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

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3. Modelling procedure

The starting point for modelling double mode pulsations is a nonlinear sequence of fundamental and first overtone models corresponding to a horizontal path through the instability strip. For comparing our results with previous investigations we adopted the same input parameters used by Bono et al. (1997), namely [FORMULA], [FORMULA] and XYZ=(0.760, 0.239, 0.001). Effective temperatures range from 6000 to 7200 K in steps of 100 K and for each model we compute the nonlinear fundamental and first overtone limit cycles. The resulting variation of the pulsation amplitude with respect to the effective temperature is given in Fig. 1 and shows the nonlinear instability strip boundaries for the chosen stellar parameters. A comparison with Bono et al. (1997) yields good agreement for the boundaries although the pulsation amplitudes in Bono et al. are pronounced higher in particular for the fundamental mode pulsations near the blue edge. A detailed analysis of our light curves in terms of Fourier parameters (Simon & Teays, 1982) shows excellent agreement with observed RR Lyrae light curves and will be presented in a companion paper. From Fig. 1 it can be seen that in the range between 6750 and 7000 K both fundamental and first overtone modes are unstable so that we expect multimode pulsations in this region.

[FIGURE] Fig. 1. Bolometric amplitude versus effective temperature for the fundamental and first overtone sequence through the instability strip. The vertical grey line indicates the location of the double mode model.

Consequently in the next step we perform an iterative search for a double mode solution between 6750 and 7000 K. For the initial velocity perturbation we set a mixture of fundamental and first overtone Eigenfunctions with an amplitude ratio of 1:1 and a maximum velocity of 5 km/s. The convergence of this iterative search can be followed in Fig. 2a depicting the temporal evolution of the amplitude ratio [FORMULA] for several temperatures in the considered range. For the two sequences above 6820 K the first overtone is growing while below 6820 K the fundamental mode is dominant. At 6820 K the amplitude ratio remains constant over about 6700 overtone cycles corresponding to [FORMULA] time steps. A detailed analysis of the amplitude data reveals no trend towards fundamental or first overtone mode. Consequently we state that the model exhibits a stable double mode pulsation at least within the considered time interval of 6700 overtone cycles. The location of the multimode solution within the instability strip is indicated by a grey line in Fig. 1. Note from Fig. 2a that the region of double mode pulsation is smaller than [FORMULA] K. From stellar evolution calculations by Lee & Demarque (1990, Table 3) we estimate a duration of about [FORMULA] years for evolving through this temperature range.

[FIGURE] Fig. 2. Temporal evolution of the amplitude ratio [FORMULA] for several effective temperatures around the double mode model a and growth of the first overtone ([FORMULA]) and the fundamental ([FORMULA]) component to full amplitude for the double mode solution at 6820 K b . Please note the different scales for [FORMULA] and [FORMULA] in panel b .

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© European Southern Observatory (ESO) 1998

Online publication: August 17, 1998