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Astron. Astrophys. 335, 605-621 (1998)

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4. Period determinations of the microvariations

4.1. The period-search methods

The search for periodicities was made with two methods: the Fourier method and the Phase Dispersion Minimization Method (PDM) from Stellingwerf (1978). The two methods differ in that the Fourier method deconvolves the lightcurve in a series of sinusoidal variations whereas the PDM method makes no apriori assumptions about the shape of the variations. The results of the two methods were compared to find information about the shape of the variations and about the reliability of the derived periodicities.

During 11 of the selected intervals the microvariations occurred while the visual magnitude was slowly changing. In these intervals we subtracted the slower variation.

When the Fourier method was applied, the HWHM of the main peak in the Fourier data spectrum was taken as the uncertainty of the frequency. The PDM method usually gives a slightly different uncertainty of the frequencies as the variation is not exactly sine-shaped. We fitted Gaussians through the minima of the [FORMULA]-spectrum of the PDM method. The standard deviation of the Gaussian is taken as uncertainty in the frequency.

One example of a microvariation of R71 is given in Fig. 2. The top part shows the observed lightcurve during an interval with a clear periodicity; the middle part shows the comparison with a sine curve of the main frequency; the lower part shows the lightcurve folded with the sine wave.

[FIGURE] Fig. 2. a The photometric data of R 71 between Nov 1984 and April 1985; b The data with a sine wave of the calculated frequency; c The data folded with a sine wave. The lightcurve is given by two different symbols because the measurements were done with two different instruments (Spoon et al., 1994)

4.2. Periods and amplitudes of microvariations

The results of the period search are listed in Table 4. For each star and each interval we give the stellar data and the data about the periodicity.


Table 4. Periods and amplitudes of microvariations.
Notes: (1): the quoted value is the half of the peak-to-peak amplitude
(2): values for [FORMULA] and [FORMULA]=-9.54
(3): values for [FORMULA] and [FORMULA]=-10.35
(4): sine of 296 days.

Columns 3 to 6 give information about the [FORMULA], [FORMULA], [FORMULA] and log  g derived in the way described in Sect. 2.4. The values of [FORMULA] were derived from the BC, using the relation between BC and [FORMULA] from Schmidt-Kaler (1982) for supergiants. For R 71 these data refer to the stellar parameters with [FORMULA] and [FORMULA] and for R 127 they refer to [FORMULA] and [FORMULA].

Columns 7 and 9 give the dominant period of the microvariations and its half-amplitude. The period of the dominant variation is derived from the highest peak in the Fourier data spectrum, the frequency of the minimum value of [FORMULA] and from the aliases and the subharmonics. The quoted uncertainty is the 1[FORMULA] error, that was derived by fitting the dip in the [FORMULA]-spectrum by a Gaussian profile and from the HWHM of the peak in the Fourier spectrum. In Column 10, we give the probability (in percentage) that the minimum in [FORMULA], corresponding to that period, is due to a random fluctuation. In all cases, except for R71, this probability is less than 20 percent. Column 11 gives the number of periods covered by the interval. If the photometry shows a gradual trend during the interval, we subtracted this trend from the data before determining the period. This correction is indicated in the last column, where "up" and "down" indicate a trend of increasing or decreasing visual brightness respectively.

The Q-values are listed in Column 8. They were derived from the mass listed in Table 2 and the radius. They will be discussed in Sect. 5.2.

We briefly describe the results for each of the stars individually:

  1. R 71 : Eight intervals with a sufficient density of photometric data points were selected. The first one is during a phase of decreasing brightness. A rather long interval of 450 days was selected in order to have enough data points for a meaningful search for periodicities. We corrected for the linear decrease during this period before we searched for periodicity of the microvariations. The other intervals are during visual minimum. The periods of the microvariations varies between 76 days and 19 days, and the amplitude varies between 0.031 and 0.058 mag. The longest period occurred when the star was brightest. The periods are shorter during subsequent intervals, when the star has about a constant visual magnitude. All but two periods have a probability of less than 20 percent for being due to random fluctuations. The variation in the interval from JD +6010 to +6160 was chosen for the example in the previous section (Fig. 2).

  2. HR Car : Five intervals were selected. The first one was when the star was at a visual minimum of V=8.4 mag.; the other ones when V[FORMULA]8.1 mag. The period of the microvariations ranges from 18 to 41 days, and the amplitudes vary between 0.010 and 0.059 mag.

  3. 164 G Sco : We selected two intervals: one of 545 days and a shorter one of 115 days. Both occurred when the star was in a relative minimum. During the first interval the star was slowly decreasing in visual brightness, whereas it was at about a constant magnitude during the second interval. The microvariations had periods of 55 and 45 days. The amplitudes are 0.043 and 0.037 mag. respectively. Sterken et al. (1991) determined a mean period of 57 days during a long time interval from about JD +5500 to +7500 during a period when V decreased from 6.5 to 6.9 mag.

  4. S Dor : We found only two intervals of sufficient density of data points to search for periodicities of the microvariations. The first coincides with the visual minimum, the second one with the phase of increasing brightness. The microvariations during visual minimum have a period of 195 days. This is the longest period of microvariations found in our sample. The amplitude is 0.098 mag. During the second interval the period is considerably shorter, 131 days, and the amplitude is also smaller.

  5. R 127 : Two intervals were selected. The first interval of 215 days coincides with a phase of visual brightening. The microvariations during that time have a period of 35 days and an amplitude of 0.008 mag. The second interval occurs when the star is close to maximum visual brightness. The period of the microvariations has increased to 111 days and the amplitude has also increased to 0.031 mag.

  6. AG Car : Three intervals were selected, all of them during the phase of visual minimum. The microvariability period varies from 41 to 11 days, but the amplitude remains approximately constant near 0.027 mag.

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Online publication: June 18, 1998