Amplitude saturation in β Cephei models

Although the driving mechanism acting in β Cephei pulsators is well known, problems concerning the identification of the amplitude limitation mechanism and the non‐uniform filling of the theoretical instability strip remain to be solved. In the present analysis, these problems are addressed by non‐linear modelling of radial pulsations of these stars. In this approach radial modes are treated as representative of all acoustic oscillations. Several models of different masses and metallicities were converged to limit cycles through the Stellingwerf relaxation technique. The resulting peak‐to‐peak amplitudes are of the order of Δ V = 0.3 mag . Such amplitudes are significantly larger than those observed in β Cephei pulsators. Assuming that all acoustic modes are similar, we show that collective saturation of the driving mechanism by several acoustic modes can easily lower predicted saturation amplitudes to the observed level. Our calculations predict a significant decrease in saturation amplitudes as we go to high‐mass/high‐luminosity models. However, this effect is not strong enough to explain the scarcity of high‐mass β Cephei variables. A possible weakness of the collective saturation scenario is that the estimated line‐broadening, resulting from excitation of many high‐ l modes, might be higher than that observed in some of the β Cephei stars. We argue that this difficulty can be overcome by allowing g‐modes to participate in the saturation process. We also discuss robust double‐mode (DM) behaviour, encountered in our radiative models. On a single evolutionary track we identify two DM domains with two different mechanisms responsible for DM behaviour. The non‐resonant DM domain separates the first overtone and fundamental‐mode pulsation domains. The resonant DM domain appears in the middle of the first overtone pulsation domain. Its origin can be traced to the 2ω 1 =ω 0 +ω 2 parametric resonance, which destabilizes the first overtone limit cycle.