Pulsational Instabilities in Accreting White Dwarfs

(Abridged) The Cataclysmic Variable (CV) population harbors a diverse range of donor stars and accreting white dwarfs (WDs). A range of WD masses is expected, from low mass Helium core WDs, to massive WDs which have previously accreted at rates high enough for Hydrogen to burn steadily. Furthermore, a wide range of Helium enrichment is expected in the accreted material depending on the degree to which the donor star is evolved. We investigate the impact of this diversity on the range of effective temperatures ($T_{\rm eff}$) for which g-modes are unstable. The critical $T_{\rm eff}$ below which modes are unstable ("blue edge") depends on both surface gravity, $g$, and He abundance, $Y$. The Hydrogen/first Helium ionization instability strip is more sensitive to $g$ than $Y$. We find that (for solar composition envelopes), relative to a fiducial WD mass $0.6 M_\odot$, the blue edge for a $0.4 M_\odot$ He core WD shifts downward by $\approx 1000 {\rm K}$, while that for a massive $\approx 1.2 M_\odot$ WD shifts upward by $\approx 2000 {\rm K}$. The second Helium ionization instability strip exhibits strong dependences on both $g$ and $Y$. Surprisingly, increasing $Y$ by only 10% relative to solar creates an instability strip near $15,000 {\rm K}$. Hence CV's below the period gap with evolved donor stars of $Y\ga 0.4$ may have an "intermediate" instability strip well outside of the isolated DA and DB variables. This "intermediate" instability strip also occurs for low mass He WD with solar composition envelopes. The lack of pulsations in CV's with $T_{\rm eff}$ in the pure Hydrogen ZZ Ceti instability strip is also easily explained.