] [FORMULA] (Table 2). Hence [FORMULA] was not rejected for V 1794 Cyg.

The [FORMULA] data quality prevented [FORMULA] and [FORMULA] estimates with the weighted WSD- and WK-methods. The two largest rescaled [FORMULA] weights (Jetsu 1996: [FORMULA] in Eq. 1) are [FORMULA], the smallest being [FORMULA]. The ratio [FORMULA] exceeds those in Jetsu (1996: Table 1), while the WSD-method "breakdown" parameter is [FORMULA] (Jetsu 1996: Eq. 5). Both [FORMULA] and [FORMULA] indicate unreliable [FORMULA] and [FORMULA] estimates. However, the order of significance and the [FORMULA] estimates for the P detected with these weighted methods are correct in Table 2 (see Paper iii: Sect. 5).

No significant active longitudes were detected for V 1794 Cyg, unlike in an equivalent analysis for four RS CVn binaries (Jetsu 1996). This absence of [FORMULA] periodicity verifies the inappropriateness of a traditional constant P ephemeris for V 1794 Cyg. Several phenomena complicate active longitude detection, strong differential rotation being the most probable one (Sect. 4.2). V 1794 Cyg had several low amplitude light curves with inaccurate [FORMULA] and [FORMULA]. The SD- and K-methods analyse all [FORMULA] with unity weights. Hence these low amplitude light curves deteriorate the [FORMULA] statistics. Because [FORMULA], [FORMULA] and [FORMULA] already eliminate a sizeable fraction of data, an additional rule would have been questionable (e.g. rejecting [FORMULA] when [FORMULA] goes below some fixed limit). On the other hand, no quantitative significance estimates for the active longitudes in chromospherically active stars have been published, except for those given here and in Jetsu (1996). For example, Eaton & Hall (1979), Hall (1987, 1991) or Zeilik (1991) do not quantify the statistics of this phenomenon. This may justify our subsequent qualitative [FORMULA] study with the best [FORMULA] and [FORMULA] of Table 2, not forgetting that these periodicities do not reach [FORMULA].

Although [FORMULA] has the best critical level ([FORMULA]), the [FORMULA] with [FORMULA] are discussed first, this periodicity being within [FORMULA] the best one detected with the SD-, WSD- and WK-methods (Table 2). Fig. 4 displays the [FORMULA] with [FORMULA] and an arbitrary zero epoch. The long-term [FORMULA] changes appear irregular, and even the yearly changes are rapid (Fig. 4a). Note that the dark and white distributions denote the analysed and rejected [FORMULA], respectively. The dark [FORMULA] distribution is broad and unimodal, the white rejected one being uniform, i.e. random (Fig. 4b). The dark and white [FORMULA] distributions are nearly uniform (Fig. 4c). The combined dark unimodal [FORMULA] and [FORMULA] distribution is too dispersed to reach [FORMULA].

[FIGURE] Fig. 4. a  The primary and secondary minima phases ([FORMULA], [FORMULA]) with [FORMULA]. The rules [FORMULA], [FORMULA] or [FORMULA] for rejecting/not rejecting some of these [FORMULA] and [FORMULA] estimates are explained in the beginning of Sect. 4. The [FORMULA] and [FORMULA] rejected/not rejected with [FORMULA], [FORMULA] or [FORMULA] are denoted with open/closed squares and open/closed triangles, respectively. b  White/dark distributions denote [FORMULA] rejected/not rejected with [FORMULA], [FORMULA] or [FORMULA]. c  As in b for [FORMULA]. d  Combined [FORMULA] and [FORMULA] distributions

A similar discussion is unnecessary for [FORMULA], but some striking features in Fig. 5 deserve a few comments. The [FORMULA] and [FORMULA] distributions are similar (Fig. 5bc). Furthermore, the rejected/not rejected distributions resemble each other, especially for [FORMULA]. Although the rejected data were not analysed, they fit the analysed data. Thus [FORMULA] might represent a real structure having rotated with a constant angular velocity for about two decades. After all, the dark distribution in Fig. 5d is quite improbable ([FORMULA]). Finally, Figs. 4 and 5 connect through [FORMULA], which is relatively close to the [FORMULA] window. But is [FORMULA] or [FORMULA] "real" or "spurious", remains unsolved. In conclusion, the active longitudes of V 1794 Cyg are a controversial subject, although we reject this hypothesis with [FORMULA].

