6 The frequency f$_\mathsf{1} = 1.03$ c d $^\mathsf{-1}$

The observed variations can be interpreted in terms of a model with corotating spots or clouds (B01) or with the presence of a NRP mode. In this section we investigate whether the observed frequency f₁ could be interpreted in the frame of NRPs.

The frequency f₁ has also been detected in photometry (e.g. B01). From $f_{\rm rot}$ determined in Sect. 2, we obtain f₁/ $f_{\rm rot} = 1.4$. This is in favor of the pulsation model as Zorec et al. (2002) showed that $f_{\rm photometry}$ can hardly represent $f_{\rm rot}$.

6.1 Phase diagrams

Figure 15: Power (solid line) and phase (cross symbols) for the frequency f1 = 1.03 c d-1 and for its first harmonic at 2.06 c d-1 (power as dashed line and phase as plus symbols). Vertical dotted lines show the limits of the domain determined from the variance; vertical dashed lines show the domain $\pm$$v\sin i$.

For each studied line the phase and power diagrams for the frequency f₁ are shown in Fig. 15. The $\pm$$v\sin i$ domain and the velocity domain derived from the variance (see Sect. 5) are shown. The phases were recomputed with a LS technique, as cleaning methods such as RLC give less accurate phase values. Greyscale dynamic spectra as a function of phase are presented in Fig. 16, showing absorption and emission features travelling across the line profiles. These features are clearly seen on the blue side of all lines. They can also be seen travelling back on the red side of strong lines such as He I 6678. This can be explained with the low inclination angle i of the star, so that we view the pulsations on the far side of the star, similar to what has been observed in other Be stars (see $\upsilon$ Cyg in Floquet et al. 2000b, 48 Per in Hubert et al. 1997, $\mu$ Cen in Rivinius et al. 1998b).

Note that both the He I and the purely photospheric lines of other species (C II 4267, Mg II 4481 and Si III 4553) show pulsations, which are all in phase with each other. The slope of the phase variation is the same for each line, except for C II 4267, but this line is the weakest one and its phase is less well defined. Once again, note that the power of the pulsations is generally higher at the blue side of the line, especially in the He I lines which show stronger emission. This asymmetry, which has also been observed in other Be stars (see EW Lac in Floquet et al. 2000a, $\mu$ Cen in Rivinius et al. 2001), is especially strong in $\omega$ Ori but remains unexplained. Townsend (2000) suggested that trans-photospheric wave leakage may play a role.

6.2 Mode determination of the NRPs

Figure 16: The greyscale plots of all spectra for each studied line normalized by its mean profile folded in phase with the frequency f1 = 1.03 c d-1. The velocity range corresponds to the one determined by the variance of He I lines.

 

 

Table 5: NRP parameters (pulsation degree l and azimuthal order m) for the frequency f₁ = 1.03 c d^-1 determined from different methods. Cols. 2-3: phase slope determination; Cols. 4-5: same with T&S correction coefficients; Cols. 6-7: FDI determination.

	Phase			T&S			FDI
Line	l	|m|		l	|m|		l	|m|
He I 4471	1-2	1		2	0-1		2
He I 4713	2	1		2	0-1		3
He I 4921	2	1-2		2-3	1		3
He I 5876	2	1-2		2-3	1		3
He I 6678	2	1-2		3	1		3	2-3
C II 4267	1?	1		1?	0-1		3
Mg II 4481	1	1		2	0-1		3
Si III 4553	2	1-2		2	1		3

The frequency f₁ = 1.03 c d^-1 found by the time-series analysis can be associated with NRP modes. The slope of the phase diagram gives an estimate of the pulsation degree l. The slope of the phase diagram of the first harmonic provides an estimate of the azimuthal order |m| (Fig. 15). For numerous model fits Telting & Schrijvers (1997, hereafter T&S) derived that for l-|m| < 2 the following corrections apply:

$${l = 0.076 + 1.110 * \frac{\vert\Delta\Phi_0\vert}{\pi}}$$ (1)

$${\vert m\vert = -1.028 + 0.613 * \frac{\vert\Delta\Phi_1\vert}{\pi}\cdot}$$ (2)

The obtained values, however, still have uncertainties of $\pm$1 for l and $\pm$2 for |m|. The results, obtained with and without these correction coefficients, are shown in Table 5.

The mode parameters of the pulsations have also been determined by Fourier Doppler Imaging (FDI, see Kennelly et al. 1992, 1996). In a rapidly rotating star, the pulsation velocity field and the temperature perturbations are mapped onto a wavelength position corresponding to the rotationally induced Doppler shift. When the oscillations are confined to the equatorial region, the obtained normalized wavelength frequency corresponds to |m|, otherwise it represents l. The FDI method is based on the number of travelling bumps and therefore the mode with |m| = 0 cannot be detected.

