We present a detailed calculation of the evolution of low–mass ( < 0.25 ~ { } M _ { \odot } ) helium white dwarfs .
These white dwarfs ( the optical companions to binary millisecond pulsars ) are formed via long–term , low–mass binary evolution .
After detachment from the Roche lobe , the hot helium cores have a rather thick hydrogen layer with mass between 0.01 to 0.06 ~ { } M _ { \odot } .
Due to mixing between the core and outer envelope , the surface hydrogen content is 0.5 to 0.35 , depending on the initial value of the heavy element ( Z ) and the initial secondary mass .
We found that the majority of our computed models experience one or two hydrogen shell flashes .
We found that the mass of the helium dwarf in which the hydrogen shell flash occurs depends on the chemical composition .
The minimum helium white dwarf mass in which a hydrogen flash takes place is 0.213 ~ { } M _ { \odot } ( Z=0.003 ) , 0.198 ~ { } M _ { \odot } ( Z=0.01 ) , 0.192 ~ { } M _ { \odot } ( Z=0.02 ) or 0.183 ~ { } M _ { \odot } ( Z=0.03 ) .
The duration of the flashes ( independent of chemical composition ) is between few \times 10 ^ { 6 } years to few \times 10 ^ { 7 } years .
In several flashes the white dwarf radius will increase so much that it forces the model to fill its Roche lobe again .
Our calculations show that cooling history of the helium white dwarf depends dramatically on the thickness of the hydrogen layer .
We show that the transition from a cooling white dwarf with a temporary stable hydrogen–burning shell to a cooling white dwarf in which almost all residual hydrogen is lost in a few thermal flashes ( via Roche–lobe overflow ) occurs between 0.183–0.213 ~ { } M _ { \odot } ( depending on the heavy element value ) .