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

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 ( = ), 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, , and pulsation velocity semi-amplitude . For each grid element we calculated specific intensities and limb darkening of the HeI 4713 line for the given stellar parameters ( and ) 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 were varied in the domain of Fig. 8 and the maximum amplitude was measured. We applied this method to Per using = 36 000 K and = 3.4 (Puls et al. 1996, case 2 in the above). We computed 25 models in the - i plane between 5 and 25 km s^-1 and from 10 to 90^o respectively. The resulting semiamplitudes relative to the line depth, , are shown as contours in Fig. 8.

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

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

For  Cep the stellar parameters are not too different from those of  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 and the amplitude 0.11, implying = 0.06. For an inclination angle of 90^o this gives 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 Cep has nearly the same limb darkening and width of the intrinsic profiles as for  Per, which are the main quantities on which the NRP amplitude depends. In spite of all these approximations we consider the derived value for  Cep to be quantitatively justified.

© European Southern Observatory (ESO) 1999

Online publication: April 12, 1999
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