3 Analysis of observations

3.1 Algorithm used

The search for lpv multiperiodicities in HD 127972 spectra was made by means of the Fast Fourier Transform (FFT) with the "Cleanest" algorithm (Foster 1995; Emilio 1997). It is known that the use of FFT without a frequency search criterion yields periodograms with a very large number of frequencies, where the signal appears convolved with the time sampling (aliasing), and the detection of periodicities becomes somewhat risky and uncertain. However, some constraints on the selection of the detected periods can be imposed by filtering signals from the data sampling with well-founded criteria. One of such methods is the Cleanest algorithm which considers the series as data vectors represented linearly on a given vectorial base. The method proceeds by steps by subtracting sequentially from the residuals, obtained once the FFT was applied to the vector data, a model function derived using each new frequency peak considered as statistically significant by a $\chi^2$ test. Once the last significant peak in a step cycle has been found, a new vectorial base is constructed with the last vectors plus the contribution of the last detected signal. The process goes on until there are no more statistically significant peaks or simply because the process reaches a stop condition, such as the imposed maximum number of possible frequencies. The last model function is formed by a basis with 2n+1 tentative functions. An important feature of Cleanest is that it does not assume the average signal present in each time residual to be zero, as does the "Clean" method (Roberts et al. 1987). The assumption of zero-averaged signals in time residuals leads to the unprobable fact that the sample data are modulated by signals with an integer number of cycles which can hinder the performance of signal detection (Emilio 1997; Levenhagen 2000).

The reliability of the resulting power spectrum can be questioned by the time data sampling. It implies a finite extent of observational missions, an unequal time distribution of data and the strong 24 hours periodicity of data sampling. It is thus important to test the performance of the algorithm of time analysis used in adverse sampling conditions. This can be achieved in principle by applying the method to a set of synthetic spectra affected by the same noise and time distribution as the real spectra. In this way one can infer the effects of the convolution of the NRP signals with the spectral window. For this purpose, we created a set of 652 synthetic spectra composed of four sinusoidal signals, whose frequencies are the same as those detected in real data (see Sect. 3.2), namely $\nu$ = 0.6, 1.5, 3.8 and 5.3 c/d, with the same time distribution as the 1996-2000 LNA data set. These spectra have been affected by random noise with a Gaussian distribution, whose FWHMrepresents about 30% of total amplitude of the input signal. The frequencies used in this test were the same as those found in the actual spectra, and all the synthetic spectra were generated with phase-dependence along the wavelength bins. From this set, we constructed 241 time series formed at each 0.1 Å across the line profile in the same way as we had done with real data. The series were then analyzed with the Cleanest algorithm with appropriate frequency step, of around 0.0005 c/d, corresponding to the total time span. The resulting periodogram is shown in Fig. 2.

In order to discover whether the frequencies detected by this method are actually related to the star signal or whether they simply reflect the time sampling of data, we carried out to another test by shuffling the synthetic intensities at random, though taking care that the original time sampling was preserved. The resulting periodogram is shown in Fig. 3. A study of the frequency of detected signals in Fig. 3 shows that none is statistically significant. Further analyses taking different combinations of the data sampling also yielded similar results. This shows that all periods displayed in Fig. 2 are robust against random selection of the data points. On the other hand, as the signals displayed in the periodogram of Fig. 2 as well as the amplitude of the simulation noise are close to those of the real data, our results should not be strongly affected by the window spectrum.

3.2 Results on lpv

LNA spectra were arranged in four main sets: 1996, 1997 to 1998, 2000 and 1996 to 2000. Each data set was divided into 241 time series, from 6664.9 Å to 6688.9 Å with steps of 0.1 Å. All series were analyzed with frequency steps in the range 0.2 to 0.0006 c/d, depending on the sample to be studied. The lpv frequencies with highest significance ($\geq$75% confidence level) detected with Cleanest in all data sets concerning He  I, Fe  II, Mg  II and Si  II lines are shown in Table 2. Figure 4 shows the resulting periodogram for the 1997 to 1998 data. The confidence diagram resulting from a $\chi^2$ test for the periodicities found in the He  I $\lambda$6678 Å line is shown in Fig. 5. We can readily see that, besides a signal with $\nu = 2.5$ c/d, frequencies greater than 6 c/d are of lower significance, so they are less trustworthy. The He  I $\lambda$6678 Å lpv was analyzed using only LNA spectra, since ESO spectra presented problems related to bad columns of CCD in that wavelength region. Results concerning lines other than He  I $\lambda$6678 Å correspond to the 2000-2001 epoch.

