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Astron. Astrophys. 362, 1020-1040 (2000)

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5. Time series analyses (TSA)

First, it should be emphasized that a period search was done, although the complex profile of He I 6678 includes various types of variation. Indeed, one of the main goals of this study was the confirmation of frequencies detected in 1989 data (Floquet et al. 1992). The main typical quantities of He I 6678 and H[FORMULA] were investigated first, such as EW, FWHM, the centroid velocity of He I 6678 absorption (RV), the maximal intensity and equivalent width of weak variable V and R emissions flanked on outer edges of the line, along with EW and maximal intensity of V and R components of H[FORMULA] double emission line. Line profile variability (lpv) analysis was also performed at each resolution element across the He I 6678 line and H[FORMULA] line profiles. Investigation of multi-periodicity in main quoted variable quantities and in lpv was done using the same methods as in Hubert et al. (1997). Results deduced from 1989 analyses are compared with 1993 data (see Table 5).


[TABLE]

Table 5. Frequencies (c/d) deduced from the time-series analysis of He I 6678 and H[FORMULA] profiles in 1993 and He I 6678 in 1989 and common to the two methods. The asterisk indicates that the frequency was obtained only with the Clean method


Fourier analysis and CLEAN algorithm (method 1) were applied to the measured quantities and to time-series of spectra in a similar manner to that introduced by Gies and Kullavanijaya (1988). The least-squares sinusoidal fitting (method 2) with the AIC criterion (Kambe et al. 1993 and references therein) was also used in the analysis of lpv. In both methods, weighting by the signal to noise ratio was introduced in the calculation of averaged data. The duration of the observing run was 8.6 days in 1993 and 6.3 days in 1989 and the corresponding frequency resolution (i.e. [FORMULA]) was 0.12 and 0.16 c/d respectively. Periods shorter than 0.11 day ([FORMULA] c/d), which corresponds to four times the most frequent exposure time, could not be detected. Mathematical tests were done concerning the value of the frequency over-sampling factor and the best results were obtained for [FORMULA]. For this value small amplitude frequencies are better recovered. Other tests were done by introducing white noise on a sinusoidal signal with an amplitude varying with wavelength. It was found that the TF+CLEAN method gives correct restoration of frequency values, but the phase determination across the line profile is more accurate with the least-squares sinusoidal fitting method. He I 6678 profiles were scanned every 0.242 Å between 6665 and 6690 Å for time-series analyses.

5.1. He I 6678 line

5.1.1. EW, FWHM and RV variations

The initial profiles normalized to the stellar continuum were used for this investigation.

Equivalent width and FWHM. As indicated above, EW and FWHM were measured in the absorption part of the profiles and did not include blue (V) and red (R) emissions. Due to contamination by these weak emissions, the equivalent width of the absorption is slightly underestimated. The general behaviour of this quantity is seen in Fig. 5a. A relatively large variation, with [FORMULA]EW/EW = [FORMULA], is nicely detected over the observational run on a time-scale of about 9-10 days. From Fig. 3, Fig. 4 and Fig. 5a it can be concluded that the gradual decrease in EW comes mainly from the vanishing of its broad component (a). Short-term variations are clearly superimposed on the longer one with a semi-amplitude which goes up to 7%. Main frequencies detected in the EW variations are 1.20, 1.51 and 0.89 c/d. Periodograms of EW, RV, V and R emission peaks are given in Fig. 7. This figure also displays the window function.

[FIGURE] Fig. 7a-e. Periodograms obtained for He I 6678 with Clean (full line) and Least Squares (dashed line) methods. a: Window function; b: Equivalent width; c: Radial velocity; d: V emission peak intensity; e: R emission peak intensity

The central depth of the line is weakly variable around the mean value 0.16 and is deeper than in 1989. Nevertheless this quantity is affected by narrow absorption features (shell contribution) and thus could not be analyzed.

The FWHM parameter was determined with the help of a Gaussian fit to the individual profiles. FWHM variation is dominated by the 9-10 day variability. Its semi-amplitude is 13% over the run and 10% on short-time scales. A 1.20 c/d frequency is detected with method 2 but method 1 gives 2.20 c/d. FWHM and EW are nicely correlated for each data set (site) (Fig. 8). FWHM and EW decrease as R emission increases in the first half of the run (Fig. 5a and f). Moreover, at the beginning of the run they are influenced by broad additional "pseudo-photospheric" absorption (see Sect. 4.2) which was strong from August 30 to September 2, and decreased afterwards. At the end of the run this broad additional absorption was very weak. Conversely, the blended absorptions at -13 and +50 km s-1 were strong and narrow and their increase drastically modified the shape and the FWHM of the profiles, as their equivalent width decreased to a lower value close to that of 1989.

