5 Line-profile analysis

Figure 3: Examples of typical variations for each studied line. The different normalized profiles have been vertically shifted by 0.05 to facilitate comparison.

Several lines were examined: He I 4471, 4713, 4921, 5876, 6678, C II 4267, Mg II 4481 and Si III 4553. They all show variations. In Fig. 3 we present characteristic examples of the studied line profiles. Note that the red wing of the He I 5876 line is corrupted by an unreliable continuum determination due to a blend of telluric lines. After the mean profile is subtracted, LPVs are very similar for all lines. When emission is present, the variations are seen on a broader domain than $\pm$$v\sin i$: Fig. 4 shows the variance of the He I 6678 line, affected by emission, and the unaffected Si III 4553 line. The He I 4713, 4921 and 5876 lines show variations similar to the He I 6678 line, whereas the C II 4267 and Mg II 4481 lines behave similar to the Si III 4553 line. The red wing of the He I 4471 line and the blue wing of the Mg II 4481 line contaminate each other.

Figure 4: Variance of the He I 6678 and Si III 4553 lines. The limits of the He I 6678 line are indicated as solid lines and of the Si III 4553 line as dotted lines, on both panels.

5.1 Time series analysis of LPVs

Figure 5: Top: the spectral window of the MuSiCos 98 campaign for the He I 6678 line. Bottom: residual window after applying the RLC method.

We applied different methods to search for periodicity in the LPVs. The results of the two best methods, the Restricted Local Cleanest (RLC, based on Foster 1995; Foster 1996, developed by Emilio 1997 and Domiciano de Souza Jr. 1999 and applied in Domiciano de Souza et al. 2000) and the Least Squares (LS) methods, are presented here.

We have used the RLC method to search, in each wavelength bin, for 30 frequencies in a predefined range from which it computes all possible models with 4 frequencies. Comparing the power of each of these models, it selects the 7 optimal frequencies, while suppressing the aliases. A Local Cleanest (Foster 1995) is then applied, i.e. the RLC looks for 7 points around each of these 7 frequencies to finetune the final frequencies.

With this method the window aliases are removed. However, in case some aliases may remain, the cleaned spectral window is checked and the remaining frequencies are not considered as real (e.g. Fig. 5). Periods longer than the total duration of the observing run of 22 days ( $f \le 0.047$ c d^-1) cannot be detected. The mathematical frequency resolution of the RLC method is better than 0.01 c d^-1, but the accuracy of the results due to the length of the observing run is 0.047 c d^-1. We also assigned weights to each spectrum according to the S/N ratio measured in the continuum next to the line, but this did not change the results.

The LS method, as applied by Kambe et al. (1993a), considers the whole line profile at the same time, i.e. all wavelength bins together, to determine which frequencies describe the LPVs in the best way. After the first frequency is found, it is removed from the data (prewhitening). Then the program searches for the next frequency in the residual spectra and the procedure is repeated several times.

   

Table 3: Main frequencies (in c d^-1) deduced from the analysis of several lines for all sites for which data was available, indicated in Col. 2, with the number of spectra in Col. 3. In Cols. 4 and 5 the frequencies found by the RLC method are shown in normal fonts while the results of the LS method are shown in boldface. The resolution is 0.01 c d^-1 whereas the accuracy of the frequencies is 0.047 c d^-1.

Line	Sites	Spec.	Frequencies (c d-1)
			f1	f2
He I 4471	3, 4, 5, 8	125	1.04	0.50/ 0.50
He I 4713	2, 3, 4, 8	131	1.01	0.43
He I 4921	2, 3, 4, 6, 8	130	1.03/ 1.03	0.47
He I 5876	2, 3, 4, 5, 6, 8	147	1.03/ 1.04	0.48/ 0.48
He I 6678	1, 3, 4, 5, 6, 7, 8	198	1.03/ 1.04	0.46/ 0.46
C II 4267	2, 3, 4, 8	120	1.03/ 1.04
Mg II 4481	2, 3, 4, 5, 8	125	1.04/ 1.03	0.48
Si III 4553	2, 3, 4, 5, 8	125	1.04/ 1.03	0.48

Sites: 1 = OHP152, 2 = OHP193, 3 = Kitt Peak, 4 = ESO,
5 = Xinglong, 6 = Mt. Stromlo, 7 = MCT/LNA, 8 = INT.

Figure 6: For each line, the mean line profile is plotted in the bottom panel for the velocity range with a well defined continuum level. The corresponding greyscale periodogram, obtained with the RLC method, is shown in the upperleft panel. The associated power spectrum, smoothed with a gaussian filter and summed over the domain [-350, 400] km s-1 when possible, is shown in the right-hand panel. When the edges of the domain are perturbed by other lines, the sum is taken over the plotted domain (see mean line and greyscale panels).

The results of the time-series analysis for each line with both methods are recorded in Table 3. The search has been done simultaneously for all sites where data for the specific line were available. This gives strong confidence in the frequencies detected in many lines: they cannot be attributed to window aliasing as the database for each line differs. The spectral window for the He I 6678 line is shown as an example in Fig. 5 (top).

