Carbon Shell or Core Ignitions in White Dwarfs Accreting from Helium Stars
White dwarfs accreting from helium stars can stably burn at the accreted rate and avoid the challenge of mass loss associated with unstable Helium burning that is a concern for many Type Ia supernovae scenarios. We study binaries with helium stars of mass $1.25 M_\odot\le M_{\rm{He}} \le 1.8 M_\odot$, which have lost their hydrogen rich envelopes in an earlier common envelope event and now orbit with periods ($P_{\rm orb}$) of several hours with non-rotating $0.84$ and $1.0 M_\odot$ C/O WDs. The helium stars fill their Roche lobes (RLs) after exhaustion of central helium and donate helium on their thermal timescales (${\sim}10^5$yr). As shown by others, these mass transfer rates coincide with the steady helium burning range for WDs, and grow the WD core up to near the Chandrasekhar mass ($M_{\rm Ch}$) and a core carbon ignition. We show here, however, that many of these scenarios lead to an ignition of hot carbon ashes near the outer edge of the WD and an inward going carbon flame that does not cause an explosive outcome. For $P_{\rm orb} = 3$ hours, $1.0 M_\odot$ C/O WDs with donor masses $M_{\rm He}\gtrsim1.8 M_\odot$ experience a shell carbon ignition, while $M_{\rm He}\lesssim1.3 M_\odot$ will fall below the steady helium burning range and undergo helium flashes before reaching core C ignition. Those with $1.3 M_\odot \lesssim M_{\rm He} \lesssim 1.7 M_\odot$ will experience a core C ignition. We also calculate the retention fraction of accreted helium when the accretion rate leads to recurrent weak helium flashes.