[FIGURE] Fig. 8. Correlation between EW and FWHM for He I 6678 in August-September 1993

Radial velocity. The radial velocity (RV) of the He I line centroid was also measured in the absorption part of the initial profile. The general behaviour of RV can be seen in Fig. 5b. It is clear that rapid variations exhibit a beat phenomenon superimposed on the 9-10 day variation. Radial velocities range between -31 km s-1 and -3 km s-1. Frequencies of 1.55 and 1.37 c/d are detected without ambiguity (see Fig. 7c). It is thought that the contribution of weak V and R emission components has no influence on the determination of the line centroid, as frequencies detected in RV variations are also derived from lpv. Indeed, the emission contribution of each component represents only 0.10 to 0.15% of EW.

5.1.2. V and R emissions

Red emission (R) is essentially always present in the He I profiles, and is variable in intensity, with maxima close to [FORMULA]; it remains generally stronger than violet emission (V), a which is more difficult to measure due to the strong variability of the blue wing of the photospheric line when the absorption wing reaches -420 km s-1. The daily difference in radial velocities between the V and R peaks decreases over the run (Fig. 5e). The V emission peak is more stable in radial velocity than the R one and lies around -400 km s-1 which is greater than [FORMULA] km s-1, a value adopted by Floquet et al. (1992) for EW Lac. To investigate the importance and the variability of these emissions we measured the maximal intensity of each peak and their equivalent width measured above the continuum. The variations of equivalent width of the R emission peak are reported in Fig. 5f. In the beginning of the run it shows a slight but gradual increase superimposed on rapid variability. Though the same trend is just slightly detected in the V emission peak, it is found that the sum of the V and R equivalent widths globally increased over the observational run indicating a change over a few days in some physical parameters of the involved layers as in the EW, FWHM and RV of the absorption component. So it can be thought that the behaviour of V and R components is linked to the variation of the additional pseudo-photosphere (components (a) and (b)) and that they are conceivably formed in the same layer.

Periodicities were sought on time-series of each quantity (see Fig. 7d and e). In the R component the dominant frequency is 2.76 c/d. In the V component frequencies 1.24 and 2.77 c/d have been found. In each quantity the amplitude of the signal is strongly modulated over the run by the mid-term oscillation. The 1.22 c/d frequency is mainly detected in the V/R ratio.

To summarize, we find that the V and R emission components generally strengthened, while the FWHM and EW of the photospheric line decreased during the first week of the campaign. Though these quantities are correlated, mid-term variations over 9-10 days detected in FWHM and EW do not result from a combination of V and R emissions with the photospheric profile, but are quite real. Sporadic expulsion of matter from the star would enlarge the extent of the photosphere and could mimic a "photospheric" profile with a lower FWHM value according to computations by Collins et al. (1991); as He I 6678 is particularly sensitive to local formation conditions and to NLTE effects, small opacity variations in outer layers of the photosphere are able to induce detectable changes in line parameters.

5.1.3. Line profile analyses

Intensity variations of He I 6678 were investigated at fixed wavelengths separated by the sampling interval (0.242 Å for 1993 data and 0.101 Å for 1989 data). Frequency analyses were performed in the 0.12-9 c/d frequency range on two sets of data (1993 and 1989) by using two methods described in Sect. 5: Fourier Transform with Clean algorithm (method 1) and the Least Squares sinusoidal fitting with the AIC criterion (method 2). Results are given in Table 5.

We only retained frequencies detected by both methods, powers being somewhat different according to the method used (see Fig. 7). Our reinvestigation of frequencies in 1989 data, with better value of oversampling in the search for frequencies, led to similar values as those contained in Floquet et al. (1992), in which we only used Fast Fourier Transform with Clean algorithm. The 1.60 c/d frequency is highly dominant with both methods; 1.42 c/d is found over the whole profile with method 2 and the 0.42 c/d detected with method 1 is considered to be a one-day alias of 1.42 c/d. Other frequencies with lower power are 2.76, 3.17 and possibly 1.25 c/d detected only with method 1.

