We have calculated the mass accumulation efficiency during helium shell flashes to examine whether or not a carbon-oxygen white dwarf ( C+O WD ) grows up to the Chandrasekhar mass limit to ignite a Type Ia supernova ( SN Ia ) explosion .
It has been frequently argued that luminous super-soft X-ray sources ( SSSs ) and symbiotic stars are progenitors of SNe Ia .
In such systems , a C+O WD accretes hydrogen-rich matter from a companion and burns hydrogen steadily on its surface .
The WD develops a helium layer underneath the hydrogen-rich envelope and undergoes periodic helium shell flashes .
Using OPAL opacity , we have reanalyzed a full cycle of helium shell flashes on a 1.3 M _ { \odot } C+O WD and confirmed that the helium envelope of the WD expands to blow a strong wind .
A part of the accumulated matter is lost by the wind .

The mass accumulation efficiency in helium shell flashes is estimated as \eta _ { He } = -0.175 \left ( \log \dot { M } +5.35 \right ) ^ { 2 } +1.05 , for -7.3 < \log \dot { M } < -5.9 , and \eta _ { He } = 1 for -5.9 \leq \log \dot { M } < \kern - 12.0 pt \lower 4.73 pt \hbox { $ \sim$ } ~ { } ~ { } -5 , where the mass accretion rate \dot { M } is in units of M _ { \odot } ~ { } { yr } ^ { -1 } .
In relatively high mass accretion rates as expected in recent SN Ia progenitor models , the mass accumulation efficiency is large enough for C+O WDs to grow to the Chandrasekhar mass , i.e. , \eta _ { He } = 0.9 for \log \dot { M } = -6.3 , and \eta _ { He } = 0.57 for \log \dot { M } = -7.0 .
The wind velocity ( \sim 1000 km s ^ { -1 } ) is much faster than the orbital velocity of the binary ( < \kern - 12.0 pt \lower 4.73 pt \hbox { $ \sim$ } ~ { } ~ { } 300 km s ^ { -1 } ) and therefore , the wind can not be accelerated further by the companion ’ s motion .
We suggest observational counterparts of helium shell flashes in relation to long term variations in super-soft X-ray fluxes of SSSs and symbiotic stars .