Astron. Astrophys. 362, 189-198 (2000)

4. Frequency analysis

4.1. Low frequencies

Three different methods, adapted to unequally spaced data, were used to perform the frequency analysis: Fourier, CLEAN (Roberts et al. 1987) and PDM algorithms (Stellingwerf 1978). Only the Fourier searches (Breger 1990) are presented; the PDM method confirms this analysis, while the CLEAN method leads to doubtful results in some cases.

We first concentrated on typical SPBs behaviour i.e., we looked for frequencies in the range [0;5] c.d^-1. However, because no significant peak is present above 2 c.d^-1, and for clarity, we presented only periodograms in the [0;2] c.d^-1 frequency range (Fig. 7). Once the main frequency peak was detected, a sinusoid was fitted to the different data sets. It appeared that sometimes the maximum amplitudes (and the minimum residuals) did not correspond to the main peak. In this case, we retained only the frequency provided by the sine-fit (i.e., having the minimum residuals). A summary of this analysis is given in Table 2. A frequency around 0.29 c.d^-1 is common to all our observation series. Paper I was the only previous study to report such a frequency. We thus decided to perform the same analysis on different data sets available in the literature:

• the 408 radial velocities of Paper I

• the 58 radial velocity measurements of Mulliss (1996) scattered over one year. The considered line is SII (), for which the measurements are the most numerous

• the Hipparcos data, which consist of 105 observations recorded through both and filters (close to Johnson B and V)

• the 161 UV radial velocities obtained from a scan of Rogerson's (1984) observations.

Fig. 7a-d. Fourier periodograms corresponding to different data sets. Power is normalized to unity. a: 1985 spectroscopic data. b: 1987 spectroscopic data. c: 1995 spectroscopic data (AURELIE). d: 1987 photometric data. The vertical dashed line represents the position of the frequency (see text)

Table 2. Results of the frequency analysis concerning different data sets. Epoch is related to the observation campaigns [years], columns "Nights" and "Ranges" represent respectively the number of nights (when meaningful) and the total range (in nights) of the observations, while N is the number of observations in the considered data set. Then and represent respectively the main detected frequency [c.d^-1] and its amplitude [km s^-1 in the first part of the table, mmag in the second part]. Finally, in the last column the reference from which the data sets have been obtained is given.

Results are listed in Table 2. The previously noted frequency around 0.29 c.d^-1 appears also in Paper I and Hipparcos data. In Mulliss's data (1996), the 1.287 c.d^-1 detected frequency can be interpreted as a 1 day alias of the 0.29 c.d^-1 frequency. As for the 0.76 c.d^-1 frequency detected in Rogerson's (1984) data, it will be discussed below. Therefore, we can retain a main frequency around 0.29 c.d^-1 present in both photometric and spectroscopic observations. Hipparcos provides the most precise value:  c.d^-1, because its measurements are spread over a large time range and have the better distribution (in particular no alias). However, since the very wide observing window leads to a Fourier periodogram with many peaks, it is the CLEAN result that is represented on Fig. 8.

Fig. 8. Periodogram corresponding to the Hipparcos filter. For clarity, only the CLEAN periodogram is represented. The gain used is 0.5, and 100 iterations were performed. Power is normalized to unity. The maximum peak corresponds to the frequency (see text)

The frequency is not detected in Rogerson's (1984) observations while the periodograms, obtained from the 3 different methods cited above show a main peak at 0.766 c.d^-1, associated with an amplitude of 1.92 km s^-1. A prewhitening of Rogerson's data with this frequency leads to a second one at 0.47 c.d^-1, with an amplitude of 0.99 km s^-1. In order to improve this result, we computed a sine-fit with these two frequencies as starting values. However our analysis diverged.

The algorithm used by Rogerson in a similar analysis, with the same two frequencies (0.766 and 0.47 c.d^-1) as starting values, converged to the 0.618 and 0.660 c.d^-1 frequencies. We do not agree with these values, because the corresponding beat-period is 24 d, a value much larger than the narrow observing window that spans only 3.6 d. There is also a significant gap between the starting and the final solution. Finally, the corresponding amplitudes are of the order of 10 km s^-1, a factor 5 times greater than the values obtained for the starting frequencies. Unfortunately, only the 0.618 and 0.660 c.d^-1 frequencies are mentioned in Rogerson's abstract and thus became the only cited result. Our interpretation is that the UV variations are best explained by the 0.77 and 0.47 c.d^-1 frequencies.

From Fig. 7, it is obvious that additional frequencies are present in the different data sets. As can be seen in Table 3, the variance associated with the frequency ranks from 19 (Mulliss 1996) to 87 % (Paper I) of the total variance, with a mean value of 56 %. Therefore, other frequencies must be present.

Table 3. Results of the frequency analysis concerning the different data sets prewhitened with  c.d^-1. Epoch is related to the observation campaigns [years], represents the amplitude associated with [km s^-1 in the first part, mmag in the second part of the table], while is the percentage of the variance that accounts for. Then, and represent respectively the main detected secondary frequency together with its amplitude and its variance percentage . Finally, in the last column the reference from which the data sets have been obtained is given.