For the 1993 data, frequencies obtained with both methods are 0.12, 0.92, 1.22, 1.39, 1.55, 2.76 and 3.20 c/d; most of them were found for the EW (Table 5). With method 2 the AIC criterion decreased steeply for the first 3 detected frequencies (0.12, 1.55 and 1.22 c/d) then varied slightly. Fig. 9 shows the Clean periodogram for 1993.

[FIGURE] Fig. 9. Clean periodogram obtained from 1993 He I 6678 line profile. Gray scale indicates the power.

The 0.12 c/d frequency is due to the mid-term variation whose amplitude dominated throughout the run. It corresponds, in fact, to an upper limit in frequency for this mid-term variation. This time-scale is probably representative of the relaxation time of the outburst which occurred prior to the beginning of the run, since at its end the mid-term variations (EW) were back to the 1989 level.

The 4 frequencies found at 1.55, 1.39, 2.76 and 3.20 c/d deserved careful attention, and were also detected (within the frequency resolution) in the 1989 data. Apart from frequency resolution considerations, the peculiar behaviour of the value of some of these frequencies across the line profile in 1993 data is noteworthy; according to results obtained by method 1, the mean frequency 1.55 c/d, which extends from -400 to [FORMULA] km s-1, varies continuously from 1.60 c/d in the photospheric wings where the power is higher to 1.50 c/d towards the core of the line where the signal is lower. Similarly the mean frequency 1.39 c/d, whose extension is [FORMULA] km s-1, varies from 1.39 c/d in the wings to 1.32 c/d towards the core. This trend depicted from wings to core could explain the lack of coherence in the velocity phase near the center of the profile for both frequencies. This could be due to oscillations produced at different depths or to a superposition of unresolved frequencies. We did not find such behaviour in the 1989 data.

A period variation across the He I 6678 line was also reported for another Be star ([FORMULA] CMa) by Stefl et al. (1999). These authors found a period value (1.49d) in the far wings of the line profile, well beyond the limit defined by the projected velocity of the stellar limb, slightly greater than the main one (1.37d) found in the photospheric part. A possible origin in CS layers was attributed to the longer period. The behaviour of period variation across the line profile in EW Lac is different from [FORMULA] CMa. Indeed, both frequencies (1.55 and 1.39 c/d) were found varying continuously from the line center to the [FORMULA] limit in the outer wings.

The 3.20 c/d appears as a harmonic of the fundamental 1.55 c/d, while the 2.76 c/d is more puzzling, though it could be considered as the first harmonic of the fundamental 1.39 c/d. For this frequency, we should note the strong increase of the power and the abrupt change in the phase velocity at the level of the R emission. This frequency has been detected in IUE spectra obtained on the last day of our campaign on UV photospheric lines using a cross-correlation technique (Peters & Gies 2000).

The 0.92 c/d frequency is mainly detected on the blue part of the profile and extends between -300 and [FORMULA] km s-1. The corresponding amplitude is higher around -200 km s-1, i.e. in the wavelength region dominated by the blue component (b) (see Sect. 4.2). This frequency falls in the error range for the stellar rotation (see Sect. 2). The 1.22 c/d frequency is weakly present in the red wing of the line, but extends widely towards short wavelengths (-480 km s-1). For these two last frequencies we did not find any coherent phase variation.

nrpinvestigation. The two frequency groups of 1.55 and 3.20 c/d and of 1.39 and 2.76 c/d (Fig. 10 and Fig. 11), which are present in both 1989 and 1993 data, may be associated with nrp modes. To determine [FORMULA] and [FORMULA] values we used the method of Telting & Schrijvers (1997) and Schrijvers et al. (1997) applied to the 1993 data. The blue-to-red phase difference for the full wavelength range over which variability is found, [FORMULA], allows determination of the [FORMULA] value from the fundamental and a rough determination of the [FORMULA] value from the first harmonic. More precise values were calculated with the new coefficients given by Schrijvers & Telting (1999) for models taking into account temperature and velocity effects. But as has been shown by Telting & Schrijvers (1997), determination of [FORMULA] and m is not accurate for low degree modes. The errors [FORMULA] and [FORMULA] are estimated to be [FORMULA] and [FORMULA] respectively.

[FIGURE] Fig. 10a and b. Power distribution a and phase distribution b corresponding to the fundamental frequency 1.55 c/d and its first harmonic 3.20 c/d across He I 6678 in August-September 1993.

[FIGURE] Fig. 11a and b. Power distribution a and phase distribution b corresponding to the fundamental frequency 1.39 c/d and its first harmonic 2.76 c/d across He I 6678 in August-September 1993.

