Turbulent Convection and pulsational Stability of Variable Stars I. Oscillations of Long-Period Variables

We have performed a linear pulsational stability survey of 6 series of long period variable models with M = 1.0M⊙, L = 3000 − 8000L⊙, and (X,Z) = (0.700, 0.020), (0.735, 0.005). The dynamic and thermodynamic couplings between convection and oscillations are treated by using a statistical theory of nonlocal and time-dependent convection. The results show that the fundamental and all the low overtones are always pulsationally unstable for the low-temperature models when the coupling between convection and oscillations is ignored. When the coupling is considered, there is indeed a ”Mira” pulsationally instability region outside of the Cepheid instability strip on the H-R diagram. The coolest models near the Hayashi track are pulsationally stable. Towards high temperature the fundamental mode first becomes unstable, and then the first overtone. Some one of the 2nd -4th overtone may become unstable for the hotter models. All the modes higher than 4th (n > 4) are pulsationally stable. The position and the width of such an instability region on the H-R diagram critically depends on the mass, luminosity and metal abundance of the star. The overall properties of the dependence are: 1) For the same mass and luminosity the instability region becomes slightly wider and moves to lower effective temperatures as the metal abundance increases; 2) For a given chemical abundance, the instability region becomes wider and moves to the lower effective temperature as their luminosity increases or their mass decreases For the luminous red variables seated outside the instability strip the dynamic coupling between convection and oscillations balances or may even overtake the thermodynamic coupling. Turbulent viscosity can no longer be ignored for the pulsational instability of the low-temperature red variables. The effect of turbulent viscosity becomes more and more important for higher modes, and may finally become the main damping mechanism of the pulsation.