Astron. Astrophys. 345, 172-180 (1999)

3. Period analysis

We performed a standard CLEAN analysis (Roberts et al. 1987, Gies & Kullavanijaya 1988) of the HeI 4713 Å line. The resulting power spectra are shown as grey-scale representations in Figs. 4 and 6. There is considerable power near periods around 1 day and longer, which for both stars we can mostly attribute to wind contaminations, rather than windowing aliases. These low-frequency signals may also partly reflect unaccounted systematic differences between observatories and slight differences in processing of flat fields, normalization, etc. between nights, in spite of our efforts to remove them. In the following we ignore this frequency range. In the uncontaminated areas (above 1.5 c d^-1) significant power is found across the line profile at several frequencies, visible as horizontal grey areas in the figures, which we attribute to NRP. The frequencies quoted in Sect. 4 were determined by first fitting in each velocity bin a Gaussian function to the peak in the power (f):

This yields three parameters: the maximum power , the central frequency and the width of the peak . To determine the uncertainty in the height of the peak relative to the noise level, defined as , should be taken into account. For this purpose we use the width of the peak at , following Schwarzenberg-Czerny (1996). Applying Eq. 1 gives:

We use a conservative method to calculate the noise level, which is a critical parameter: we defined as the power below which 95% of the datapoints fall in the histogram of the periodogram between 5 and 10 c d^-1. For  Per we find and for  Cep . We have chosen this procedure because white noise gives low and high peaks at all frequencies and apart from this it is not always clear whether a peak is only noise or contains a weak periodic signal.

Fig. 4. Grayscale representation of the periodograms as a function of velocity of  Per spectra. The side panel shows the power summed over all velocities. The highest peak is normalized to unity. The top panel displays the ratio of the observed standard deviation to the expected standard deviation

Finally, the average frequency across the line profile was computed in two ways: a normal average with its standard deviation and an error-weighted average. In the latter method the uncertainty in the average is directly derived from the errors, , using the following equations:

In both cases we computed for every datapoint the deviation where is the standard deviation. Secondly, we computed for each point the probability of being a statistical fluctuation assuming a Gaussian distribution. We discarded all points for which is smaller than 0.5 (Chauvenet's criterion). In a sound statistical distribution both methods should give the same values. In case of differences we have chosen the method giving the largest error.

The phase of the signal can be extracted from the CLEANed Discrete Fourier Transform (CDFT), using the following representation:

Here is the flux as a function of time, t, is the frequency, is the frequency interval , is the power, is the phase of component i, and is the average time of the sample, used to relate the calculated phases to the Barycentric Julian Date (2447818.698 for  Per and 2447818.532 for  Cep). By convention is defined between 0 and 1. Note that features with larger phase arrive earlier in time.

As an independent check we derived the phase information also from least- multiple sine fits (components in Eq. 5) with frequencies fixed on the main peaks in the periodograms. We used 0.997, 9.138 and 7.956 c d^-1 for  Per and 0.424, 1.96, 2.54 and 3.67 c d^-1 for  Cep. Error bars on the phases could then be derived by means of a Monte Carlo method in which artificial noise was added within the S/N limits of the data. In Figs. 5 and 7 we show the standard deviations and of the phases and amplitudes resulting from 25 fits for each point. Before computing the phases () were anchored to the first calculated value using . This assures that any value of differs less than 0.5 from . The error bars are therefore not expected to exceed 0.28 which is the standard deviation of a uniform distribution with phases between 0 and 1 in absence of any signal. The apparently non-random distribution of the values of some phases outside the line profiles in Figs. 5 and 7 is unclear. We think that this might be caused by very small residual periodic variations in the continuum as seen in Figs. 2 and 3, rather than by a statistical dependency of the different points.

The quoted amplitudes and phases are determined from these multiple sine fits. These amplitudes are more reliable than from the CDFT's. This is because CLEAN removes the window-function effects component by component from the DFT, regardless whether the peak being removed contains some power of a real frequency. Consequently amplitudes in a CDFT may be smaller than they truly are. The phases determined by the two methods agree within the error bars.

As a comparison we reconstructed the pulsation patterns by inverting the DFTs and added all the components of Eq. 5 within the frequency range of interest, thereby filtering out a large fraction of the noise. We also folded the data with the detected periods. These two resulting reconstructions, in a grey-scale representation and appropriately rebinned, are shown in Figs. 2 and 3 along with the original data.

Fig. 5. Amplitude and phase (in 2 radians) of the signal at = 3.45 h (7 c d-1) and = 2.63 h (9 c d-1, not believed to be real). The vertical dashed lines are drawn at sini. The top scale is the observed wavelength

Fig. 6. Periodogram of  Cep spectra. The layout is the same as in Fig. 4

As an independent method for finding periods, complementary to the CLEAN analysis, we also performed a minimum-entropy method (Cincotta et al. 1995). In this method the time series are folded with trial periods, i.e. the phase of each observation is computed as . The phase/flux space is then divided into a number of grid elements in which the number of datapoints is counted. By defining the probability of finding a datapoint in element i as the entropy is calculated as . The lower the value of S, the higher the probability that the trial folding period corresponds to a true period. The significance of the considered period follows from , where the index c refers to the continuum. In order to deal with the window function caused by the inhomogeneous data sampling, the minimum-entropy method was first performed after shuffling the data in a random manner, yielding . Subtraction of from S removed in this way the periods caused by the window function.

© European Southern Observatory (ESO) 1999

Online publication: April 12, 1999
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