We explore the outcome of mass transfer via Roche lobe overflow ( RLOF ) of M _ { He } \lesssim 0.51 M _ { \odot } pure helium burning stars in close binaries with white dwarfs ( WDs ) .
The evolution is driven by the loss of angular momentum through gravitational wave radiation ( GWR ) , and both stars are modeled using Modules for Experiments in Stellar Astrophysics ( MESA ) .
The donors have masses of M _ { He } = 0.35 , 0.4 , \& 0.51 M _ { \odot } and accrete onto WDs of mass M _ { WD } from 0.6 M _ { \odot } to 1.26 M _ { \odot } .
The initial orbital periods ( P _ { orb } ) span 20 to 80 minutes .
For all cases , the accretion rate onto the WD is below the stable helium burning range , leading to accumulation of helium followed by unstable ignition .
The mass of the convective core in the donors is small enough so that the WD accretes enough helium-rich matter to undergo a thermonuclear runaway in the helium shell before any carbon-oxygen enriched matter is transferred .
The mass of the accumulated helium shell depends on M _ { WD } and the accretion rate .
We show that for M _ { He } \gtrsim 0.4 M _ { \odot } and M _ { WD } \gtrsim 0.8 M _ { \odot } , the first flash is likely vigorous enough to trigger a detonation in the helium layer .
These thermonuclear runaways may be observed as either faint and fast .Ia SNe , or , if the carbon in the core is also detonated , Type Ia SNe .
Those that survive the first flash and eject mass will have a temporary increase in orbital separation , but GWR drives the donor back into contact , resuming mass transfer and triggering several subsequent weaker flashes .