3 Frequency analysis

A power spectrum obtained through a simple Fourier transform of a data string contains not only "true" frequencies present in the physical phenomenon under examination, but also many aliases introduced by data sampling. Combination of aliases can produce strong spurious peaks, and it is a difficult task to disentangle them from the "true" frequencies.

In our search for multiperiodicity in the selected $\eta$Cen data sets we applied the Fourier analysis and two methods that try to get rid of aliases due to data sampling. They are the CLEAN (Roberts et al. 1987) and the CLEANEST (Foster 1995) algorithms implemented in a SUN workstation by Emilio (1997). Both methods are based on modeling the observed data in Fourier space as a combination of trial trigonometric functions. CLEANEST is more powerful, because it uses more than one trial function simultaneously. In the case of our spectroscopic data, frequency analysis was performed on each of the individual time series formed by data at each resolution step across the He I $\lambda$667.8 nm line profile. The lpv were also analysed with the Fourier-Doppler Imaging technique (FDI) (KWM). In that method, the temporal variations of the line profile series are transformed in both time and Doppler space; a two-dimensional Fourier representation of the variations is thus obtained, where the frequencies along the velocity axis can be identified with the degrees l rather than with the (apparent) azimuthal orders (Gough, private communication to Kennelly et al. 1997).

3.1 Spectroscopic data

We searched for periodicity in the lpv of the spectra taken at LNA. The CLEAN, CLEANEST and FDI algorithms were applied to 385 time series formed at each line-profile resolution element inside the profile. CLEAN and CLEANEST were also applied to the time series of RV, EW and FWHM measurements. Periods longer than about the total duration of the observing run, $\sim$7.16 d ($\nu$=0.14 c/d, which is also the formal frequency resolution) cannot be detected. The same procedure was applied to the RV, EW and FWHM data of SBHB.

3.1.1 Line-profile variability

High order lpv is quite conspicuous in $\eta$Cen. Additional low-order lpv is also strongly present, as it can be seen in the peculiar average stellar line profile (see Fig. 1).Variations in line depth at the $\sim$ 1-2% level in time scales of $\sim$ 5 min are also observed in the central region of the HeI $\lambda$667.8 nm line. They are probably due to intrinsic Be-type variations probably arising in the circumstellar disk, not linked to NRP.

  

Figure 1: Upper panel: global CLEANed periodogram at each position across the line profile. The first and second frequencies (by order of amplitude) found at each pixel are represented by a barred ellipse and a square, respectively. Lower panel: mean HeI 667.8 nm line profile of $\eta$Cen in 1995 March (Heliocentric wavelength)

The CLEAN algorithm was applied to each of the 385 residual time series formed by subtracting the global mean spectrum of the observing run from each individual spectrum, and following the procedure of Gies & Kullavanijaya (1988). The gain was set at 0.8. The main peaks were identified after a few iterations, and there were no significant changes afterwards; we thus limited to 10 the number of iterations. A CLEANed periodogram at each position across the line is shown in Fig. 1 with the global average profile in the lower panel. Frequencies separated by 1 c/d can be combinations of a true frequency and an alias that CLEAN could not eliminate.

Indeed, the sequences $\sim$0.3/1.3/3.3 and 1.7/2.7/3.7 c/d never occur together at any wavelength; only one of the frequencies of each group is probably present in the star.

We also applied the CLEANEST algorithm (Foster 1995) to our 385 time series across the line profile. This algorithm requires the number of frequencies to be defined a priori. We found that the clearest two-dimensional pattern is obtained by searching for just two frequencies. Since the method is applied to each of the 385 wavelength bin time series, more than two frequencies will show up (if more than two are present). In fact, we constructed two distinct residuals time series. A first one was formed by subtracting the global mean spectrum of the observing run from each individual spectrum; for the second one, we subtracted the mean nightly profile from each spectrum. For the first time series, lower frequencies are enhanced, while higher frequencies are more easily distinguished in the second series. Figure 2a shows the results of CLEANEST using the global mean spectrum; the corresponding periodogram summed across the wavelength axis is shown in Fig. 2b. The same is presented in Figs. 3a,b, but now using the nightly mean spectrum to form the residuals. Note that the algorithm was not able to eliminate completely the 1-day aliases.

  

Figure 2: a) Global CLEANESTed periodogram at each position across the line profile. The first and second frequencies (by order of amplitude) found at each pixel are plotted. Residuals were calculated with the global mean spectrum. The limits of the HeI line are indicated. b) Corresponding periodogram summed across the wavelength axis inside (solid curve) and outside (dashed curve) the HeI line

  

Figure 3: Same as Fig. 2, but using the nightly mean spectra to construct the residuals

The FDI method (KWM) was also applied to the 385 time series across the line profile. This technique maps the residuals onto a space where the coordinates are the time frequency and the normalized wavelength frequency, provided that the wavelength is mapped onto stellar longitude $\ell_i$ using:

$$\ell_{i}=\sin^{-1}\left(\frac{v_i}{v_{\rm e} \sin i} \right)$$ (1)

where v_i is the velocity corresponding to the wavelength bin i with respect to the center of the line (KWM). It can be shown that the normalized wavelength frequency corresponds to the pulsation degree l. In the case of sectoral modes, it will naturally be equal to $l=\vert m \vert$. The main advantage of this technique is that it decomposes complex patterns of multiple modes. However, great care must be taken in the interpretation of the results, since simulations indicate that there is not a one-to-one relationship between an oscillation mode and its representation in Fourier space (e.g. Kennelly et al. 1997). The FDI was applied to the two residual time series as was done with CLEANEST. The projected rotational velocity is determined in Sect. 4.1. The variations of the line profile overall shape were decomposed by the FDI method into very strong low-degree pulsation modes (l = 2, 4), although higher degree modes up to $l \sim$15 can also be seen. In spite of the fact that the two-dimensional map is hampered by one-day aliases in the time frequency direction, the main frequencies coincide with that produced by CLEANEST.