The 2.76 c/d frequency can be considered as the first harmonic of 1.39 c/d (case A in Table 6). However, its power distribution does not show the same behaviour over the line profile as the fundamental, as can be expected for the harmonic even in the case of non-adiabaticity effects (Schrijvers & Telting 1999). So we have also considered the possibility that 2.76 c/d is an independent signal (case B in Table 6). Amplitudes for all frequencies, derived from 1989 and 1993 data respectively, are also reported in Table 6. All amplitudes were weaker in 1993, except for the 2.76 c/d signal; the amplitude of the latter was measured outside the red abnormal intensified portion (RV [FORMULA] km s-1).


[TABLE]

Table 6. Amplitude of common frequencies detected in 1989 and in 1993, and determination of [FORMULA] and m. Case A: considering two groups of frequencies as fundamental and first harmonic; Case B: 2.76 c/d considered independently


As for the 0.92 and 1.22 c/d frequencies found with TSA in 1993 data, we did not find any coherent phase variation over the profile, and we consider that they are not related to nrp.

5.1.4. Evidence of orbiting circumstellar clouds

There is clearly a non-periodic component of the line profile variability which can only be seen on residues formed by subtraction of the run mean profile from individual spectra (Fig. 12). On September 3, a sharp absorption component crosses He I 6678 slower than the other (broader) blue-to-red moving patterns. This sharp feature, which appears first in the OHP spectra, can be followed in the Kitt Peak and DAO spectra; it is seen during 14 hours (HJD 2449233.44-2449234.02) crossing from -200 to +160 km s-1, followed unfortunately by a gap in observations. On September 5, a sharp but weaker feature appears again in OKAO spectra (HJD 2449236.0-2449236.28) with about the same transit velocity and during a shorter period (7 hours), observable only near the center of the line (-60 to +40 km s-1). The time span separating the two crossings of the narrow feature at RV [FORMULA] 0 km s-1 is [FORMULA]t = 2.4 d.

[FIGURE] Fig. 12. Gray-scale picture of residues around HJD 2449233.5 showing the transit of an orbiting circumstellar cloud. Observations from DAO, OHP, KPNO and OHP are displayed from bottom to top. As DAO and KPNO observations are simultaneous, we chose to use data from only one site per night.

The acceleration of this feature across the line profile is too low (620 km s-1/d) for a corotating stellar spot. Indeed, at the stellar surface, the acceleration is about [FORMULA] km s-1/d (see Sect. 2). These non-periodic features thus originate in the innermost parts of the circumstellar layers.

First, it was assumed that both features are two successive images of the same orbiting cloud. In this picture, if an equatorial plane motion with a circular law is assumed, the circular velocity [FORMULA] and the distance r can be reduced from purely observational data from [FORMULA] at v = 0 km s-1 and P = [FORMULA]. Inserting numerical data, dv/dt = 620 km s-1/d and P = 2.4 d, the cloud is found orbiting at about [FORMULA]; nevertheless its motion is not Keplerian because the derived stellar mass would be too low (3.1 [FORMULA]) compared with the expected mass estimated in Sect. 2 (9.6 [FORMULA]).

Secondly, it was considered that both features are images of different orbiting sub-features having about the same acceleration. In this case if Keplerian motion is assumed, clouds are found orbiting at about [FORMULA]. As the peak separation of V and R emissions in He I 6678 (which varies from 800 to 720 km s-1) is always larger than [FORMULA], implying the presence of layers which reach Keplerian velocity, it was reasonable to assume that orbiting clouds detected with [FORMULA] km s-1/d have nearly Keplerian motion.

Evidently we have observed the transfer of discrete ejected material to the envelope/disc, resulting from an outflow which occurred just prior to the run. This quantity of material is so small that it induces no important changes in line profile variability; as a matter of fact it is seen when variations in equivalent width and RV of He I are weaker, when the beat phenomenon between 1.39 and 1.55 c/d frequencies has minimal amplitude between HJD 2449233.5-2449235.2.

5.2. H[FORMULA] line

H[FORMULA] was intensively monitored from KPNO and also observed from Ondejov and Okayama observatories. Mean profiles in 1989 and 1993 are given in Fig. 13. In 1993 this strong double peak emission line is slightly asymmetrical with an intensity of 3.5 and a V/R ratio oscillating between 1.02 and 1.16. The overall emission is shifted to the blue with [FORMULA] km s-1, [FORMULA] km s-1, and [FORMULA] km s-1.

