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Next: Turbulent velocity variation Up: FWHM and turbulent velocity Previous: Line profile modelling

FWHM curve

Figure[*] represents the observed variations of the FWHM with the phase over three consecutive nights (Chadid & Gillet 1997), together with our theoretical fit obtained with variable vturb for the RR Lyr model RR41. Note that the observed FWHM curve slightly varies from one pulsational cycle to another. This effect is mainly due to the poor repetition of the pulsational cycles, most probably caused by the shock propagation. The dispersion is certainly not due to the Blazhko effect because in this case the expected total amplitude of the FWHM variation would be excessively large (the Blazhko period for RR Lyrae is 40.8days). For this reason we did not try to obtain a detailed fit but a rather general one.

As seen from the figure, the FWHM curve has a complex structure. The inspection of the atmospheric structure and dynamics shows that the principal peak of the observed FWHM curve occurs exactly during the very short phase of the line doubling induced by a shock. This point was qualitatively put into evidence by Chadid & Gillet (1997). As shown by Fokin & Gillet (1997), this doubling is provoked by the shock s1, emerging at the phase 0.92 rather than by the thermal or turbulent broadening. The height of the narrow FWHM peak at $\varphi$=0.92 represents the Dopplerian separation of the two components of the splitted FeII line. On the theoretical velocity curve (Fig.[*]), this event corresponds to a drastic velocity inversion after the shock passage. Since the doubling is very short, the peak is very narrow. As shown by our numerical experiences, the intensity of the shock s1, and consequently the amplitude of the Doppler splitting, is very model dependent. Nevertheless, this shock-induced splitting, or broadening, is always much stronger than any physically realistic turbulent broadening. Thus, only a detailed treatment of the observed broadening of each line components, can provide the correct value of the turbulence at this phase. Nevertheless, this would be only possible when very high quality observations would be available. We can only suggest that, according to current theories of the turbulence amplification (Gillet et al. 1998b), vturb must increase due to compression in the shock.

The reason for the drastic decrease of FWHM after the phase 0.93 seems to be twofold. On the one hand, it is in part related to a very short decrease in the velocity gradient in the line formation region (LFR) after the first shock (s1) passage, at $\varphi\approx 0.92$.Indeed, this shock is very rapid, and crosses the whole LFR in about 0.01P due to high compression degree of the atmosphere. Consequently, the radial velocity behind the front becomes nearly equal everywhere. Because the velocity gradient in the LFR strongly contributes to the line width (FGB), its decrease causes a drop of the FWHM. Note that this local FWHM minimum is yet much higher than the basic minimum at the phase 0.45. Shortly after that, the second shock arrives (s2, $\varphi=0.95$), but it is too weak to produce a perceptible Doppler splitting. The disk integration at this phase also does not contribute much to FWHM due to relatively low radial velocity in the LFR.

On the other hand, FWHM depends on the compression state of the turbulent gas. The adiabatic turbulence amplification theories predict an increase of vturb when density increases, and vice versa (see for instance Gillet et al. 1998b). As seen from Fig.[*], after the first shock (s1) passage, the mass density first rised, quickly decreases due to rapid and homogenious expansion of the atmosphere after the shock passage. Hence, this atmospheric dilatation during the interval 0.91-0.94, can also contribute to the observed drop of FWHM after the main peak. However, the observed minimum of the FWHM corresponds to the arrival of the second shock s2 (Fig.[*]), which also provokes an appreciable compression. Thus, at this moment, we can expect that the turbulent velocity, and consequently FWHM, increase again. Actually, without turbulence enhanced at this phase, the predicted FWHM minimum would be lower. We should note, however, that the value of the calculated FWHM minimum is not only model dependent, but also depends on the accuracy of the numerical description of the physical conditions in the LFR. These conditions, in turn, are strongly influenced by the shock propagation and the gas dynamics near the hydrogen ionization front (see Fokin & Gillet 1997). We must admit that both these physical processes cannot be satisfactorily calculated at the present time Due to current limitation of the numerical approach. Finally, we conclude, that the FWHM, and consequently vturb, calculated within the phase interval 0.91-0.94 are only indicative although our result is well consistent with observations (Fig.[*]).

The second peak, that appears after the phase 1.0, is mainly due to the rapid expansion of the atmosphere. At this moment the integation over the stellar disk alone gives very broad lines which well fit to the observations (the ``de Jager'' point, see FGB). Between the phases 1.0 and 1.1, FWHM depends on the turbulent velocity only weakly, because the high expansion velocity (about 60km/s) largely exceeds the sonic velocity (about 7km/s). Indeed, due to strong dissipation effects, the turbulent velocity should be always subsonic. The discrepancy between the predicted and observed FWHM at this phase is mainly due to the discrepancy in the dynamical characteristics of the model and the real star. Consequently, the value of vturb obtained here for this phase, is rather speculative.


  
Figure: Mass density variation for different mass zones in the hydrodymanical model of RR Lyr (RR41). The photosphere is located between mass zone number 52 and 60 depending on pulsation phase. Here only the part of the atmosphere between mass zone numbers 60 and 80 is shown. The curves are arbitrary shifted to demonstrate the relative behavior of the density for different layers. The shocks are indicated following the notation as in Fokin & Gillet (1997)
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics{fig3.ps}}\end{figure}

A small theoretical FWHM peak near the phase 1.5 (Fig.[*]) coincides with the passage of an early shock (s3), as can be seen from Fig.[*]. The Dopplerian broadening, produced by this shock in our hydrodynamical model RR41, is much larger than observed. For this reason, this theoretical peak cannot be eliminated by decreasing of the vturb . Evidently, in real RR Lyrae stars this shock must be much fainter than in our nonlinear models. The origin of this shock, as discussed in Fokin & Gillet (1997), is related to a stop of the hydrogen recombination front at the end of the expansion phase. Since, in our Lagrangian code, this front is badly resolved with respect to the mass grid, its calculated motion through the mass is certainly approximative. Yet, there are now strong theoretical evidences that such predicted type of shocks, first studied by Hillendahl (1970), should exist in real stars.


next up previous
Next: Turbulent velocity variation Up: FWHM and turbulent velocity Previous: Line profile modelling

8/13/1998