5 Non-radial pulsations in the primary

We have analysed the line-profile variations of some of the absorption lines in the spectra of the primary of $\psi ^2$ Ori, focussing on lines that are least affected by blending. We have tried to analyse some HeI lines (4387, 5015, and 5047 Å), but found that the relatively strong, broad and moving profile of the secondary leads to inaccuracies in the analysis. These lines show orbital-phase dependent normalization errors, leading to Fourier spectra dominated by the orbital frequency and its first eleven (or so) harmonics. We found that for the absorption lines of heavier elements we had to prewhiten the data with the orbital frequency and its first four harmonics, in order to diminish the effects of the orbit. By doing so, however, we lose the ability to study line-profile variations of the primary that have frequencies similar to that of the orbit. The spectra and periodograms we discuss below all have these five frequencies removed.

Figure 5: Power and phase diagram of frequencies 10.49 and 10.71c/d in the wavelength region of the SiIII 4574 Å line. The black line (power) and the triangles (phase) represent the diagnostics as derived from the multi-sinusoid fit. The grey line and the circles are from the CLEANed periodogram. Phases are plotted as small dots for bins with insignificant power values. The top panel displays the mean spectrum.

In Fig. 1 we display the line-profile variations of the 4552 Å SiIII line in different stages of the analysis. One can clearly see a moving bump pattern remeniscent of that of non-radial pulsations. After shifting the spectra to the velocity frame relative to the primary, it is evident that the bump pattern is not constant. The distance between consecutive bumps varies; beating of two similar moving-bump patterns seems present. This indicates that probably more modes than just one pulsation mode are responsible for the profile variations. The profiles of all other investigated absorption lines show variations very similar to those in the 4552 Å profile.

To study the temporal behaviour of the line-profile variations we have analysed them in the way described by Gies & Kullavanijaya (1988) and Telting & Schrijvers (1997). Figure 4 displays the periodogram resulting from CLEANing the Fourier transform (Roberts et al. 1987) of the signal in each wavelength bin in the profiles of the 4590 and 4596 Å OII lines. We used 400 CLEAN iterations with a gain of 0.2. A double power peak is found at frequency 10.5c/d; the duplicity of this peak explains the apparent beating of the moving-bump pattern. One-day aliasing is still present in the periodogram; the CLEAN algorithm has not been able to fully correct for the window function.

We have tried to resolve the two frequencies responsible for this double peak by first prewhitening the data with the frequency of the combined peak (10.5c/d) and then redoing the Fourier analysis, in order to find the frequency of the lower-amplitude peak of the double peak. We find this peak to be at $\sim$10.73c/d (see Table 3). Then we prewhitened the original signal with this frequency, and did a Fourier analysis to recover the frequency of the higher-amplitude peak of the double peak: $\sim$10.48c/d (Table 3). Note that the HWHM of the main peak of the window function is 0.083c/d, which means that our frequency resolution is about 0.17c/d, corresponding to a time base of the observations of 6 days.

 

 

Table 3: Observed frequencies in c/d and blue-to-red phase differences in $\pi$ radians, of spectral line variations in $\psi ^2$ Ori.

	CII	SiIII	SiIII	SiIII	OII	OII
	4267	4552	4567	4574	4590	4596
f1	10.46	10.48	10.47	10.49	10.48	10.48
$\Delta$$\Psi$	3.0	3.5	3.5	4.8	3.9	4.5
f2	10.73	10.76	10.74	10.71	10.73	10.73
$\Delta$$\Psi$	4.1	5.1	4.7	4.9	4.8	4.5

The amplitude and phase of the variations with the above two frequencies, as a function of position in the line profiles, give information about the pulsations that are responsible for these variations. We have analysed the amplitude and phase diagrams as obtained from the CLEANed periodogram, as well as those obtained from multi-sinusoid fits (in this case two sinusoids) to the data in each wavelength bin. For the multi-sinusoid fits we used the frequencies as derived from the CLEAN analysis (as listed in Table 3), and created amplitude and phase diagrams from the fitted amplitudes and phases (see our forthcoming paper on $\nu$ Cen, Schrijvers & Telting, for a further description of this method). In Fig. 5 we plot the resulting amplitude and phase diagrams for the SiIII 4574 Å line. As the multi-sinus method is not affected by one-day aliasing, higher power is found than in the case of the CLEANed periodogram in which power has leaked to one-day aliases. From the overplotted phase diagrams we have determined the blue-to-red phase differences $\Delta$$\Psi$, which are a measure of the degree $\ell$ of the modes (see Telting & Schrijvers 1997). The resulting phase differences are listed in Table 3.

We use the linear representation for modes with $\ell - \vert m\vert < 6$from Telting & Schrijvers (1997), $\ell=0.015+1.109\Delta\Psi$with $\Delta$$\Psi$ expressed in $\pi$ radians, to derive the degree of the modes. For this representation the chance of correctly identifying the modal degree within an interval of $\ell \pm 1$ is about 84%.

In most lines the amplitude of the 10.48c/d frequency was not detected in the blue wing. For this reason the phase diagram spans only about 75% of the total line width, which means that the derived values of $\Delta$$\Psi$ are lower limits for the $\ell$ value of the responsible mode.

The amplitude of the 10.73c/d frequency is more symmetrically distributed over the line profiles. From general line-profile modelling it has become clear that the phase diagrams run along beyond $v\sin i$ of the star, albeit with low corresponding amplitude. As our dataset does not provide the accuracy to measure the full extent of the phase diagrams, the derived $\ell$ values will be lower limits.

Taking the largest values of $\Delta$$\Psi$ from Table 3 we find $\ell = 5.3$ for the 10.48c/d frequency, and $\ell = 5.7$ for the 10.73c/d frequency. Given the fact that our determinations are likely to be lower limits, we estimate the true value for both modes to be $\ell=6$. In Fig. 5 one can see that the slope of the phase diagrams of both frequencies is very similar, which supports the possibility of both modes having the same degree.

We stress that, because of the limited time base of our data set and the corresponding limiting frequency resolution, the derived phases of the variations at these frequencies might be affected by each other. More data, spanning a longer time base, are needed to confirm our mode identifications.

We did not find significant power at the harmonics of the frequencies derived above, which means that we cannot estimate the m-values of the modes directly from the Fourier analysis (Telting & Schrijvers 1997). It is likely that due to the limited sampling of the variational signal, and the phase smearing during the individual exposures, it has not been possible to detect the harmonic variability in our dataset.