Gravity-Modes in ZZ Ceti Stars: I.Quasiadiabatic Analysis of Overstability

We analyze the stability of g-modes in variable white dwarfs with hydrogen envelopes. In these stars, the radiative layer contributes to mode damping because its opacity decreases upon compression and the amplitude of the Lagrangian pressure perturbation increases outward. The overlying 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 two provided $\omega\tau_c\geq 1$, where $\omega$ is the radian frequency of the mode and $\tau_c\approx 4\tau_{th}$ with $\tau_{th}$ being the thermal time constant evaluated at the base of the convective envelope. As a white dwarf cools, its convection zone deepens, and modes of lower frequency 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 DAVs. The linear growth time, ranging from hours for the longest period observed modes ($P\approx 20$ minutes) to thousands of years for those of shortest period ($P\approx 2 $ minutes), probably sets the time-scale 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.