Figure 6 pictures in a grey scale the dynamic spectra of the pulsation cycles for He  I $\lambda$6678 Å line. A total of 652 spectra from 1996 to 2000 were sorted. All spectra falling into the same phase bin were averaged to minimize the influence of other variabilities and noise (i.e. no prewhitening was applied). They are presented as residuals from the respective mean profiles and folded with frequencies ${\nu }_{2}$ (left), ${\nu }_{3}$ (center) and ${\nu }_{4}$ (right).

Figure 1: Spectroscopic sampling of LNA spectra, centered on He  I 6678 Å  line profile.

Figure 2: Periodogram of synthetic residuals with time distribution (window spectrum) equal to that of LNA spectra, from 1996 to 2000.

Figure 3: Periodogram of synthetic residuals with the same time distribution as Fig. 2, but the input signals were randomly shuffled. Notice that none of the previous frequencies were found, suggesting that they are not strongly dependent on data sampling.

Figure 4: Periodogram of He  I 6678 Å line profiles obtained at LNA in 1997/1998.

Figure 5: Diagram of confidence level for periodicities found in He  I 6678 Å line profiles. The horizontal line indicates 75% confidence level.

The signals with high degree of confidence are $0.61 \pm 0.05$ c/d (${\nu}_{1}$), $1.48 \pm 0.05$ c/d (${\nu }_{2}$), $3.81 \pm 0.28$ c/d (${\nu }_{3}$) and 5.31 $\pm 0.19$ c/d (${\nu }_{4}$). The highest considered frequencies, $9.24 \pm 0.19$ c/d (${\nu}_{5}$) and $10.35 \pm 0.13$ c/d (${\nu }_{6}$) were found only in He  I $\lambda$6678 Å. The uncertainty related to time sampling for these signals is of the order $\sim$0.05 c/d, considering the averaged data span of 1996, 1997-1998 and 2000 sets, weighted by the number of observed spectra. A number of frequencies displayed in Table 2 are in agreement with our previous results for this star, like 1.48 c/d, 1.78 c/d, 5.31 c/d (Janot-Pacheco et al. 1999). As in Janot-Pacheco et al. (1999), we have also found a signal at 4.52 c/d, but only in the He  I $\lambda$4471 line. Since this signal can be an alias of 3.52 c/d, we do not report it in Table 2. The frequencies found in the lpv analyses (${\nu }_{2}$ to ${\nu }_{6}$) are attributed to NRP modes. The 0.61 c/d signal (${\nu}_{1}$) is discussed in Sect. 6, where it is shown that it is compatible with the presence of an ejected orbiting shell. In Sects. 4.1 and 4.2 it is shown that the 1.3 c/d signal can be associated with stellar rotation, since it was determined from the continuum and line spectra, and was also found in the lpv analyses of He  I  $\lambda\lambda$4026, 4388 Å and Si  II $\lambda$4131 Å  whose detection could be assured through the presence of inhomogeneities such as spots (Balona 1990).

Figure 6: Grey-scale of spectroscopic residuals centered at He  I 6678 Å, folded with ${\nu }_{2}$ (left), ${\nu }_{3}$ (centre) and ${\nu }_{4}$ (right).

3.3 Global line profile variations

Besides the lpv analyzed in the previous section, variabilities in radial velocity (RV), equivalent width (EW) and full width half maximum (FWHM) measurements in He  I $\lambda$6678 Å line profiles were also detected. Table 3 compares the frequencies obtained from these global line profile variations with those found in the previous section. Figure 7 shows the RV, EW and FWHM variations of the He  I 6678 line from 1996 to 2000 where the mid-term variation of the line profile can be seen. There is a noticeable anticorrelation between the EW and the FWHM of the line.

3.4 Photometric variations

Photometric data of HD 127972 from the Hipparcos satellite (ESA 1997) obtained from 1990 to 1992 were also analyzed. In this case, the time series analysis with Cleanest indicated a strong signal with frequency $\nu_{\rm phot} = 1.55$ c/d. The photometric data folded with this frequency are shown in Fig. 8. These data cannot be recast into a neat phase-dependent diagram with a lower frequency, in particular with 1.3 c/d, since the amplitude of the photometric signal is much more significant than the last one. We argue that 1.55 c/d could be associated with NRPs rather than with stellar rotation (see Sect. 4.1).

