Astron. Astrophys. 363, 568-574 (2000)

4. Comparison with observations and conclusion

The differential velocity , radius and acceleration between observed Fe I and Fe II lines are given in Paper I. Contrary to theoretical models, the rebound effect on and -curves is not detected. Moreover, only one large perturbation is visible at the minimum radius. This means that only one shock has a high enough amplitude to affect the Fe I and Fe II layers, like in RR7b. In this case, only s1 would play a role. Alternatively it may not be possible to distinguish the effect of s1 and s2 due to their small phase separation.

Data plotted in these figures suggest that the shock effects in the models do not occur at the same phase. This shift can be explained by the fact that the effective temperatures of the two adopted models vary from 6900 K for RR7b to 7175 K for RR41. As mentioned in Sect. 2, such temperature differences may lead to different Fourier spectra of the limit cycle pulsation due to the nonlinear influence of the overtones. In turn, this may result in different amplitudes of shocks and alter their dynamics.

The discrepancy between theory and observations suggests that the adopted models do not account for the details of the atmospheric dynamics. Even though a thorough analysis of this mismatch is beyond the aim of this investigation, we briefly mention some plausible reasons. Apart from the uncertainty of the main stellar parameters, the most critical point is the description of the shock waves. Two aspects seem to be important. The first one concerns the numerical modelling of the shock interaction with the HIZ. This problem is typical for all the Lagrangean codes, in which the HIZ is not well spatially resolved.

It is probable that the convection may also alter the result. For example, as shown by the convective models of Bono et al. (1994a), the pulsation amplitudes of these models are smaller that those in the radiative models, which in turn reduce the shock amplitudes. The detailed calculations of the Van Hoof effects using the convective models are needed to study the effect of convection.

The second aspect concerns the thermal structure of shocks, which is important for the line formation. For instance, the shock heating provokes different variations in the number of absorbers of the Fe I and Fe II lines. Thus, an incorrect calculation of the shock heating/cooling may result in a wrong estimation of the level of formation of each of these lines.

To conclude, this work clearly shows that the study of the Van Hoof effect provides a useful tool to understand the detailed dynamics of atmospheric layers of pulsating stars. It is also a good test for the consistency of the existing (unfortunately not numerous) nonlinear models for pulsating atmospheres. The two models presented, that we have successfully used up to now in less detailed studies of the RR Lyrae, reveal difficulties in reproducing correctly the Van Hoof effect. This means that further improvement of the numerical method, as well as more sophisticated physics, is needed to allow one to interpret the new high-resolution observational data.

© European Southern Observatory (ESO) 2000

Online publication: December 11, 2000
helpdesk.link@springer.de