For each line, the greyscale periodogram, mean line profile and power spectrum are shown in Fig. 6. Several frequencies are detected:
(a) a powerful frequency was clearly found at f₁ = 1.03 c d^-1 ( P₁ = 0.97 d), identical to the one published by B01;
(b) a second frequency is present at f₂ = 0.46 c d^-1 ( P₂ = 2.17 d);
(c) a frequency f₃ = 0.56 c d^-1 ( P₃ = 1.78 d) is also detected, together with its first harmonic 1.12 c d^-1. This frequency is probably a combination of f₁ and f₂ as f₁ - f₂ = 0.57 c d^-1;
(d) a frequency around f₄ = 0.82 c d^-1 ( P₄ = 1.22 d) is seen in the stronger lines, similar to the one detected in UV data of February 1996 (Peters & Gies 2000), but the signal is too weak to be studied here. Note that this frequency is close to the rotational frequency determined in Sect. 2.
Finally, a slow variation ($\sim$weeks) is seen over the duration of the observing run.

5.2 Centroid velocity

For several lines we measured the centroid velocity of the line profiles, which corresponds to the first velocity moment (Balona 1986). All lines show a periodic variation with a frequency of 1.03 c d^-1 corresponding to f₁, shown as a function of phase in Fig. 7, together with a best sine fit. For the C II 4267 line, the high noise level prevented a good fit. The mean centroid velocity $V_{\rm mean}$, the amplitude of velocity variations A and the phase shift $\Phi$ of the sine fits compared to phase 0 vary for the different lines. Although such differences have already been detected in other Be stars (Stefl et al. 2000), the ones seen here could be due to the limited accuracy of the velocity determinations and the fact that the variations may not be sinusoidal. On average for He I lines, except the He I 4471 line, we obtain $V_{\rm mean} = 34.3$ km s^-1, A = 3.7 km s^-1 and $\Phi = 0.2$. For purely photospheric lines, except the C II 4267 line, we obtain $V_{\rm mean} = 28.3$ km s^-1, A = 4.0 km s^-1 and $\Phi = 0.1$. We consider the differences as not significant.

5.3 v sin i and apparent variations

The high sampling rate and long duration of the campaign cancel all short-term variations (days) in the mean spectrum of each line. These mean spectra are then not perturbed by pulsational variations if present, but the mean line width will be different from the one of a similar non-pulsating star because of the pulsational velocity component.

However, $v\sin i$ can be well determined by applying a Fourier transform analysis (Gray 1976) to the mean line profiles of a pulsating star. An additional broadening can introduce new minima in the Fourier transform of the profile, but the determination of vsini is unaffected since the position of the first minimum of the rotation profile cannot be modified. Thus, using this method, the estimate of $v\sin i$ is not significantly affected by emission in the line wings (Jankov et al. 2000).

The result for each line (He I 4009, 4026, 4144, 4471, 4713, 4921, 5876, 6678, C II 4267, Mg II 4481 and Si III 4553) is reported in Table 4 and plotted in Fig. 8 for the eight main lines studied in this work. The values are consistent with each other, except for the He I 5876 line, but recall that its red wing is not reliable, and for the He I 6678 line at a lower degree. The $v\sin i$ value of the pulsating star can then be averaged, using all lines in Table 4 except the He I 5876 line, giving $178.8 \pm 3.5$ km s^-1.

Figure 7: Centroid velocities (in km s-1) of the different lines folded in phase with frequency f1 = 1.03 c d-1.

Figure 8: Amplitude (in logarithmic scale) of the reduced Fourier frequency for each line. The first minimum gives $v\sin i$ in km s-1.

 

 

Table 4: Determined values of v sin i (km s^-1) for each individual line using the Fourier transform analysis, compared to the values given by B01.

Line	v sin i
	this paper	B01
He I 4009	180.8
He I 4026	180.9	177
He I 4144	179.0	173
He I 4471	176.6	174
He I 4713	179.4	180
He I 4921	179.5	183
He I 5876	159.8?	156
He I 6678	170.6	173
C II 4267	177.7	214
Mg II 4481	185.1	233
Si III 4553	178.8	223

B01 determined v sin i for most of these lines (see Table 4), using calculated intrinsic line profiles. They found similar results for the He I lines, but a discrepancy between He I (173 km s^-1) and purely photospheric (226 km s^-1) lines and proposed to explain this difference as being due to the presence of circumstellar material in the He I lines.

Although the emission intensity during the MuSiCos 98 observations ($I_{\max}$(H $\alpha) = 1.4$) was higher than during their observations ($I_{\max}$(H $\alpha) = 1.1$), this discrepancy is not seen in this work, thanks to the Fourier method, since the position of the first minimum of the rotation profile is not very sensitive to emission in the line wings. From the results shown here, we conclude that $v\sin i$ is of the same order for He I and purely photospheric lines.

Figure 9: Apparent vsini (in km s-1) for each line folded in phase with frequency f1 = 1.03 c d-1. A best sinusoidal fit is overplotted.

