4. Mode identification
4.1. Pulsational degree of the mode
Apart from direct model fits to the spectra (e.g. Campos & Smith 1980, Vogt & Penrod 1983, Smith 1986), two distinct techniques of deriving a mode identification from time series of spectra have been developed: the analysis of the velocity moments of the variable profiles (Balona 1986; Aerts et al. 1992; Aerts 1996), and the analysis of the variations of the normalized intensity as a function of position in the line profile, through either one-dimensional Fourier analyses (Gies & Kullavanijaya 1988) or a single two-dimensional Fourier transform (Kennelly et al. 1992, Kennelly et al. 1998). The moment method can primarily be used for the identification of low-degree modes . As the number of bumps in the profiles of Sco suggests a higher value, and as we do not measure significant profile shifts (Sect. 2.1), we have chosen to use the method described by Gies & Kullavanijaya (1988) and further explored by Schrijvers et al. (1997) and Telting & Schrijvers (1997a).
Fig. 6 shows the distribution of Fourier phase and power at frequency 15.0 cycles/day in the periodogram of Fig. 4. The power distribution is very noisy, which is probably due to the poor time sampling of the pulsation period (only 30 spectra, with hardly enough time-resolution). Nevertheless, the distribution of the phase is well determined. We read off the phase difference between the blue and the red wings of the profile as 81 radians, which with Eq. 9 of Telting & Schrijvers (1997a) corresponds to a pulsational degree . For the different lines we get consistent results. As the line-profile variations are mainly caused by a single pulsation mode, the blue-to-red phase difference corresponds to the number of bumps and troughs in the individual line profiles (see Fig. 1).
4.2. The corotating frequency: p-mode or g-mode?
The amplitude distribution of the line-profile variability at frequency 15.0 cycles/day (Fig. 6) is consistent with that expected for a p-mode oscillation (Kambe & Osaki 1988). However, this information is not conclusive as modes with high temperature variations and also inclined tesseral g-modes can give rise to similar amplitude distributions (Lee et al. 1992; Schrijvers et al. 1997). Therefore we attempt to determine the nature of the mode from the observed oscillation frequency; we show that the observed variability is probably due to a p-mode oscillation.
De Geus et al. (1989) have analysed Walraven photometry of the members of the Sco OB 2 association, and found 5.5Myr for the age of the Upper Scorpius subgroup. For Sco they derived the following parameters: , , and . The above parameters, the age of the subgroup, and the spectral type of Sco are all indicative of a star with and .
Dziembowski & Pamyatnykh (1993, see their Figs. 5 and 6 for ) have shown that for a star like Sco one may expect both p-mode and g-mode pulsations. They show that the excitable high- g-modes have a maximum corotating frequency of about (where we used and to transform their dimensionless frequency to a corotating frequency).
From the current dataset we cannot derive a value of the azimuthal quantum number m of the pulsation. This number is needed to transform the observed pulsation frequency to that in the corotating frame of the star: (where the minus sign enters for retrograde modes, i.e. ). Nevertheless, we can constrain the corotating pulsation frequency by assuming that the mode is prograde and sectoral, i.e. . This allows us to compute a lower limit for the corotating pulsation frequency, to see whether we can exclude a g-mode origin of the pulsation.
The combination of (observed), (theoretical g-mode limit), and (assumed), gives a stellar rotation rate of =1.2 cycles/day. With the observed =110 km/s and with , this implies that a g-mode can only be the origin of the dominant pulsation in Sco if the inclination of the star is . Such a pole-on orientation of the star is unlikely, as only 5% of all stars are expected to have , the star would rotate at more than 57% of critical, and since for smaller inclination we would not expect to see such prominent line profile variations due to a non-radial pulsation in Sco. For any other possible values of the inclination , and of m , the corotating frequency would be higher, and hence the mode cannot be of g-mode origin.
We note that, considering the model calculations of Dziembowski & Pamyatnykh (1993), Sco is a young pulsating star that must have entered the instable region of the HRD quite recently.
4.3. Velocity amplitude of the mode
As demonstrated by Lee et al. (1992) the line-profile variations are the results of both Doppler shifts, due to the pulsational velocity field, and of pulsational temperature effects. The temperature variations lead to local brightness variations, and to local changes of the equivalent width. Here, we use our model (Schrijvers et al. 1997; Telting & Schrijvers 1997a; Schrijvers & Telting 1998) in order to estimate the amplitude of the pulsational velocity of the dominant mode in Sco. In order to do so, we estimate the magnitude of the temperature effects from observations of other stars.
