The helium white dwarf in two pulsars: too cool in PSR J0751+1807 and too hot in PSR J1012+5307?

We discuss the cooling history of the low-mass, helium-core white dwarfs in the short orbital period millisecond pulsars PSR J0751+1807 and PSR J1012+5307. The revised cooling age estimated in 1996 by Alberts et al. agrees with the age estimation for PSR J1012+5307, removing the discrepancy between the spin-down age and the cooling age. However, if we accept this model, then the helium white dwarf in the binary pulsar system PSR J0751+1807 must be much hotter than is observed. We propose that this discrepancy may be resolved if, after detachment of the secondary star from its Roche lobe in PSR J0751+1807, the star loses its hydrogen envelope as a result of pulsar irradiation. When hydrogen burning stops, the white dwarf will cool down much more quickly than in the case of a thick hydrogen envelope with a hydrogen-burning shell. We discuss several possibilities to explain the different cooling histories of the white dwarfs in the two systems.