Determination of the turbulent velocity

As discussed in FGB, due to the difficulty of modelling non-LTE effects and due to an underestimation of opacities currently used which lead to deeper line profiles (see FGB Fig.7) in the coolest shocked regions, the determination of the turbulence based on complete line profiles are presently not relevant. Because the FWHM of the line does not depend much on the non-LTE effects (see FGB), it appears as the best reliable parameter to estimate the turbulence. Thus, following the method developped in our previous paper (FGB), we have determined the turbulence velocity assuming a rotational velocity $v_{rot}\sin\,i=7.5$km/s (see FGB). We have calculated 85 FeI profiles over one pulsation period to obtain a good time resolution of the turbulent velocity with respect to our new observations. As in FGB, we have used the two nonlinear nonadiabatic pulsational models A and B characterized by the following parameters: $T_{\mbox{\scriptsize eff}}=6056\,$K, $L\,=3100\,$L$_{\odot}$,$M\,=7.0\,$M$_{\odot}$, C_q=1.0, $\alpha\,=0.01$, LAO (model A), and $T_{\mbox{\scriptsize eff}}=5750\,$K, $L\,=3000\,$L$_{\odot}$,$M\,=5.7\,$M$_{\odot}$, C_q=1.0, $\alpha\,=0.1$, OP (model B). To compute each FWHM, we have considered the thermal broadening mechanism, the projection effect, the stellar rotation and the pulsation including velocity gradients which are enhanced by the presence of the three shock waves s1, s2 and s3. Because our hydromodels do not take into account turbulence, our theoretical FWHM are smaller than the observed ones. Their difference gives us an estimation of the turbulence velocity at each calculated phase point. Figure shows a comparison between observed (points) and calculated (line) FWHM for model A. A similar curve was also obtained for model B. We estimate that our ``fit'' is good enough to give an accuracy between 0.3-0.5km/s on the turbulent velocity v_turb.

  

Figure: Comparison between observed (points) and theoretical (continuous line) FWHM of FeI $\lambda\lambda$5576.0883 for $\delta$ Cephei. The calculated FWHM (model A) includes pulsation, rotation, gradient velocities and projection effects, assuming v_turb=0km/s. The continuous line adds the expected turbulence shows in Fig. which is given by the root mean square of the difference between observed and calculated square FWHMs

Figure presents the resulting turbulent velocity curves for models A and B respectively. In addition to the main peak centered at phase 0.84, a few secondary peaks or humps are clearly put into evidence. The main peak is associated with the global compression of the atmosphere. Thus, at the minimum radius, the turbulence becomes almost supersonic because it reaches a value near 7.5km/s. The minimum turbulence, occurring around the maximum radius ($\varphi=0.37$), is equal to 2.2km/s. This value is two times larger than that deduced by FGB which was only based on a single observed profile.

Near phase 0.9, our theoretical turbulent velocity reveals a violent oscillation with an unphysically sharp decrease (not shown on Fig. but phase interval marked by a dashed line-segment). At the same phase the theoretical light curve (Fig.1 of FGB) also shows an unrealistic minimum. As shown by a detailed inspection of the pulsating model, both these results are numerical provoked by a compression/shock wave passing through the hydrogen ionization zone (HIZ). Unfortunately, in our Lagrangian code the spatial resolution of this zone is rather poor which does not allow to correctly compute the rapid variation in the state of gas when a shock propagates through this zone. As a result, the opacity of the neutral gas just above the HIZ grows too rapidly, which blocks the radiative flux for a while and leads to an unrealistic light minimum. In the meanwhile, the artificial and gas kinetic pressures increase as well due to the shock passage and ionization. All this increases the line width, so that to fit the observed FWHM one must strongly diminish the turbulent velocity during this short phase interval (0.89-0.94). Keeping this in mind, we have removed a few points near phase 0.9 from our analysis as artefacts. This also means that it is not possible to determine the turbulent velocity within this phase interval and consequently to know the effect of the strongest shock s2 on the turbulence because it is crossing the relevant part of the FeI line formation region at this time.

  

Figure: Calculated turbulent velocity vs. the pulsation phase for model A (thick line) and model B (thin line). The dashed line-segment indicates the region in which a numerical artefact does not permit us to determine the shape of the turbulence curve

For model A, at phase 0.65, a large bump with an intensity around 3.5km/s appears when the shock s3 is crossing the FeI formation region while for model B, only a hump is present (Fig.). At phase 0.76, a weak hump is detected both in models A and B. It occurs when the shock s2 is propagating in this layer. Finally, a small (near 3km/s) but large bump takes place when ``buzz waves'' are observed in models A and B ($0<\varphi<0.3$). Of course our detection of the turbulence increases, occurring when shocks are traversing the FeI layer, are questionable because the shock signatures were already manifest in our two pulsating models. Especially, if these shocks are also present in observations at the same phases, a part of the deduced turbulence can be due to an underestimation of the shock wave intensity. Also, if this latter is overestimated by the model, the turbulence peak can be larger than in reality. Nevertheless, we never find that our shock intensities are excessive because the calculated FWHM are always smaller than the observed ones.