We perform binary evolution calculations on helium star - carbon-oxygen white dwarf (CO WD) binaries using the stellar evolution code MESA. This single degenerate channel may contribute significantly to thermonuclear supernovae at short delay times. We examine the thermal-timescale mass transfer from a 1.1 - 2.0 $M_{odot}$ helium star to a 0.90 - 1.05 $M_{odot}$ CO WD for initial orbital periods in the range 0.05 - 1 day. Systems in this range may produce a thermonuclear supernova, helium novae, a helium star - oxygen-neon WD binary, or a detached double CO WD binary. Our time-dependent calculations that resolve the stellar structures of both binary components allow accurate distinction between the eventual formation of a thermonuclear supernova (via central ignition of carbon burning) and that of an ONe WD (in the case of off-center ignition). Furthermore, we investigate the effect of a slow WD wind which implies a specific angular momentum loss from the binary that is larger than typically assumed. We find that this does not significantly alter the region of parameter space over which systems evolve toward thermonuclear supernovae. Our determination of the correspondence between initial binary parameters and the final outcome informs population synthesis studies of the contribution of the helium donor channel to thermonuclear supernovae. In addition, we constrain the orbital properties and observable stellar properties of the progenitor binaries of thermonuclear supernovae and helium novae.
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