Gravity Modes in ZZ Ceti Stars. I. Quasi-adiabatic Analysis of Overstability

We analyze the stability of g-modes in white dwarfs with hydrogen envelopes. All relevant physical processes take place in the outer layer of hydrogen-rich material, which consists of a radiative layer overlaid by a convective envelope. The radiative layer contributes to mode damping, because its opacity decreases upon compression and the amplitude of the Lagrangian pressure perturbation increases outward. The convective envelope is the seat of mode excitation, because it acts as an insulating blanket with respect to the perturbed flux that enters it from below. A crucial point is that the convective motions respond to the instantaneous pulsational state. Driving exceeds damping by as much as a factor of 2 provided ωτ_c≥1, where ω is the radian frequency of the mode and τ_c≈4τ_th, with τ_th being the thermal time constant evaluated at the base of the convective envelope. As a white dwarf cools, its convection zone deepens, and lower frequency modes become overstable. However, the deeper convection zone impedes the passage of flux perturbations from the base of the convection zone to the photosphere. Thus the photometric variation of a mode with constant velocity amplitude decreases. These factors account for the observed trend that longer period modes are found in cooler DA variables. Overstable modes have growth rates of order γ~1/(nτ_ω), where n is the mode's radial order and τ_ω is the thermal timescale evaluated at the top of the mode's cavity. The growth time, γ⁻¹, ranges from hours for the longest period observed modes (P≈20 minutes) to thousands of years for those of shortest period (P≈2 minutes). The linear growth time probably sets the timescale for variations of mode amplitude and phase. This is consistent with observations showing that longer period modes are more variable than shorter period ones. Our investigation confirms many results obtained by Brickhill in his pioneering studies of ZZ Cetis. However, it suffers from two serious shortcomings. It is based on the quasiadiabatic approximation that strictly applies only in the limit ωτ_c≫1, and it ignores damping associated with turbulent viscosity in the convection zone. We will remove these shortcomings in future papers.