[FIGURE] Fig. 5a-d. As in Fig. 4 for [FORMULA]

4.2. Differential rotation: the P changes

The short-term P changes indicate detectable differential rotation in V 1794 Cyg. Fig. 6 displays the rejected and nonrejected P from Table 3 with [FORMULA] errors. Their respective weighted means, [FORMULA] and [FORMULA], give [FORMULA]% and 7.5% (Jetsu et al. 1993: Eq. 3). These Z confirm that a reliable TSPA requires [FORMULA]. Nevertheless, even [FORMULA]% implies three times stronger differential rotation in V 1794 Cyg than in FK Com (Jetsu et al. 1993: [FORMULA]%). These rapid irregular P changes interfere with active longitude detection and verify the uselessness of a constant P ephemeris. The detailed example for SET=114 with nts=16 revealed that [FORMULA] by itself is not a sufficient condition for a reliable TSPA (Paper i: Sect. 6.2). The TSPA of low amplitude light curves is especially unstable, e.g. the P difference certainly exceeds [FORMULA] for the simultaneous SET=61 and 62 (nts=16 and 23). Since P is unique in time, these data must contain unidentified errors.

[FIGURE] Fig. 6. P rejected (open squares) and not rejected (closed squares) with [FORMULA] and [FORMULA] (Table 3: [FORMULA] errors!). The dotted and continuous horizontal lines are the respective [FORMULA] limits

Latitude does not determine the angular velocity of an individual sunspot. One should not forget the "sobering reminder" about the solar differential rotation law (Howard 1994: his Fig. 2). This law, frequently cited in connection with stellar differential rotation (e.g. Hall & Busby 1990: Eq. 2), relies on a gigantic sample (e.g. Howard 1994: 36708 sunspot groups). Were the solar-stellar-connection literally valid, regular P changes would be hard to detect in the meagre stellar data samples. Does our empirical Fig. 6 then support this conclusion? Suppose a stellar counterpart for the solar butterfly diagram, where the latitude of main activity correlates with the starspot cycle phase. This might be detectable as regular yearly P changes, assuming no abrupt longitudinal shifts for the activity centres. Fig. 7 displays the P obtained with TSPA for the yearly [FORMULA]. This analysis was restricted between 1985 and 1995, because at least four [FORMULA] subsets per year ([FORMULA]) were available during this interval. Fig. 8 shows the [FORMULA] with these yearly ephemerides. Their predictability, as well as the two ephemerides for 1990, are the subject of our next Sect. 4.3.

[FIGURE] Fig. 7. P for the yearly [FORMULA] (Table 4: [FORMULA] errors!). Note the two P values in 1990 when the only observed abrupt phase shift of [FORMULA] occurs

[FIGURE] Fig. 8. The n yearly [FORMULA] for [FORMULA] subsets and their ephemerides. Note that 1990 is divided into phot1990a and phot1990b

Unlike the seasonal (Fig. 6), the yearly P changes are surprisingly regular (Fig. 7). These P are very accurate (Fig. 7: [FORMULA]), and might even follow some cycle, except for those at 1986 and 1994. They fulfill [FORMULA]% with [FORMULA], and suggest that our nonparametric [FORMULA] analysis in Sect. 4.1 could have been restricted within [FORMULA]% ([FORMULA]) around 3.30. These yearly P might represent the butterfly diagram of V 1794 Cyg, but reveal no significant correlation with M. Again a solar analogy could explain this, because the sunspot maximum coincides with the main activity on the butterfly diagram mid-latitudes. This could induce a phase shift of several years between P and M. But V 1794 Cyg has had only one well established M minimum at 1991 (i.e. spot maximum), and two uncertain ones at 1984 and 1994 (Fig. 3). Hence we do not speculate about the connection between these regular yearly P changes, and latitudinal migration or activity cycle (if there is any).

