3 Analysis of the time-series

Both the spectroscopic as well as the photometric time-series were analysed with the program package TRIPP. It enables the calculation of periodograms, confidence levels and fits with multiple sine functions (see Dreizler et al. 2002 for more details).

3.1 Spectroscopy

We were able to derive accurate radial velocity curves for the Balmer lines ${\rm H}_{\beta }$ and ${\rm H}_{\gamma }$ (see Fig. 2). The frequency resolution for this run is calculated to be 51 $\mu$Hz. Trailing mode observations require stable weather conditions so that the intensity of the star's signal is almost constant during one exposure. Thus bad seeing and transparency changes due to the passage of small clouds influence the quality of the data rather strongly. Because of these disturbances the quality of the radial velocity curves varies with time. The radial velocity curves of the other Balmer lines and the He lines turned out to be too noisy for a quantitative analysis and are not further discussed. Better conditions should allow a more extensive study of radial velocity changes in this star.

The Lomb-Scargle periodograms calculated from the radial velocity curves (see Fig. 3) show peaks clearly surpassing the 3$\sigma$ confidence level. Applying a prewhitening procedure to the data three frequencies and their amplitudes (see Table 1) were found for both Balmer lines investigated. Other peaks that seem to lie above the 3$\sigma$ level could not be extracted by our procedure. The dominant frequency at 2.076 mHz has the largest velocity amplitude with 12.7 km s^-1 for ${\rm H}_{\beta }$ and 14.3 km s^-1 for ${\rm H}_{\gamma }$. The amplitudes of the other two frequencies are lower than the strongest one with values ranging from 6 to 8 km s^-1. We determined the amplitude accuracy $\Delta A$ by calculating the median value of the white noise in the frequency range 3-7 mHz where almost no power arises in the frequency spectrum. According to this, we found $\Delta A$ = 1.5 km s^-1 for ${\rm H}_{\beta }$ and $\Delta A$ = 1.0 km s^-1 for ${\rm H}_{\gamma }$.

Figure 2: Radial velocity curve for ${\rm H}_{\beta }$ (upper panel) and ${\rm H}_{\gamma }$ (lower panel); artefacts due to poor weather conditions were not removed.

3.2 Multi-band photometry

BUSCA is a unique instrument which enables the measurement in four different wave bands simultaneously. As a result we obtained four light curves.

Figure 4 shows the Lomb-Scargle periodograms of all BUSCA bands. They are quite similar to the periodograms derived from the radial velocity curves (Fig. 3). This data set spans over three nights so that daily aliases are clearly visible and the frequency resolution is much better at $\Delta\nu = 5.68~\mu$Hz (compared to 51 $\mu$Hz for the spectroscopy). Again we applied the prewhitening technique in order to remove all significant peaks from the periodogram and to obtain the amplitudes for each frequency. As before, the horizontal line in the diagrams represents the 3$\sigma$ confidence level above which we assume the detected frequencies to be real. In all four wave bands five peaks with the same frequencies can be identified (see Tables 2 and 3). The dominant frequency is again found at 2.076 mHz and therefore confirms the results from spectroscopy. Furthermore, additional frequencies were found in the region around 2.74-2.78 mHz but these peaks are closely spaced so that a corresponding identification in all BUSCA bands due to the medium frequency resolution was not possible. Peaks that fall below the 3$\sigma$ confidence level were not removed from the periodograms.

Figure 3: Lomb-Scargle periodogram of the radial velocity curve of ${\rm H}_{\beta }$ (upper panel) and ${\rm H}_{\gamma }$ (lower panel). The power spectral density (psd) is a measure for the probability that a period is present in the radial velocity curve or light curve. The horizontal line represents the confidence level of 99% (3$\sigma$-level).

Figure 4: Lomb-Scargle periodogram of the light curves of the " $UV_{\rm {B}}$'', " $B_{\rm {B}}$'', " $R_{\rm {B}}$'' and " $NIR_{\rm {B}}$'' band (from top to bottom) of the BUSCA camera. The horizontal line in each panel represents the confidence level of 99% (3$\sigma$-level).

The amplitudes of the brightness variations are measured in fractional intensity. The light curves are normalized to the fraction of intensity $\delta{I}/{I}$ and the amplitudes were then converted to mmag. This unit will be used throughout the whole paper. The accuracy of these amplitudes are calculated in the same way as we did it for the radial velocity amplitudes. Here we used the median level of the white noise in the range 3-5 mHz. The accuracies for the BUSCA wavebands are 1.52 mmag for " $UV_{\rm {B}}$'', 1.53 mmag for " $B_{\rm {B}}$'', 1.12 mmag for " $R_{\rm {B}}$'' and 1.37 mmag for " $NIR_{\rm {B}}$'', respectively.

Figure 5 shows the semi amplitudes of four selected frequencies as a function of effective wavelength of the bands. In Fig. 6 we display the relative change of the semi amplitude of each waveband. The deviation with respect to the mean is largest for the " $UV_{\rm {B}}$'' band. The other channels, considered separately, behave rather similar showing much smaller deviations from the mean brightness. This is explained through the fact that the " $UV_{\rm {B}}$'' band lies blueward to the Balmer jump and the other redward of it. The opacity changes a lot across this wavelength range and thus the stellar flux originates from different atmospheric depths.

Furthermore, we used the phases which are delivered by the sine fit procedure (see Tables 2 and 3) in order to test whether there is any wavelength dependency. The phase values are normalized to unity. Figure 7 shows the deviations of the phases with respect to the mean value of all four wavebands for the four selected frequencies of Fig. 5. The error bars are calculated from

$${A} = {\rm Re}\{{A}\}~{\rm sin}\phi.$$ (1)

The phase error $\Delta\phi$ can be calculated from the amplitude uncertainty $\Delta A$ according to

$$\Delta\phi = {\rm arctan}\left(\frac{\Delta{A}}{{A}}\right)\cdot$$ (2)

The two strongest frequencies at 2.076 and 1.986 mHz show no wavelength dependent behavior within the error margins. The deviations for the other two frequencies from the mean value are slightly larger than the formal errors. However, we regard this as insignificant, and conclude that we cannot find any wavelength dependency of the phases of the modes.

Figure 5: Semi amplitudes of four frequencies as a function of effective wavelength. The error bars are 1$\sigma$ errors and are calculated by means of a $\chi ^{2}$ method.