Independently of the results shown above, we determined vsini using a selection of the 98 available IUE spectra of $\omega$ Ori. We convolved the spectrum of a slowly rotating ( $v\sin i=40$ km s^-1) reference star of the same spectral type ($\pi^4$ Ori) with a rotational profile. Using a least-squares method and about 800 lines, the best fit is achieved at $v\sin i = 169$ km s^-1. This implies that the best value of $v\sin i$ for $\omega$ Ori is $\sqrt{169^2 + 40^2} = 174$ km s^-1. This result is in agreement with the one obtained by Fourier analysis, giving strong confidence in the obtained value.

In the rest of this work we adopt the value determined by the Fourier method: $v\sin i = 179$ km s^-1.

Inspecting individual spectra of the MuSiCoS campaign, a variation in time is observed in the apparent $v\sin i$ with frequency 1.03 c d^-1 corresponding to f₁. An overplot with a best fit sinusoid is diplayed in Fig. 9. In the frame of NRPs, such variations can be interpreted as a consequence of a horizontal velocity field and/or temperature oscillations.

5.4 Slow variation ($\sim$weeks)

A variation over the length of the run is found in the line-profile analysis, but is hard to characterize from the LPV as its duration is comparable to the length of the MuSiCoS 98 campaign. However, this variation can clearly be seen from other parameters such as the peak intensities or the equivalent widths of the lines.

5.4.1 Peak intensities

Figure 10: Example of variable emission peaks in the C II doublet at 6578 and 6583 Å near the H$\alpha$ line at HJD 2451142.6 and at HJD 2451147.6 (dotted, and shifted by -0.02 in intensity for clarity). The emission is less strong in the red wing of the H$\alpha$ and C II doublet lines.

Figure 11: Variations of the summed Violet + Red emission peaks of the H$\alpha$, He I 5876, 6678, 7065 and Si II 6347 lines during the MuSiCoS 98 campaign.

Figure 12: Variations in km s-1 of the emission peak separation of the He I 5876, 6678 and H$\alpha$ lines during the MuSiCoS 98 campaign.

All the lines affected by emission (H$\alpha$, H$\beta$, He I 5876, 6678, 7065, Si II 6347, C II 6578, 6583) show the same kind of peak variations. Figure 10 shows an example of the H$\alpha$ line and the C II doublet at 6578 and 6583 Å: blue and red emission peaks are seen. Figure 11 shows the summed Violet + Red emissions of the H$\alpha$, He I 5876, 6678, 7065 and Si II 6347 lines, while Fig. 12 shows the separation in the peaks of the He I 5876, 6678 and H$\alpha$ lines.

The peak separation of the He I lines fluctuates during the first part of the run, with a daily difference up to 60 km s^-1 between HJD 2451144.5 and HJD 2451148, but the temporal distribution of data between those days is unequal. Then it progressively decreases again until the end of the observing campaign.

The peak separation of the H$\alpha$ line slowly varies over the run; a minimum occurs when the peak separation of the He I lines fluctuates more conspicuously, followed by a gradual increase until HJD 2451155 and finally a slow decrease similar to the He I lines. Nevertheless, a fluctuation at HJD 2451147-48 is still visible.

This evolution is typical for an emission line outburst in a Be star: precursor phase, outburst phase and relaxation phase (see R1), which suggests that an outburst occurred around HJD 2451147. Note that there is no emission in the Si II 6347 line at the beginning of the observing campaign (V + R = 2, corresponding to the continuum level) and that its maximum in emission does not occur simultaneously with the maxima in emission of the He I lines, but rather precedes them.

Figure 13: Variations of the ratio of the Violet over Red emission peaks of the C II doublet at 6578 and 6583 Å and the He I 5876, 6678 and 7065 lines during the MuSiCoS 98 campaign.

The ratio of the Violet and Red emission peaks of all the lines affected by emission also shows strong variability during the first half of the campaign and becomes more stable during the second half. The results for the He I 5876, 6678, 7065 and C II 6578, 6583 lines are shown in Fig. 13. The similarity between the C II 6578 line and the He I lines is clear. For the C II 6583 line, the variations look different, likely due to the difficulty of the determination of R for this line as this critically depends on the placement of the continuum.

5.4.2 Equivalent widths

The variation over the run is also reflected by the change in equivalent width of the red He I lines. Because the variation at the center is in antiphase with respect to the wings of the lines, its effect is cancelled when looking at the equivalent width of the whole line. However, studying the wings (blue: [-350, -100] km s^-1 and red: [180, 400] km s^-1) and the center ([-100, 180] km s^-1) of the line separately allows to recover the variation, as shown in Fig. 14.

Figure 14: Equivalent widths variations (in Å) of the center and the blue and red wings of the He I 6678 line during the MuSiCoS 98 campaign. See symbol caption in Fig. 12.

We also studied the H$\delta$ line but no significant changes indicative of an outburst have been detected. A weak tendency of narrower wings and less deep line core is observed between HJD 2451148 and 52. Note, however, that there is almost no emission in H$\delta$ during the campaign.