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 wit h helium cores in the mass range from 0.179 to 0.414 M-circle dot. The lower mass limit is well-suited for comparison with white dwarf compa nions of millisecond pulsars. The tracks are based on a 1 M-circle dot model sequence extending from the pre-main sequence stage up to the t ip of the red-giant branch. Applying large mass loss rates at appropri ate positions forced the models to move off the giant branch. The furt her evolution was then followed across the Hertzsprung-Russell diagram and down the cooling branch. At maximum effective temperature the env elope masses above the helium cores increase from 0.6 to 5.4 . 10(-3) M-circle dot for decreasing mass. We carefully checked for the occurre nce of thermal instabilities of the hydrogen shell by adjusting the co mputational time steps accordingly. Hydrogen flashes have been found t o take place only in the mass interval 0.21 less than or similar to M/ M-circle dot less than or similar to 0.3. The models show that hydroge n shell burning contributes significantly to the luminosity budget of white dwarfs with helium copes. For very low masses the hydrogen shell luminosity remains to be dominant even down to effective temperatures well below 10 000 K. Accordingly, the corresponding cooling ages are significantly larger than those gained from model calculations which n eglect 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 M-circle dot and a co oling age of 6 +/- 1 Gyr, in good agreement with the spin-down age, 7 Gyr, of the pulsar.