Detection

In Paper I, we had not detected a Van Hoof effect between the metallic lines FeII$\lambda\lambda$4923.921 and FeI$\lambda\lambda$4920.509 using observations done with the AURELIE spectrograph (resolving power of 17,000 and a temporal resolution of 5%). Our new observations with the spectrograph ELODIE are shown Figs.-. From the velocity-velocity diagram, it is clear that a variable phase shift is present between these two lines and consequently, for the first time, we confirm the detection of a metallic Van Hoof effect.

Note that to obtain Fig., we have done a selection of our data, which were collected over three consecutive nights, to make up a complete pulsation period: from $\varphi=0.774$ to 0.108 for August 3rd, 1994, from 0.197 to 0.700 for August 5th, 1994 and from 0.714 to 0.766 for August 4th, 1994 (see Fig.1 of Paper II). Due to the presence of a Blazhko effect a small velocity shift must be expected from one night to the following. For instance, it is around 0.6km/s from August 4th to August 5th, 1994 for the FeII line. Consequently, this effect must be smaller than the amplitude of the observed shifts in Figs.2-5. Another effect, due to the non-strict reproducibility of the atmospheric dynamics above all the photosphere, is certainly the main cause of dispersion of radial velocities from one cycle to the following.

  

Figure: Heliocentric radial velocity curves of FeII$\lambda\lambda$4923.921 (black points) and FeI$\lambda\lambda$4920.509 (white points) obtained over three consecutive nigths 3rd, 4th and 5th August, 1994 with the spectrograph ELODIE

  

Figure: The heliocentric radial velocity curve of FeII$\lambda\lambda$4923.921 is represented versus that of FeI$\lambda\lambda$4920.509. On the resulting curve is indicated the pulsation phase, according to the ephemeris given in Sect.2

In Fig., each point represents an individual spectrum. The average accuracy is certainly better than 500m/s. Consequently, the two closed loops are certainly real. From approximately phase 0.43 to 0.94 i.e., during the atmospheric compression, the upper loop is described anti-clockwise. This means that the variation of the FeII velocity curve is late compared to those of FeI one (see Fig.). Then, during the expansion, from phase 0.94 to 0.43, the lower loop is described clockwise, showing that now the FeI velocity curve is late with respect to the FeII one. Note that the maximum amplitude of these two velocity curves are about the same.

Contrary to pulsating stars with very weak amplitude and showing almost sinusoidal velocity curves (see for instance Mathias & Gillet 1993), the complicated shape of the velocity-velocity diagram for these iron metallic lines indicates that the velocity shift between these two lines is phase-dependent. This is consistent with the fact that, for large amplitude pulsating stars, the motion of atmospheric layers above the photosphere is far away from that of a standing wave (Fokin & Gillet 1997). Consequently, except the detection of the Van Hoof effect, it is difficult to deduce additional reliable information from this kind of velocity-velocity diagrams.

  

Figure: Same as Fig.2, but for FeII$\lambda\lambda$4923.921 and BaII$\lambda\lambda$4934.076

  

Figure: Same as Fig.2, but for FeII$\lambda\lambda$4923.921 and TiII$\lambda\lambda$5188.700

Figures- show respectively the velocity-velocity diagrams for BaII$\lambda\lambda$4934.076 and TiII$\lambda\lambda$5188.700 lines always with respect to the FeII$\lambda\lambda$4923.921 line. These curves are similar to the FeI-FeII curve (Fig.).

Note that the line doubling (or broadening) phenomenon (see Paper II), occurring within the phase interval 0.90-0.95, has an effect on the shape of the velocity-velocity curve. In all above figures, the velocity was measured with a gaussian fit over the full line without taking into account the doubling. Figure shows how the velocity-velocity curve changes when only the blueshifted line component is considered. A gaussian fit with two components was performed on the observed profile during this phase interval. We know that, when the line forming region is traversed by a shock wave, the relevant layers are divided in to two groups of opposite motion. Moreover, when the shock is propagating outward in the atmosphere, their two density columns are extremely variable, like the corresponding absorption components. Thus the last diagram (Fig.) always allows to follow only one type of the atmospheric motion (expansion or contraction). Three loops become visible showing again that in fact the atmospheric motion is very complicate.

  

Figure: Same as Fig.2, but when only the blueshifted absorption component is taken into account when the line doubling is present