Before closing the argument, let us discuss if and how the theoretical scenario could be varied in the attempt to match the theoretical results with observations. The theoretical framework outlined in the present paper is based on numerical experiments supplied by three different fields: 1) stellar evolution, 2) stellar pulsation, 3) stellar atmospheres. Based on proper physical assumptions, all of them simulate the physical processes which take place in different regions of the stellar structure. Unfortunately, any observational test involves in the same context the results of the quoted scenario so that it appears difficult to point out where and why the theoretical outcomes could be affected by systematic errors. In order to give some more light on this scenario, let us briefly discuss in the following the three above quoted theoretical ingredients.
1) Several and thorough papers have been recently devoted to the evolutionary properties of HB stars and to their dependence on astrophysical parameters and physical ingredients (CCP; Dorman et al. 1993 and references therein) and therefore they will not be discussed here.
2) The theoretical treatment of radial stellar pulsations is based on two main ingredients: the physical inputs necessary to produce an envelope model and the set of convective hydrodynamic equations adopted for following the nonlinear coupling between dynamical and convective motions.
Among the physical inputs, both radiative opacities and equations of state are worth being checked, since the radial pulsational instability is driven by the destabilizing effects operated by kappa and gamma mechanisms. Numerical tests performed to verify the dependence of the nonlinear pulsation characteristics on the new OP (Seaton et al. 1994) and OPAL (Roger & Iglesias 1992) opacities show almost no difference in comparison with models based on the old Los Alamos opacities. A similar result was obtained by Buchler & Buchler (1994) in their investigation on BL Herculis variables. Therefore this ingredient, at least for low metallicity stars, has no effect on the amplitude problem.
Concerning the equation of state we did not perform any specific calculation since recent nonlinear results on radiative models of bump Cepheids provided by Kanbur (1992) show only negligible physical differences on the adopted equations of state. However, we plan to test the new Livermore equation of state as soon as it becomes available for metal contents suitable to metal poor stars.
Although the assumptions we adopted to describe the turbulent field in variable stars are physically plausible, the convective transport equation has been derived by using three different free parameters (for a detailed discussion see Stellingwerf 1982 and BS). A more formal and rigorous approach for deriving the set of radial pulsation equations was adopted by Kuhfuß (1986) and by Gehmeyr (1992 and references therein). However, only few full amplitude nonlinear models have been computed by adopting the latter formalism and therefore a trustworthy comparison between hydrocodes which involve different physical and numerical approximations cannot be provided.
The self-consistent inclusion of nonlocal, nonlinear and time-dependent effects may help to improve long-standing pulsation questions (BS; BCM; Bono et al. 1995). In particular, the nonlinear convective models are characterized by pulsation amplitudes that, although still too large, are roughly half magnitude smaller than the amplitudes of radiative models.
3) The transformation of the light curves into the observational plane is one of the most thorny problems of radial stellar pulsations. At present two different routes can be followed to accomplish this transformation. The former involves the calculation of hydrodynamic pulsating atmosphere models which take into account a multifrequency radiation transport in the outermost optically thin layers. This approach not only provides useful insights on the dynamical properties of the surface regions but can also predict broad-band colors. Until now only few investigations have been devoted to soundly solve this problem for RR Lyrae variables (Keller & Mutschlecner 1970; Bendt & Davis 1971).
Davis & Cox (1980) following this route computed few models for evaluating the RR Lyrae mean colors but no theoretical study has been undertaken so far for estimating the RR Lyrae light curves in specific photometric bands.
The latter route relies on the use of bolometric corrections and color-temperature relations provided by static atmosphere models. This approach is based on two weak physical assumptions: the stellar effective temperature over the full cycle is derived by assuming that the surface zone is always in radiative equilibrium, whereas the surface gravity is evaluated by assuming both radiative and hydrostatic equilibria (Bono et al. 1994). Although the quoted approximations were proved to be suitable for evaluating the RR Lyrae mean colors (Bono et al. 1995) and for reproducing the ionization and excitation equilibrium conditions of metal lines close to the phase of minimum light (Clementini et al. 1995), the evaluation of pulsation amplitudes may require a more cautious analysis.
We conclude that the present theoretical scenario proves to be successful to shed light on many pulsation features of cluster variables and, in particular, on long-standing questions such as the transition between FO and F pulsators and the origin of the Oosterhoff dichotomy. However, predictions concerning the pulsation amplitudes are far from reaching the required compatibility with observational data. The possible origins of such a discrepancy were discussed, but firm conclusions have not been reached.
We can only draw attention on such a situation, waiting for further improvements either of the theory of radial pulsation or of the theory of stellar atmospheres and/or of stellar evolution which may succeed in reconciling theory with observation. This will reach the relevant goal of using the pulsational parameters of RR Lyrae variables to gain firm information on their evolutionary status and, in particular, on their masses, temperatures and luminosities. The use of these pulsators will allow us to solve many open questions concerning the evolution of our Galaxy and, in particular, they might become robust standard candles for mapping the distances inside the Galaxy and among the galaxies belonging to the Local Group.
Acknowledgements
It is a pleasure to thank P.A. Mazzali for useful conversations on stellar atmospheres. We wish also to acknowledge A. Balestra and C. Vuerli of the OAT Technologic Division for their warm support in computing facilities. We are also grateful to C. Aerts for his valuable comments as referee on an early draft of this paper which have improved its readability. This work was partially supported by MURST, CNR-GNA and ASI.