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Astron. Astrophys. 345, 172-180 (1999)

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4. Application and results

4.1. [FORMULA] Per (HD 24912)

In the periodogram (see Fig. 4) the most significant power outside the contaminated frequency range is found at [FORMULA] = 6.96(4) c d-1 (or [FORMULA] = 3.45(2) h) and at [FORMULA] = 9.14(7) c d-1 (or [FORMULA] = 2.63(2) h). The corresponding amplitudes and phases are displayed in Fig. 5. The phases for amplitudes less than 2[FORMULA] above the continuum are not significant and are plotted as open symbols. We argue below why we do not consider [FORMULA] as a significant period, and first consider [FORMULA]. Whereas the phase behavior of [FORMULA] is clearly characteristic for a NRP, the asymmetry of the amplitude around line center is not, unless non-adiabatic temperature variations are important. In the profiles of [FORMULA] Per we find small equivalent-width (EW) changes, which could be in principle due to temperature effects. A CLEAN analysis of the EW variations show a few weak peaks in the power diagram, none of which correspond to any other known frequency in this star. The source of the EW variability is therefore probably noise. Temperature effects generate in general larger amplitudes in EW than we observe (although cancellations cannot be excluded), which makes us conclude that these effects are not likely to play a significant role (see Schrijvers & Telting 1998). Data with a higher S/N and better time resolution are needed to establish the (expected) first harmonic and to determine the azimuthal number m.

We investigated whether the relatively low power on the blue side at [FORMULA] could be due to wind contamination. Simultaneously taken UV spectra (Kaper et al. 1999) show that a new DAC developed at low velocity around BJD 2447818.9. Similar simultaneous observations from another campaign on this star (see Henrichs et al. 1998a) showed that the new development of a DAC is accompanied by enhanced blue-shifted absorption in H[FORMULA]. The HeI line presently studied is probably also partly formed in the wind and therefore should in principle display similar kind of extra absorption, which would disturb the period analysis. We therefore excluded the 24 spectra between BJD 2447818.975 and BJD 2447819.1 in our further period analysis. These spectra showed extra blue-shifted absorption which was not present in other spectra, and which we attribute to wind effects. This procedure indeed decreased somewhat the asymmetry in amplitude, although not completely. Some wind absorption is undoubtedly still present in a number of spectra, but a thorough elimination is beyond hope. A second cause might be a blending effect by the partly overlapping (unidentified) weak lines in the alleged blue continuum which may have distorted the normalization. We consider this as less important.

We furthermore folded the spectra with [FORMULA] (right panel in Fig. 2). This shows a clear NRP pattern in the range from -100 to 160 km s-1, less asymmetric around zero velocity than Figs. 4 and 5 suggest. Although the asymmetry in power remains worrisome, more evidence for [FORMULA] being a true period in this star comes from observations of the OIII  [FORMULA]5592 line taken during the MUSICOS November 1996 campaign (see Henrichs et al. 1998b), which also clearly show the NRP pattern over the whole line profile. This dataset was however insufficiently sampled to determine any NRP frequencies.

Considerable power is also found at [FORMULA] = 9.14(7) c d-1 (or [FORMULA] = 2.6(2) h) over almost the whole line profile (see Fig. 4). Several reasons argue against an interpretation in terms of NRP, however. First, the minimum-entropy analysis (see Sect. 3) did not reveal any signal around [FORMULA], whereas [FORMULA] was detected with 15[FORMULA] significance. Secondly, the phase behaviour is suspiciously similar to that of [FORMULA] (see Fig. 5). In addition, the amplitude increases where the amplitude of [FORMULA] decreases, which may point towards interference. For these reasons we do not consider [FORMULA] as a NRP frequency.

From the phase diagram belonging to [FORMULA] we can derive the azimuthal degree, [FORMULA], according to Telting & Schrijvers (1997, hereafter TS) by measuring the difference in phase at [FORMULA]230 km s-1, i.e. just outside [FORMULA]sini of the star. We note that our adopted vsini value is in accordance with Penny (1996), although Howarth et al. (1997) find 213 km s-1. From Fig. 5 we derive [FORMULA] = 2[FORMULA]1.6(2), where we had to use a linear extrapolation of a fit through the region between -125 and 180 km s-1 because outside these regions the phase is likely to be poorly defined due to the very low amplitudes.

Table 2 of TS lists coefficients of empirical linear fits to input and output [FORMULA] values of synthetic data generated by Monte Carlo calculations for various pulsation parameters. We show below that in our case the ratio of horizontal to vertical motions, k, is small, certainly less than 0.3. This constraint gives for [FORMULA] the result [FORMULA] = 3.4(5) with 88[FORMULA] confidence that the [FORMULA] value is correct within [FORMULA]1. We adopt [FORMULA] = 3 as the most probable value for the harmonic degree. A higher or lower [FORMULA] value would show up as one more or one less absorption feature in the line profile, which is not seen (Fig. 2). Since the first harmonic of this signal (at 14 c d-1 or 1.75 h) could not be detected in our dataset because the sampling rate was too low, we have no means to derive a value for the azimuthal parameter m.

