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

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5. Model calculations

From the observed amplitudes in the line profiles one can derive the velocity amplitude of the pulsation. To obtain a crude estimate we considered for simplicity a single sectoral p-mode ([FORMULA] = [FORMULA]), i.e. without horizontal velocity component (k = 0), and no temperature variations. We divided the visible stellar surface in about 2500 elements. The grid is further defined by the inclination angle, i, rotational velocity, [FORMULA], and pulsation velocity semi-amplitude [FORMULA]. For each grid element we calculated specific intensities and limb darkening of the HeI [FORMULA]4713 line for the given stellar parameters ([FORMULA] and [FORMULA]) using atmosphere models created by TLUSTY and SYNSPEC by Hubeny & Lanz (1992). For each grid element the radial velocity and angle towards the observer is computed. Using these values, Doppler-shifted specific intensities as function of v are interpolated from the model atmosphere. Finally, the line profile was calculated by integrating the intrinsic line profiles of all grid elements over the visible surface. For a given inclination angle and pulsation amplitude 20 profiles were calculated spread over a full period and the profile with the largest amplitude at line center was selected. At that particular phase the inclination and [FORMULA] were varied in the domain of Fig. 8 and the maximum amplitude was measured. We applied this method to [FORMULA] Per using [FORMULA] = 36 000 K and [FORMULA] = 3.4 (Puls et al. 1996, case 2 in the above). We computed 25 models in the [FORMULA] - i plane between 5 and 25 km s-1 and from 10 to 90o respectively. The resulting semiamplitudes relative to the line depth, [FORMULA], are shown as contours in Fig. 8.

[FIGURE] Fig. 8. Contour plot of calculated relative amplitudes of the HeI [FORMULA]4713 line profile as a function of the pulsation velocity semi-amplitude [FORMULA] and inclination angle i. The curves are based on 25 model calculations with stellar atmosphere parameters corresponding to [FORMULA] Per with [FORMULA] = 3.4 and [FORMULA] = 36 000 K (Puls et al. 1996)

In [FORMULA] Per the central depth of the HeI line is 4[FORMULA] and the amplitude only 0.12[FORMULA] of the local continuum (see Fig. 5), which means that the NRP signal is weak with [FORMULA] = 0.03. Following the corresponding contour in Fig. 8 we derive that [FORMULA] can be at most about 5 km s-1 for the adopted [FORMULA] 40o.

For [FORMULA] Cep the stellar parameters are not too different from those of [FORMULA] Per (although k is larger), and we simply applied the same calculations for this star as a first approximation. The central depth of the line is 2[FORMULA] and the amplitude 0.11[FORMULA], implying [FORMULA] = 0.06. For an inclination angle of [FORMULA] 90o this gives [FORMULA] 6 km s-1 for this star. The inclination cannot be much lower according to the stellar parameters (see Sect. 4.2). From a sample calculation we found that a model atmosphere for [FORMULA] Cep has nearly the same limb darkening and width of the intrinsic profiles as for [FORMULA] Per, which are the main quantities on which the NRP amplitude depends. In spite of all these approximations we consider the derived value for [FORMULA] Cep to be quantitatively justified.

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© European Southern Observatory (ESO) 1999

Online publication: April 12, 1999