Shock wave origins

As said in the previous section, the maximum extension of the atmosphere occurs at $\varphi=0.55$.After this phase, the layers start their infalling motion due to gravity. In the absence of opposite forces (small density gradients or radiation pressure), the atmosphere would be continuously accelerated. This is in accordance with the motion of the red core component during the largest discontinuity, which can be represented by a straight line (Fig.a). In this case, the slope of this line (about 29m.s^-2) should approximately corresponds to the gravity of the star. As for the blue component, the associated deceleration is much lower, being around 20m.s^-2. Therefore, there is a mechanism which stops the free fall motion. Indeed, when the bottom of the layer is falling, it encounters the deepest photospheric layers which are denser. Consequently, a strong pressure gradient develops between the bottom and the top of the line formation region. This gradient becomes so large ($\varphi=0.87$) that the induced compression wave front breakes into a shock wave. Indeed, at this phase, the line doubling shows that there exist two distinct velocity fields, for which the velocity difference is directly related to the shock amplitude. As shown on Figs., and , the post-shock velocity is negative. Therefore, for the observer, (Eulerian coordinates), the shock is falling down. Thus, the Eulerian velocity of this receding front is smaller than that of the layers.

This is shown by comparing the velocity related to both H$\alpha$and Siiii line formation regions. Figure represents the heliocentric radial velocity curve associated to Siiii as a function of that associated to the H$\alpha$ line.

  

Figure: The radial velocity of Siiii is represented versus that of H$\alpha$ for the night August 1$^{\rm st}$, 1994. Note that these velocities are associated to the blue component for both lines. Black dots represent the ascending branch of the heliocentric velocity curves while the open circles stand for the descending branch. The arrows indicate the way the loops are drawn (clockwise), whereas the numbers give the pulsation phase of the closest point.

Since loops develop in this diagram, a phase lag exists between the considered line formation regions. Namely, as already mentioned in the previous section, the variation of H$\alpha$ is late compared to that of Siiii. This Van Hoof effect is interpreted in terms of outward propagating wave (Mathias & Gillet [1993]). Therefore, if the H$\alpha$ line formation region is located above that corresponding to the Siiii one, the wave propagates outward. Thus, although the shock is falling down within the atmospheric layers, it propagates outward in mass (Lagrangean coordinates). This means that the shock front comes close more and more to the H$\alpha$ layer during the infalling motion of the atmosphere. This shock, which will be called the ``infalling shock'', desappears before the end of the ballistic motion of the atmosphere, depending of its intensity.

Then, a rapid expansion phase takes place during the second velocity discontinuity. Shortly before this phase, the atmosphere is nearly at rest. Suddenly, the profile becomes strongly distorted. Despite no visible line doubling, the line shows a well marked asymmetry. This infers that a shock wave, originating from the subphotospheric regions, sweeps the layer outward such as in the classical Schwarzschild ([1954]) mechanism. This wave is likely caused by the iron opacity $\kappa$-mechanism which has been recently shown as the true engine of the pulsation of $\beta$Cephei stars. Note that, as for the infalling shock, the Van Hoof effect is also detected here and have the same interpretation.

Thus, the first discontinuity and the associated infalling shock would be a consequence of the wave induced by the $\kappa$-mechanism. This second shock will be called the ``outward-shock''. We could not determine any appreciable differences between the velocity separation between the two components for both lines considered, implying that the shock energy, in the framework of the precision of our data, remains almost constant during its propagation from Siiii to H$\alpha$ layers.

Having this scenario in mind, it is now interesting to consider the stillstand. Indeed, if the upper atmosphere was completely stopped by the denser bottom layers and before the arrival of the outward shock, the stillstand would be at the systemic velocity of the star. However, it is well known that the stillstand is variable in shape (Odgers [1955]). Our data show for the two considered nights, that, immediatly after the infalling-shock, from the beginning of the stillstand, the velocity is slightly below the $\gamma$-axis, which represents an expansion phase (Figs. and ). This can be interpreted as a bounce on the photospheric layers. From both figures, the larger the intensity of the infalling-shock, the larger the bounce. Then, because gravity acts, the atmosphere slows down and then falls back toward the photosphere, as shown by the velocity value above the $\gamma$-axis. This is well shown by the small and continuous decrease of the velocity during the 3/4 of the stillstand phase. In this view, the stillstand can be considered as the final damping step of the infalling atmospheric motion. Its variable shape is the consequence of different amplitudes of the ballistic motion which must changes from cycle to cycle. Indeed, as noted in the previous section, although the above scenario is valid for both studied nights, the line profiles and the velocity curves do not present the same pattern. This may be due to a different intensity of the successive shocks.