4.3. Migration

The yearly P gave the [FORMULA] ephemerides in Fig. 8. Although the inadequacy of a constant P ephemeris for the whole data has been frequently emphasized, this does not necessarily indicate disruption for the regions of main activity. For example, latitudinal migration might induce apparent disruption. The [FORMULA] modulations for the very same long-lived active region could resemble the continuous O-C changes of the primary and secondary minima of eclipsing binaries (e.g. Jetsu et al. 1997: AR Lac). Such lagging or running ahead of [FORMULA] with a constant P would only be more prominent in V 1794 Cyg, and could display phase shifts. Assuming yearly phase coherence, this problem resembles that in Jetsu et al. (1997), except that the predictive periods are known (i.e. [FORMULA] below). All [FORMULA] and [FORMULA] from Table 3 were transformed into phases with the ephemeris of Fig. 4, where [FORMULA]. Only [FORMULA] were analysed, assuming that the main region of activity crosses the centre of the visible stellar disk at these epochs. Since the ephemerides in Fig. 8 relied on all available yearly [FORMULA], both rejected and not rejected [FORMULA] were studied, which ensured adequate yearly [FORMULA] data. We denote the yearly [FORMULA] averages with [FORMULA]. Our prediction begins from the six [FORMULA] of phot1985 with an average [FORMULA] (Table 4). The periods [FORMULA] (phot1985) and [FORMULA] (phot1986) determine a predictive period [FORMULA]. The [FORMULA] difference [FORMULA] between phot1985 and phot1986 predicts a phase shift of [FORMULA], where [FORMULA]. Hence the phot1986 prediction is [FORMULA], where [FORMULA]. The observed [FORMULA] for phot1986 yields a prediction residual [FORMULA]. Table 4 summarizes all predictions from 1985 to 1995. This unique [FORMULA] path determined by the [FORMULA] minimization rule is outlined in Fig. 9. The cases [FORMULA] are uncertain, e.g. RES=0.46 could be -0.54 or 1.46 (from phot1987 to phot1988). If [FORMULA] represents unpredictability, Table 4 contains 7/11 such cases. Thus these yearly predictions fail. Yet, the yearly [FORMULA] curves are continuous, the phase coherence for high amplitude light curves being excellent (Fig. 8). The [FORMULA] dispersion is largest in phot1991 when the brightness of V 1794 Cyg was nearly constant (Figs. 1 and 8h). Furthermore, the observed yearly [FORMULA] changes are discontinuous only once (phot1990a and phot1990b). All other [FORMULA] discontinuities coincide with observational gaps. Although these yearly [FORMULA] changes display complex shapes with [FORMULA], they retain continuity (e.g. Fig. 9: "s"-shape during phot1989).

[FIGURE] Fig. 9. The predictions for the yearly means of the primary minima [FORMULA] from Table 4. The beginning of Sect. 4.3. contains a detailed example of how the yearly P of phot1985 and phot1986 provide one of these predictions. The [FORMULA] and [FORMULA] rejected/not rejected with [FORMULA], [FORMULA] and [FORMULA] are denoted by open/closed squares and open/closed triangles, respectively. The horizontal dashed lines are the yearly [FORMULA] limits. The [FORMULA] path (thick continuous line) connects earlier observed yearly [FORMULA] to the predicted ones [FORMULA]. The narrow continuous lines denote the error limits [FORMULA]. Note that 1990 consists of phot1990a and phot1990b

The only observed abrupt longitudinal activity shift occurs between SET=56 and 57 on September, 1990 (about [FORMULA]). Hence the predicted pairs were from phot1989 to phot1990a, from phot1990a to phot1991, and from phot1990b to phot1991 (Table 4 and Fig. 8). Because the other [FORMULA] discontinuities within observational gaps can not be verified, we conclude that only one activity shift is observed in V 1794 Cyg. Thus the identification of "new" starspots after observational gaps is not trivial with a constant P, e.g. in the linear approach by Henry et al. (1995: their Figs. 4, 9, 15 and 20). Whether the [FORMULA] changes in V 1794 Cyg resemble the continuous O-C changes of eclipsing binaries, remains unsolved. The analysis by Berdyugina & Tuominen (1998) reveals such continuous longitudinal migration in four RS CVn binaries. Such continuous longitudinal migration in chromospherically active stars would mean that activity centres are not frequently disrupted. This [FORMULA] continuity could be resolved with uninterrupted long-term photometry of an active star sufficiently close to [FORMULA], say permanently [FORMULA] above the horizon. An automated telescope at higher latitudes could perform this task.

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Online publication: November 2, 1999