## 4. Mode identification## 4.1. Pulsational degree of the modeApart 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 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 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 modeAs 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 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
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 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
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 |