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Astron. Astrophys. 337, L29-L33 (1998)

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5. Discussion and conclusions

We present a stable nonlinear double mode pulsation with parameters comparable to observations. Concerning the fundamental mode period of 0.5279 days the model lies well within the range of both field stars and metal poor OoII cluster stars (M15 and M68, see Fig. 4a). The period ratio of 0.7468 is located near the upper boundary for OoII stars and slightly too high compared with the field star sample. However, this comparison is only based on 4 objects! Concerning intermediate- metal OoI stars Fig. 4a reveals no agreement with the double mode parameters of IC 4499.

An interesting property of the nonlinear double mode solution is its occurence in a very narrow temperature range. More precisely speaking we state that for the chosen set of stellar mass, luminosity and composition there exists only a very small effective temperature range of maximum [FORMULA] K where a double mode pulsation can be found. This is not in contradiction with observational results predicting a wider period and temperature range of double mode pulsations as these stars clearely disperse over a range of stellar parameters. We suppose our model to indicate that for a particular star evolving with constant mass and composition exactly on a horizontal path through the instability strip only one effective temperature exists where stable double mode pulsation is possible. However, a more detailed investigation of the longterm double mode stability within the given narrow temperature range is necessary to corroborate this statement.

A distinct property of all RR Lyrae double mode stars is the dominance of the first overtone amplitude which is also reproduced by our model. Fig. 4b shows good agreement with observed amplitude ratios found in the literature. In this context it is important to discuss the dissipation properties of the model and their influence on the double mode solution. Kovács (1993) found the amplitude ratio of nonlinear radiative and Lagrangean double mode models to be strongly influenced by the zone number and the choice of the artificial viscosity necessary to treat shock waves. He worked out that for the standard viscosity parameters the amplitude ratio [FORMULA] becomes lower than one while a reduced viscosity reverses this feature. One major advantage of adaptive methods is the high spatial resolution provided by tracing ionisation zones and other important features throughout the computation. Consequently the artificial viscosity length scale can be reduced to values smaller than the important physical length scales of the problem given e.g. by the the steep decline of the hydrostatic background structure in the photosphere. For the presented double mode model we adopt an artificial viscosity length scale of [FORMULA] times the local radius. In order to illustrate the situation we note that the ratio between the hydrostatic pressure scale hight and the local radius is about [FORMULA] at the bottom of the hydrogen ionisation zone and about [FORMULA] at the bottom of the HeII zone. In fact we do not observe an influence of the artificial viscosity on the pulsation properties. The radiative models in Feuchtinger and Dorfi (1994) use an additional linear viscosity to adjust the amplitude of the pulsations. In the case of our convective model this linear viscosity is replaced by the turbulent Eddy viscosity which serves as a free but physically motivated parameter and is used to tune the pulsation amplitudes. Note that the Eddy viscosity length scale of 0.6 times the typical convective lengthscale (mixing length) adopted for obtaining reasonable fundamental and first overtone amplitudes and instability strip boundaries (cf. Fig. 1) automatically leads to the right amplitude ratio.

Beginning with the work of Petersen (1973) double mode pulsations are used to determine so called beat masses by fitting the observed period ratios with linear pulsation models in the Petersen diagram. In this context the well known beat mass discrepancy was solved reasonably by employing the new opacities provided by the OPAL and OP projects (see Bono et al. 1996 and references therein). However, the uncertainty of lacking nonlinear confirmation of the linear results remains. From Fig. 4a it turns out that our [FORMULA] model lies within the range of OoII double mode stars and slightly above the field star range. Cox (1991) uses the new OPAL opacities to determine a value around [FORMULA] both for OoI and OoII stars. Bono et al. (1996) uses nonlinear fundamental and first overtone surveys to estimate mass and luminosity of cluster double mode stars by fitting the Petersen diagrams with models unstable both in the fundamental and first overtone mode, ending up with [FORMULA] for OoI and [FORMULA] for OoII stars. On the other hand, we use Z=0.001 which is not suitable for OoII stars but a common value used in models for RR Lyrae field stars. Thus our [FORMULA] double mode model indicates a discrepancy to the quoted investigations based on linear and nonlinear single mode calculations. However, we present only one nonlinear double mode model and it is not clear how in particular the free parameters of the convection model alter the scenario.

Obviously further computations are necessary to answer several questions arising from the above considerations. The major uncertainty of nonlinear pulsation calculations is the time-dependent convection model introducing a number of free parameters. In our case two parameters, namely the convective length scale and the Eddy viscosity length scale are specified and we expect an influence of these parameters on the properties of the double mode solution. However, these parameters also influence the general properties of single mode pulsations, in particular the amplitudes, instability strip boundaries and the shape of the light and radial velocity curves. Consequently, a comprehensive and satisfactory nonlinear picture of pulsating stars has to agree with observations in all these aspects. On the other hand, the huge and accurate observational material available for pulsating stars offers the unique opportunity to justify and constrain our theoretical understanding of stellar pulsations. As an example the influence of turbulent convection on the light curves indicates that it should be possible to investigate and calibrate time-dependent convection theories in the same way it is done for time-independent convection models using astro- and helioseismological methods but on a nonlinear and time-dependent basis. Bearing in mind the very small effective temperature range of double mode solutions nonlinear models of this phenomenon seem to provide an excellent tool for such tasks.

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© European Southern Observatory (ESO) 1998

Online publication: August 17, 1998