We can determine the direction of the pulsation mode if the stellar rotation rate is known. Periodicities in stellar wind features suggest that the rotation period, [FORMULA], is 2 or 4 days (Kaper et al. 1999). We obtain in both cases that the mode must be prograde. The value of k is in the (here justified) Cowling approximation related to the pulsation frequency, [FORMULA], in the corotating frame: k = [FORMULA]. Two rather differing values for the mass and radius are known: Leitherer (1988) finds 35 M[FORMULA] and 12 R[FORMULA] (case 1), not differing very much from earlier determinations, and also in agreement with the values given by Howarth & and Prinja (1989), whereas Puls et al. (1996) derive 60 M[FORMULA] and 25.5 R[FORMULA] (case 2) with new model-atmosphere fits, including wind effects. Even with the distance limits set by HIPPARCOS no preference can be given to either set. We can exclude a 4-day rotation period in case 1 because the implied rotation velocity would be lower than the observed vsini value, whereas a 2-day period is excluded in case 2, since the star would rotate at break-up. In both remaining combinations (case 1 with [FORMULA] = 2 d and case 2 with [FORMULA] = 4 d) the inclination angle turns out to be close to 40o. The corresponding corotating frequencies, derived from [FORMULA] = [FORMULA], are [FORMULA] = 5.4 c d-1 and [FORMULA] = 6.2 c d-1, which give [FORMULA] 0.05 and [FORMULA] 0.007, because [FORMULA]. Such small values of k imply small horizontal atmospheric motions, characteristic of a p-mode.

4.2. [FORMULA] Cep (HD 210839)

Outside the contaminated frequency range we find significant power at two different frequencies: at [FORMULA] = 1.96(8) c d-1 (or [FORMULA] = 12.3(5) h) and at about twice this frequency at [FORMULA] = 3.64(14) c d-1 (or [FORMULA] = 6.6(3) h), see Fig. 6. The minimum-entropy analysis yielded for these signals a significance of 10[FORMULA] and 8[FORMULA], respectively. The corresponding amplitudes and phases are displayed in Fig. 7. The power of the first period appears to be concentrated mostly on the blue side of the line. This could be caused by interference of the unidentified line which is displaced by about -400 km s-1 with respect to the HeI line, and which is stronger than in [FORMULA] Per. This secondary NRP pattern can clearly be seen in the folded spectra (Fig. 3). From the phase diagram of the larger period we derive an [FORMULA] value with the same method as above by measuring the difference in phase at [FORMULA]250 km s-1, again just outside [FORMULA]sini of the star. Our adopted vsini value is in accordance with Penny (1996), not very different from 217 km s-1 given by Howarth et al. (1997). We find [FORMULA] = 2[FORMULA]1.3(1). The irregular phase jumps near -160 km s-1 are likely caused by the very low power at these velocities, which are probably due to imperfectness of our dataset. We use here the [FORMULA] - [FORMULA] 2 restricted fit, which gives [FORMULA] = 2.9(3) according to TS. We adopt therefore [FORMULA] = 3 as the most probable value.

[FIGURE] Fig. 7. Amplitude and phase (in 2[FORMULA] radians) of the signals at [FORMULA] = 12.3 h and [FORMULA] = 6.6 h. The layout is similar to Fig. 5

Although [FORMULA] is nearly twice [FORMULA], it cannot be its first harmonic since the ratio [FORMULA] = 1.86(10) deviates more than 1[FORMULA] from the exact value of 2. In addition, in all velocity bins the power at [FORMULA] lies systematically below the power at [FORMULA] (see Fig. 6). This makes the probability of [FORMULA] being a harmonic of [FORMULA] less than [FORMULA]. We could also exclude [FORMULA] being a harmonic by considering the consistency check for the amplitude ratio and the phase relation of the main frequency and its first harmonic as given by TS. They find that [FORMULA] = 2[FORMULA] = [FORMULA]1.50(6), where the phase at line center of the main frequency is denoted by [FORMULA] and of the first harmonic by [FORMULA]. For [FORMULA] Cep we obtain [FORMULA] = [FORMULA]1.6(3). This would give fair confidence that the higher frequency could indeed be the first harmonic of the lower, except that in all model calculations by TS the ratio of the amplitudes of the first harmonic to the main frequency is considerably smaller than we observe, which makes [FORMULA] as a first harmonic very unlikely, which we therefore consider as a second NRP mode.

For the second mode we find [FORMULA] = 2[FORMULA]2.4(3). From this value we derive [FORMULA] = 5.2(7) using a [FORMULA] 0.3 fit with 86[FORMULA] confidence, which implies [FORMULA] = 5 as the most probable value for this second NRP mode and m remains undetermined. This mode complies with the number of bumps seen at any given time in the folded spectra in Fig. 3.

Adopting an upper limit to the rotation period of 4.5 days, using a radius of 19 R[FORMULA], a mass of 59 M[FORMULA] (Puls et al. 1996) and an inclination angle of 90o, we find that the corotating frequencies are 1.29 c d-1 for the [FORMULA] = 3 mode, implying [FORMULA] 0.38, and 2.53 c d-1 for the [FORMULA] = 5 mode, corresponding to [FORMULA] 0.1. Both modes are therefore prograde modes, and could be a p or g mode, depending on the adopted stellar model.

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© European Southern Observatory (ESO) 1999

Online publication: April 12, 1999