A frequency analysis similar to that conducted above was then undertaken on each data set prewhitened with the "Hipparcos" frequency. The corresponding periodograms are presented in Fig. 9, and results are given in Table 3. The situation is less clear than that concerning the frequency, with now the presence of 3 frequencies that will be discussed below.

Fig. 9a-d. Same as Fig. 7 after prewhitening with the frequency. The vertical dashed lines represent the position of the  c.d-1 frequency (see text)

First, a frequency around  c.d^-1 is detected in 1985, 1987 and 1995 AURELIE observations with variances ranking from 40 to 77 % in the residuals of . This frequency is identified in the periodograms represented in Fig. 9. Although it has a slightly larger value, the 0.47 c.d^-1 frequency detected in Rogerson's data (1984) may also be interpreted as the signature of . Conversely, is detected in neither Mulliss's (1996) nor in Paper I data, where its corresponding variance ranks from only 3 to 19 % of the variance. In photometric data, the frequency accounts for between 4 and 16 % of the variance (for Hipparcos and Grenade data, respectively). It should be noted that the amplitudes ratios are clearly lower in photometry than in spectroscopy (average value of respectively 0.34 and 0.77): this is why the second frequency is more difficult to detect in photometric data.

The  c.d^-1 frequency is detected only in Rogerson's data where it accounts for 56 % of the variance. Its non-detection in all the other data sets may be due either to prewhitening or to decreasing amplitude: First, the 0.5 c.d^-1 alias of the main frequency is close to , and prewhitening the data set with reduces the amplitude corresponding to below the detection threshold. Second, the amplitude associated with may have decreased (the other considered data sets are obtained later). This phenomenon has been observed for the SPB star 53 Per (Chapellier et al. 1998) and is noted for  Her itself (see Sect. 5.1.3).

Two data sets (Mulliss 1996and Paper I) present a frequency around 1.7 c.d^-1. At least three interpretations may be done: a one-day alias of the linear combination , a one-day alias of the  c.d^-1 detected in Rogerson's data (1984) or a non-real frequency since both data sets concerned here have a very bad time coverage: 58 measurements scattered over one year for Mulliss, and 408 measurements concerning 2 series of 3 consecutive nights separated by more than a month (Paper I). Thus, it seems safer to discard this frequency from the results of the present study.

Finally, Hipparcos data show, in both filters, a second frequency:  c.d^-1. Unfortunately, this frequency is too close to to be detected in any other data set, much shorter than the beat period between and (26 d). Only Mulliss (1996) had a sufficient time basis, but his data are not numerous enough.

4.2. High frequencies

As said in Sect. 1, frequencies larger than those recorded above have also been detected both in photometry and spectroscopy, ranging from 2.4 to 34.3 c.d^-1. We call them high frequency as they seem to correspond to different modes. As a matter of fact, more than one cycle can be observed within a given night. Table 4 summarizes the different results. Excepting the time scales provided by line profile studies (Smith 1981), the previous detected frequencies lie in two regions: [7;8.5] c.d^-1 and [16;24] c.d^-1.

Table 4. The different high frequencies mentioned in the literature. Epoch is related to the observation campaigns [years]; then are given the frequency scale of the variations [c.d^-1], the peak-to-peak amplitude, the used method and the reference of the concerned study.

We reanalysed the different data sets mentioned in the previous section. The measures were first prewhitened with the and frequencies. A frequency analysis was then undertaken in the range [0;30] c.d^-1. Results for different data sets are presented in Fig. 10. The main peaks are still in the [0;3] c.d^-1 frequency range, and represent non-detected frequencies of SPBs type, but also linear combinations and harmonics of the and frequencies.

Fig. 10a-d. Fourier periodogram corresponding to different data sets. Power is normalized to unity. a: 1985 spectroscopic data. b: 1987 spectroscopic data. c: 1993 spectroscopic data (Paper I). d: 1995 spectroscopic data (AURELIE)

However, most of the data sets present peaks in the [6;8] c.d^-1 frequency range (especially in the spectroscopic data obtained in 1987, 1993 and 1995 (with AURELIE)), with corresponding amplitudes between 0.15 and 0.30 km s^-1. In addition, spectroscopic data obtained in 1985, 1987 and 1993 present peaks, close to the detection threshold, in the [15;25] c.d^-1 frequency range. Results concerning the high frequency part of the AURELIE 1995 data spectrum should be treated with caution since pixel drifts occurred, especially considering the ELODIE data obtained at nearly the same epoch (see below).

In Paper I, the authors detected in their 1993 radial velocity data a high frequency, around 20 c.d^-1, associated with an amplitude of about 1 km s^-1. The correlation mode of ELODIE provides an internal precision better than 0.1 km s^-1. It is therefore possible to detect the small variation mentioned in Paper I. As can be seen in Fig. 4, the dispersion is very small, and obviously the short time-scale periodicity mentioned in Paper I is not present. A frequency analysis performed on these data shows a very flat power spectrum above a 5 c.d^-1 frequency. Therefore, no high frequency was present in the pulsation of  Her during the 1995 observing campaign.

© European Southern Observatory (ESO) 2000

Online publication: October 30, 19100
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