The results are shown in Fig. 17 for each line where the slow trend has been removed. The mode parameters are reported in Table 5. Note that when the l value was between 2 integer values, it has been averaged to the lowest integer, as our computational tests showed that, for a single mode and adopted stellar inclination, the FDI technique tends to increase the value of the mode parameters.

The frequency f₁ is then attributed to NRPs with l = 2 or 3 and |m| = 1, 2 or 3. However, the pattern of pulsations travelling across the lines seen in Fig. 16 excludes the value |m| = 1. From a more detailed modeling of $\omega$ Ori (Neiner et al. in preparation), preliminary results show that |m| = 2 is the most likely case for this star. Therefore we consider in the following the modes l = 2 or 3 and |m| = 2.

Figure 17: Greyscale Fourier Doppler Imaging (FDI) results for each studied line. The ordinates provide the time frequency (f1) while the absisses provide the wavelength frequency, which is an estimate of either the pulsation degree l or the azimuthal order |m|. The possible solutions are within an ellipsoidal zone (shown in black).

6.3 Comparison with theoretical models

The NRP frequency measured in an inertial frame can be written as (Ledoux 1951):

$$\pm f_{\rm puls} = f_0 \pm (1 - C_{nl}) \vert m\vert f_{\rm rot},$$ (3)

where $f_{\rm puls}$ is the pulsational frequency in the frame of the observer, f₀ is the frequency in the stellar frame for a non-rotating star, $f_{\rm rot}$ is the rotational frequency of the star and the signs are positive (negative) for prograde (retrograde) modes.

For high-order p modes, the higher effects of rapid rotation can be neglected (Dziembowski, private communication) and Eq. (3) transforms to:

$$\pm f_{\rm puls} = \frac{\sqrt{3 G M}}{2 \pi \sqrt{R^3}} \sigma_0 \pm \vert m\vert f_{\rm rot}.$$ (4)

The value of the positive non-dimensional frequency in the stellar frame

$$\sigma_0 = \frac{2 \pi f_0}{\sqrt{4 \pi G <\rho>}} = \frac{2 \pi \sqrt{R^3}}{\sqrt{3 G M}} f_0$$ (5)

has been determined by Balona & Dziembowski (1999, hereafter B&D), for various values of l for g and p prograde modes depending on the effective temperature and luminosity of the star. From their calculations for stars with a luminosity between 0.5 and 1.0 mag above the ZAMS, such as $\omega$ Ori, we derive, for prograde p modes, $\sigma_0 \sim 1.60 \pm 0.4$. This value is also valid in first approximation for retrograde p modes (Dziembowski, private communication).

Taking $f_{\rm puls} = 1.03$ c d^-1, l and |m| determined in Sect. 6.2, M, R and $f_{\rm rot}$ determined in Sect. 2, the value of $\sigma_0$ derived from Eq. (4) is negative for prograde p modes, which is impossible by definition. If $\omega$ Ori hosts a retrograde p mode, we obtain $\sigma_0 = 0.20 \pm 0.04$. This value is incompatible with the one found by B&D.

Taking slow rotation into account (e.g. Coriolis forces) but no higher effect due to rapid rotation (e.g. departure from sphericity), for high radial order g modes, Eq. (3) transforms to:

$$\pm f_{\rm puls} = \frac{\sqrt{3 G M}}{2 \pi \sqrt{R^3}} \sig... ...pm \left(1 - \frac{1}{l(l+1)}\right) \vert m\vert f_{\rm rot}.$$ (6)

If the star hosts a g mode, $\sigma_0$ derived from Eq. (6) is negative for a prograde mode, which is impossible by definition, but note that the results are only marginally below zero. $\sigma_0 = 0.08 \pm 0.04$ for l = 2 and $\sigma_0 = 0.14 \pm 0.04$ for l = 3 for a retrograde mode. From B&D, we derive $\sigma_0 \sim 0.2 \pm 0.05$ for prograde g modes with l = 2 and $\sigma_0 \sim 0.35 \pm 0.05$ for prograde g modes with l = 3. However the approximation of slow-rotation used here may not apply to this star, and the fact that B&D values are only calculated for prograde modes and are also used here for retrograde modes, may explain the small difference obtained for a retrograde g mode, but also prevent to rule out prograde g modes.

In conclusion, $\omega$ Ori cannot host a p mode with the determined parameters but is likely to host a g mode, possibly retrograde. Spectral modeling is necessary to confirm this result, which is the subject of a follow-up paper.