The carbon-oxygen white dwarf (CO WD) + He star channel is one of the promising ways for producing type Ia supernovae (SNe Ia) with short delay times. Recent studies found that carbon under the He-shell can be ignited if the mass-accretion rate of CO WD is higher than a critical rate (about 2 * 10^-6 Msun/yr), triggering an inwardly propagating carbon flame. Previous studies usually supposed that the off-centre carbon flame would reach the centre, resulting in the formation of an oxygen-neon (ONe) WD that will collapse into a neutron star. However, the process of off-centre carbon burning is not well studied. This may result in some uncertainties on the final fates of CO WDs. By employing MESA, we simulated the long-term evolution of off-centre carbon burning in He-accreting CO WDs. We found that the inwardly propagating carbon flame transforms the CO WDs into OSi cores directly but not ONe cores owing to the high temperature of the burning front. We suggest that the final fates of the CO WDs may be OSi WDs under the conditions of off-centre carbon burning, or explode as iron-core-collapse SNe if the mass-accretion continues. We also found that the mass-fractions of silicon in the OSi cores are sensitive to the mass-accretion rates.
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