The equivalent width (EW) of the SiIII triplet is tabulated as a function of spectral type by Kilian & Nissen (1989), who use a sample of 21 early B-type stars for their EW measurements. From their work we find that for the spectral type and luminosity class of Sco, B1V, the response of the EW to temperature varies approximately as . In terms of the parameters in our model this implies .
For radial modes, the observed integrated EW variations approximate the pulsational EW fluctuations of the local intrinsic line profile. The prototype of the Cephei stars, Cep (B2III/B1IV), shows EW variations of the order of 8% (peak-to-peak) due to its dominant radial mode (see e.g. Aerts et al. 1994a; Telting et al. 1997). The radial mode in the star Eri (B2III) results in EW variations of 14% (peak-to-peak, Aerts et al. 1994b). With the above response of the EW to temperature variations we derive that the surface temperature variations in these Cephei stars have an amplitude of 1-2%. Although the spectral types of the above stars are different of that of Sco, and although the value and the amplitude of the dominant mode in these stars is different, we assume for our modelling that the amplitude of the surface temperature variations of the mode in Sco is 2%.
In fact, Eq. 19 of Buta & Smith (1979) supports the assumption that the relation between and (the pulsational radial surface displacement) is not sensitive to the value of the mode, as the term hardly depends on for typical p-mode values of k: (Dziembowski & Pamyatnykh 1993). This means that for p-modes we should expect that, irrespective of the value, the surface temperature variations are similar for modes with similar radial displacement amplitudes.
In Fig. 7 we show our modelled time series of spectra for parameters that are suitable to Sco: =110 km/s, intrinsic line width W=10 km/s, ratio of horizontal to vertical pulsation amplitude at the surface k=0.05, maximum radial velocity amplitude at the surface =15 km/s, =3, and surface temperature amplitude =2%. We neglect the effects of the Coriolis force on the shape of the eigenfunctions. Note that we do not correct for the integration time when calculating these line profiles (for proper line-profile fits this will be necessary in the case of Sco). To investigate the value of required to produce line-profile variations with similar amplitude as that observed in Sco (see Fig. 1), we vary the value of m and i for all time series displayed in Fig. 7 while keeping the other parameters fixed.
We find a striking resemblance between the observed and modelled spectra for =9, i.e. for sectoral modes. For a near equator on inclination angle we find that a velocity amplitude of 10 km/s gives a good agreement with the observations. For we have to increase the amplitude to 15 km/s to get a similar amplitude of the line-profile variations. We also find some cases with low inclination angles, =8, and an amplitude of 20 km/s that describe the data well. For the other combinations of m and i in Fig. 7 the generated profiles do not match the observed profiles, regardless of the value of the pulsation amplitude.
In this case with small temperature variations, there is a clear difference between the line profile variations of the modes with different values of (see Fig. 7). This difference can be quantified by analyzing the harmonic content of the line-profile variations (Schrijvers et al. 1997; Telting & Schrijvers 1997a). The observed set of profiles do not have sufficient coverage of the pulsation phase to make such an analysis possible. However, visual inspection of the generated time series for non-sectoral modes suggests that the mode in Sco is (near-)sectoral.
In Fig. 8 we show the case where the temperature effects dominate the line-profile variability. We used the same set of parameters as for Fig. 7, but diminished the velocity amplitude and increased the temperature variation to =10%. One can see that for all values of m the line profiles look similar, and hence in this case it is virtually impossible to retrieve a value of from the observations. The similarity is due to the lack of harmonic terms in the line-profile variations, which in the case of high velocity amplitudes carries information on the m value of the mode.
With this high value for the temperature variations the magnitude of the line-profile variations of the near-equator on sectoral mode is similar to what we observe in Sco. For smaller inclination angles and lower values of even higher surface temperature fluctuations are needed to fit the observations.
From the magnitude of the expected variability in the EW and the centroid velocity of the profiles (see Figs. 7 and 8), it is clear that the observed constraints on these quantities (Sect. 2.2) are not strict enough to give guidance in determining whether the line-profile variations are mainly due to velocity or temperature effects.
© European Southern Observatory (ESO) 1998
Online publication: September 30, 1998