In order to examine the nightly variations, we also applied the CLEANEST and FDI methods to each night separately. In our case, the low limit time frequency that can be determined from each night is about 5.5 c/d (corresponding to $\sim$ 4.4 h nightly observing runs). The results are shown in Table 2 and Fig. 4 for nights May 15, 16. Note the changing pattern from night to night. In fact, for nights 11, 15 and 17 the two-dimensional map is apparently dominated by a mode with l = $\sim$2, whereas for nights 12, 16 and 18 a mode with $l=\,\sim$4 is by far the strongest. However, one must be cautious about the quantitative aspect of this interpretation, because signals with frequencies $\le$5.5 c/d are incompletely observed on a given night (see Sect. 4.3).

  

Figure 4: a) left: Time evolution of the residuals from March 15. a) right: Fourier Doppler imaging of the corresponding line profile variations. The gray scale is normalized to the intensity of the highest l$\simeq$2 peak. b) same as a) but for March 16. Note the dominant l $\simeq$ 4 mode

This apparent mode alternation will be discussed later (see the Discussion). For reasonable values of the stellar rotation frequency (see Sect. 4.1), high intensity modes lie generally above the non-propagating line, namely $\nu\,=\,\nu_{\rm rot}$$\vert m \vert$, which defines the position of surface phenomena that are locked to the stellar rotation. Those modes would thus correspond to prograde modes. Taking into account the extension in wavelength of occurrence of each frequency and from the power spectra summed across the wavelength axis (see Figs. 2b and 3b), the main detected periodicities are summarized in Table 2. Other weak modes also appear at higher frequencies. For the moment, we will not discuss which frequencies are really present in the star; it appears clear, however, that even CLEANEST is not capable of eliminating all aliases due to data sampling. It is worth noting that the spectroscopic coverage is 1.07 days observed in total over a time span of 7.16 days. Clearly the large gaps in sampling cause significant ambiguities to exist in extracting the actual power spectrum from the data.

   Table 2: Frequencies (c/d) found in 1995 lpv using CLEANEST, by order of decreasing amplitude. Residuals have been obtained by doing a) spectra minus global mean spectrum, b) spectra minus nightly mean spectrum (all 6 nights together), and c) spectra minus nightly mean spectrum (each night analysed separately)

3.1.2 Radial Velocity, FWHM and EW variations

The CLEAN and CLEANEST algorithms have also been applied to the time series formed by the individual measurements of the radial velocity (RV), full width at half maximum (FWHM), and equivalent width (EW) of our HeI line observations. These parameters have been measured using standard IRAF procedures. Those two algorithms have also been used to analyse the temporal series of RV, FWHM and EW data measured by SBHB in SiIII $\lambda$455.26 nm line profiles of $\eta$Cen. The main frequencies found in these analyses are shown in Table 3.

  

Table 3: RV, FWHM and EW variations

*C=CLEAN, *CL=CLEANEST.

3.2 Photometric data

We have searched for periodic variations in five different sets of photometric data (ground based using Strömgren b filter): (1) data published by Cuypers et al. (1989) corresponding to observations made in 1987 and (2) in 1988, (3) Hipparcos data, obtained between early 1990 and August 1992, (4) data published by SBHB, corresponding to observations collected in 1992 and (5), observations obtained at LNA in 1993. The temporal analysis of these data sets was equally performed using both the CLEAN and CLEANEST algorithms, except for Hipparcos data which have been analysed with CLEAN. In Fig. 5 we show the dirty spectrum and the periodogram of LNA May-August 1993 data, obtained with CLEAN. The typical uncertitude on the frequencies detected in our photometric data is $\sim$0.08 c/d, corresponding to the FWHM of the peaks in the periodogram. The main frequencies found in five data sets are:

(1)

1.36, 2.45c/d (CLEAN), 1.29c/d (CLEANEST);

(2)

1.56c/d (CLEAN), 0.56, 1.59c/d (CLEANEST);

(3)

1.56, 2.10c/d (CLEAN), 1.56, 2.13c/d (CLEANEST);

(4)

1.56c/d (CLEAN), 1.56, 3.23c/d (CLEANEST);

(5)

1.56 (CLEAN).

  

Figure 5: a) Dirty spectrum for photometric measurements obtained at LNA (May-August 1993). b) Corresponding CLEANed periodogram

These results compare well with time analysis of spectroscopic data (see above and Table 2). The 0.78 c/d quoted in the Hipparcos catalogue (ESA 1997) corresponds to a double-wave photometric curve; but in our frequency analysis the largely dominant periodicity is 1.56 c/d (=2$\times$0.78 c/d), corresponding to a single-wave curve. It can be noted from the results of the frequency analysis of spectroscopic (RV, FWHM and EW) and photometric data, that the frequencies found with the two algorithms do not differ significantly. Moreover, allowing for precision and alias occurrence, some frequencies (e.g., $\sim$1.29, 1.56 c/d) are found in independent data sets. This strongly suggests that these frequencies most probably are not aliases due to data sampling.