Unstable Helium Shell Burning on Accreting White Dwarfs
  AM Canum Venaticorum (AM CVn) binaries consist of a degenerate helium donor
and a helium, C/O, or O/Ne WD accretor, with accretion rates of Mdot = 1e-13 -
1e-5 Msol/yr. For accretion rates < 1e-6 Msol/yr, the accreted helium ignites
unstably, resulting in a helium flash. As the donor mass and Mdot decrease, the
ignition mass increases and eventually becomes larger than the donor mass,
yielding a "last-flash" ignition mass of < 0.1 Msol. Bildsten et al. (2007)
predicted that the largest outbursts of these systems will lead to dynamical
burning and thermonuclear supernovae. In this paper, we study the evolution of
the He-burning shells in more detail. We calculate maximum achievable
temperatures as well as the minimum envelope masses that achieve dynamical
burning conditions, finding that AM CVn systems with accretors > 0.8 Msol will
undergo dynamical burning. Triple-alpha reactions during the hydrostatic
evolution set a lower limit to the 12C mass fraction of 0.001 - 0.05 when
dynamical burning occurs, but core dredge-up may yield 12C, 16O, and/or 20Ne
mass fractions of ~ 0.1. Accreted 14N will likely remain 14N during the
accretion and convective phases, but regardless of 14N's fate, the
neutron-to-proton ratio at the beginning of convection is fixed until the onset
of dynamical burning. During explosive burning, the 14N will undergo
14N(a,g)18F(a,p)21Ne, liberating a proton for the subsequent
12C(p,g)13N(a,p)16O reaction, which bypasses the relatively slow alpha-capture
onto 12C. Future hydrodynamic simulations must include these isotopes, as the
additional reactions will reduce the Zel'dovich-von Neumann-Doring (ZND)
length, making the propagation of the detonation wave more likely.
