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Astron. Astrophys. 333, 565-570 (1998)

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4. Search for periodicities

The new LTPV data are not in the same photometric system as the data described in Sect.  2.1and are too sparse to be included in a frequency analysis. A period search was thus carried out using Fourier analysis on the differential P minus A [FORMULA] data in the frequency range 0.0-0.2 cd-1.

The combined V and y light curve was visually inspected, and a hand-drawn continuous enveloping curve was drawn through the minima of the microvariations. The difference between the V magnitude and the corresponding value read from the hand-drawn curve was then subtracted from the observed V, resulting in [FORMULA] which describes at each date the microvariations excluding the contribution from the developing SD phase.

In order to assess objectively the approach of eye-estimating the underlying SD variations and correcting the observed V magnitudes for the contribution of the S Dor phase, we have also made a frequency analysis of the [FORMULA] magnitudes corrected for a linear trend fitted to all LTPV data of Sect.  2.2in the [FORMULA] and v bands (this trend has a gradient of -0:m069 per year in V).

4.1. Data corrected for the long-term trend by eye-estimate

The spectral window is dominated by a strong peak at 0.00278 cd-1, due to the annual rythm of our observations. The amplitude spectrum shows its strongest peak at [FORMULA] cd-1, (a cycle of 99:d4) with weaker peaks on either side at 77:d8 and 139 days. These correspond to a difference in frequency of 0.00278 cd-1, the aliases produced by the annual cycle. A least-squares sine fit with [FORMULA] reduces the [FORMULA] standard deviation from 0:m043 to 0:m036, still more than a factor of three larger than the expected s.d. as derived from the differences between the comparison star y measurements. Fig. 3 shows the amplitude spectrum for the [FORMULA] data (middle panel) and the corresponding spectral window.

[FIGURE] Fig. 3. Spectral window (top) and amplitude spectrum (middle) of R 40 ([FORMULA]) in the frequency range 0.0-0.05 cd-1. The lower panel is the amplitude spectrum for the [FORMULA] data after prewhitening with [FORMULA]

After prewhitening for [FORMULA] cd-1, the Fourier analysis yields a number of peaks in the amplitude spectrum where the strongest is [FORMULA] cd-1 (121:d3, see Fig. 3), corresponding to the semi-period found by Szeifert et al. (1993); the residual remains at a high level [FORMULA]. Further prewhitening with [FORMULA] leads to an amplitude spectrum characterised by very strong noise.

4.2. Data corrected for for the long-term trend by subtracting a linear trend

The strongest peak in the amplitude spectrum appears at 0.01015 cd-1, but a new peak appears at 0.00077 cd-1. We call this frequency [FORMULA] ; it corresponds to a cycle of [FORMULA] 1300 d. After prewhitening with these two frequencies, the resulting amplitude spectrum shows a maximum at 0.00570 cd-1 and a number of peaks of slightly lower power, among which 0.00855 cd-1, a frequency very close to the second frequency found in Sect.  4.1. Most of these peaks are aliases of possible other secondary frequencies (see Fig. 4). Again, prewhitening with any of these frequencies does not convincingly reduce the [FORMULA] standard deviation of the fit.

[FIGURE] Fig. 4. Top to bottom: amplitude spectrum of the linearly-corrected [FORMULA] magnitudes (top), the same after prewhitening for [FORMULA] (middle), and the final amplitude spectrum after prewhitening by the two frequencies [FORMULA] and [FORMULA] (bottom)

In order to see whether shorter time intervals might reveal significant changes in the amplitude spectra, we divided the data set (corrected for the linear trend) in three more or less equal time intervals, viz. the period before JD2447500 (set 1, 109 points), the time interval between 2447500 and 2448700 (set 2, 127 points) and the remaining data (set 3, 74 points). Each such subset was submitted to a Fourier analysis in the spectral domain below 0.02 cd-1. [FORMULA] is the only frequency that appears with comparable strength in each amplitude spectrum (with amplitude peaks at 0.0102, 0.0105 and 0.0106, respectively for sets 1, 2 and 3), thus lending additional support to our conclusion that the principal cycle of microvariation is visible throughout the ascending branch of the S Dor cycle.

Furthermore, we de-trended all uvby data by removing the linear slope; Fig. 5 is the resulting phase diagram. There we see that the fitted ranges of variation in u and v (0:m090) are slightly larger than in b and y (0:m085). In addition, the scatter about the colour curves (see Fig. 2) is very much stronger around 1985 (near minimum SD phase) than several years later. The small difference in y -to-u amplitude and the fact that the amplitude of the colour variations is of the same order as the precision with which the colour can be determined, make it impossible to draw any firm conclusions about the colour behaviour when all data obtained during the ascending SD branch are combined. It should be stressed here that the individual light curves do show a correlation with colour (especially with [FORMULA]), and that the cycles of 1986-89 also allow a solution with a cycle half as long (46:d2 [FORMULA]) with a reversed colour behaviour.

[FIGURE] Fig. 5. LTPV uvby magnitudes against phase in the [FORMULA] period ([FORMULA]) covering more than 30 cycles of variation. Phase 0 was chosen arbitrarily, tick marks on the Y-axis are 0:m05 apart

We conclude our frequency analysis by accepting only [FORMULA] and [FORMULA], leaving open the possible presence of secondary frequencies in the microvariations. For the sake of argument, one could adopt [FORMULA], the cycle found by Szeifert et al. (1993), as an acceptable choice for a second frequency. That a second frequency like [FORMULA] could contribute to a better explanation of the complex light curve is shown in Fig. 6 which illustrates the impact of the combination of one slow and two fast cyclic oscillations.

[FIGURE] Fig. 6. [FORMULA] data corrected for the linear trend underlying the SD phase. The least-squares sine curve was calculated with [FORMULA], [FORMULA] and [FORMULA] (the same frequency as selected by Szeifert et al. 1993). The result of the combination of these three frequencies produces an irregular alternation of the light maxima. Medium and long gaps in the data may, at times, give the impression that the period doubles from one season to the other. Tick marks on Y-axis are 0:m1 apart

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© European Southern Observatory (ESO) 1998

Online publication: April 20, 1998