 

 

Table 2: Results of time series analysis in all profiles.

Line Profile	Epoch	Detected frequencies (c/d)
		${\nu}_{1}$		${\nu }_{2}$		${\nu }_{3}$	${\nu }_{4}$
Fe  II 5169 Å	4-6	0.61		1.51			5.33
He  I 4026 Å	"	0.62	1.30	1.48
He  I 4121 Å	"			1.49
He  I 4144 Å	"	0.58		1.47	1.78
He  I 4388 Å	"		1.28			3.52
He  I 4471 Å	"	0.61		1.48	1.79
He  I 4922 Å	"	0.57		1.50	1.82	3.51
Mg  II 4481 Å	"	0.63			1.71
Si  II 4131 Å	"	0.62	1.29	1.50	1.70	3.81
He  I 6678 Å	5	0.58		1.47	1.71	3.52	5.31
He  I 6678 Å	2-3	0.61		1.48		3.81	5.31
He  I 6678 Å	1	0.61		1.48		3.81	5.31
He  I 6678 Å	1 to 5	0.61		1.48		3.81	5.31

3.5 Characteristics of the NRPs

An approach to infer the pulsational degree $\ell$ and the azimuthal order $\vert m \vert$ was proposed by Telting & Schrijvers (1997a) with the intensity period search method (IPS). This method takes into account the phase variation across the line profile of a frequency and its first harmonic. It is mainly an empirical formulation based on analyses of phase diagrams derived from generated time series of absorption line profiles of a non-radially pulsating early-type star. For diagnostic purposes, using a Monte Carlo simulation these authors quantified the relation between $\ell$ and $\Delta\Psi_{0}$ (phase difference of main frequency), and that between $\vert m \vert$ and $\Delta\Psi_{1}$ (phase difference of its first harmonic) for spheroidal modes. They found that the fitted coefficients are remarkably stable throughout the parameter space.

From the stability of the coefficients they concluded that it is possible to derive good estimates for the pulsation parameters $\ell$ and $\vert m \vert$ by evaluating the phase differences across the line profile. The typical uncertainties on $\ell$ and $\vert m \vert$ by using the IPS method are estimated to be $\pm 1$ and $\pm 2$, respectively. Considering the previous detected lpv frequencies ${\nu }_{2}$, ${\nu }_{3}$, ${\nu }_{4}$, ${\nu}_{5}$ and ${\nu }_{6}$ as due to NRP, their pulsation parameters thus derived are given in Table 4.

Since we did not find first harmonics with significant amplitudes for most frequencies in the He  I $\lambda$6678 Å line profile, it was not possible to calculate their $\vert m \vert$ values by this method. We attempted to do this only for ${\nu }_{4}$, whose detected harmonic was supposed to be ${\nu }_{6}$ (see Table 4 and Fig. 9). Figure  9 are shown the IPS diagrams for frequencies ${\nu }_{2}$ to ${\nu }_{6}$. The upper panels show the phase diagram across the He  I 6678 line profile and the lower panels present their respective amplitudes. Figure 9 shows clearly the asymmetrical aspect of the amplitude of signals corresponding to frequencies ${\nu }_{4}$ and ${\nu }_{6}$. The same phenomenon was also seen by Floquet et al. (2000) in EW Lac. The 10.35 c/d signal could perhaps be considered the first harmonic of 5.31 c/d (scenario A). However, its power distribution does not exhibit the same behavior over the entire line profile, as can be expected for two harmonics, even when there are non-adiabatic effects (Schrijvers & Telting 1999). Thus, we also considered the possibility that the two signals are independent (scenario B). In scenario A, IPS analysis leads to a pulsational degree l = 5 and order $\vert\rm m\vert =$ 4, while in scenario B we obtain for ${\nu }_{4}$ and ${\nu }_{6}$ l = 5 and l = 8 respectively. It can also be seen from this figure that the signal ${\nu }_{4}$ is not symmetrical around the line center, which should not be the case for NRP. However, the occurrence of central quasi-emissions in He  I 6678 Å transition at the 1996-1998 epoch could be partially responsible for the assymmetry observed in ${\nu }_{4}$ around the line center.