The evolution of helium white dwarfs: I. The companion of the millisecond pulsar PSR J1012+5307

We present a grid of evolutionary tracks for low-mass white dwarfs with helium cores in the mass range from 0.179 to 0.414 Msol. The lower mass limit is well-suited for comparison with white dwarf companions of millisecond pulsars. The tracks are based on a 1 Msol model sequence extending from the pre-main sequence stage up to the tip of the red-giant branch. Applying large mass loss rates at appropriate positions forced the models to move off the giant branch. The further evolution was then followed across the Hertzsprung-Russell diagram and down the cooling branch. At maximum effective temperature the envelope masses above the helium cores increase from 0.6 to 5.4 x 10^{-3} Msol for decreasing mass. We carefully checked for the occurrence of thermal instabilities of the hydrogen shell by adjusting the computational time steps accordingly. Hydrogen flashes have been found to take place only in the mass interval 0.21 < M/Msol < 0.3. The models show that hydrogen shell burning contributes significantly to the luminosity budget of white dwarfs with helium cores. For very low masses the hydrogen shell luminosity remains to be dominant even down to effective temperatures well below 10000K. Accordingly, the corresponding cooling ages are significantly larger than those gained from model calculations which neglect nuclear burning or the white dwarf progenitor evolution. Using the atmospheric parameters of the white dwarf in the PSR J1012+5307 system we determined a mass of M=0.19 +/- 0.02 Msol and a cooling age of 6 +/- 1 Gyr, in good agreement with the spin-down age, 7 Gyr, of the pulsar.