## 3. Period analysisWe 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 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
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, 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 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.
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 © European Southern Observatory (ESO) 1999 Online publication: April 12, 1999 |