[FIGURE] Fig. 13. Mean profile of H[FORMULA]; full line: 1993 and dashed line: 1989

In 1989 H[FORMULA] is stronger, its intensity being 4.85 and V/R [FORMULA] 1. The emission line is centered at about [FORMULA], with [FORMULA] km s-1, [FORMULA] km s-1 and [FORMULA] km s-1. Note that in 1993 the behaviour of the H[FORMULA] V/R ratio is the same as for Fe II 6456 with V/R [FORMULA] 1. Nevertheless the shell is red-shifted in Fe II 6456 ([FORMULA] km s-1) and blue-shifted in H[FORMULA] ([FORMULA] km s-1). As a result, the global H[FORMULA] emission profile in 1993 is not consistent with those generally observed in V/R variables, as it is not as a whole shifted in the same direction as the weaker emission component, in agreement with optically thick line profiles from discs with [FORMULA]m[FORMULA] = 1 perturbation patterns (Okazaki 1996).

No fluctuations in the position of V and R peaks were found in H[FORMULA] and Fe II 6456.

For the H[FORMULA] line, we investigated the variation of the intensity of the blue (V) and red (R) emission peaks, the equivalent width (EW), and lpv in the same manner as for He I 6678. In H[FORMULA] the V emission peak shows a short-term variation superimposed on a monotonic increase (3%), and the R peak shows only short-term variability (Fig. 14a,b). Note that these two quantities are sometimes in phase and sometimes out of phase. Periodograms of EW, V and R, and the corresponding window function are given in Fig. 15.

[FIGURE] Fig. 14a-c. Variations in H[FORMULA] line; a: V emission peak intensity; b: R emission peak intensity; c: Equivalent width of the emission line

[FIGURE] Fig. 15a-d. Periodograms obtained for H[FORMULA] with Clean (full line) and Least Squares (dashed line) methods. a: Window function; b: Equivalent width; c: V emission peak intensity; d: R emission peak intensity

The V emitting component shows 1.61 and 2.80 c/d oscillations and the R component by 2.24 and 1.65 c/d (see Table 5). The equivalent width of the line is dominated by the 1.63 c/d frequency. It also shows a slight increase (3%) over the run (see Fig. 14c), which correlates with the V peak variation.

TSA applied to the line profiles reveals the presence of common frequencies with He I 6678 (1.60, 1.42 and to a lesser degree 0.90 c/d). These frequencies are detected within [FORMULA] km s-1 limits ([FORMULA] He I 6678 velocity range), so we consider that they are linked to subjacent photospheric variability. The amplitude of the signal corresponding to each frequency, after normalization to the intensity of the emission line in each respective scan, remains higher than in He I 6678. At first sight this effect is more important in the center than in the wings; however, it has to be considered with caution and needs to be confirmed. Indeed the uncertainty on the continuum determination was estimated at about 0.7%, which is of the same order as the amplitude of variation observed for He I 6678. Moreover, in the H[FORMULA] line, extended wings can affect the normalisation procedure and the strong emission amplifies the continuum level effects. The H[FORMULA] profiles used in this study need a careful correction in the continuum determination, which will be the object of further study. Thus, due to these uncertainties on the continuum determination, we are not able to discriminate between short-term variations due to nrp effects on the subjacent photospheric profile and a possible amplification of nrp in inner CS layers. The increase over the run of equivalent width, correlating with V emission peak intensity, can be partially explained by the greater gradual decrease of the pseudo-photospheric component in the blue part of the profile, assuming a similar behaviour in He I and in H[FORMULA] photospheric line profiles. Nevertheless episodic mass transfer from the star to the envelope should contribute to some enhancement of the line.

The H[FORMULA] profile in 1993 has a lower intensity but is wider than that of 1989. Such an effect was also observed by Hanuschik et al. (1996) in several stars such as 56 Eri, [FORMULA] Ori, HR 2284 and o Aqr. The broadening effect in emission line wings is generally attributed to electron scattering according to Castor et al. (1970) and Poeckert & Marlborough (1979). The main difficulty in explaining the H[FORMULA] line profile variation only by an increase of scattering effect is that the mean equivalent width changed from EW(1989) = -28Å to EW(1993) = -22Å while it is assumed to be conserved in the redistribution process (Mihalas 1978). It is then possible that other mechanisms involved in the long-term V/R changes also contribute to the line variation.

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Online publication: October 30, 2000
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