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We present a detailed calculation of the evolution of low-mass (,0.25 M ( ) 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 and 0.06 M ( . As a result of mixing between the core and outer envelope, the surface hydrogen content (X surf ) is 0.5±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.The duration of the flashes (independent of chemical composition) is between a few Â10 6 and a few Â10 7 yr. 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 the 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 temporarily 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 and 0.213 M ( (depending on the heavy element value). Spruit 1987;Muslimov & Sarna 1995) and, as a consequence, the system loses orbital angular momentum. For a magnetic stellar wind we used the formula for the orbital angular momentum loss _ J J MSW 23 Â 10 27 M 2 R 2 2

We present evolutionary sequences for low‐mass close binary systems in which a low‐mass (1.0–1.5 M⊙) secondary star transfers mass to a neutron star. Roche lobe filling occurs when the secondary is a turn‐off main‐sequence star (having a small helium core). We assume loss of angular momentum owing to gravitational wave radiation and magnetic braking. We have found that the loss (and the mechanism of loss) of mass and angular momentum from the system is the main factor determining the value of the bifurcation period (Pbif). The bifurcation period separates the formation of the converging systems from the diverging systems. Variations in the initial chemical composition, and in the initial mass of the secondary, lead only to minor changes in Pbif. We have also investigated how changes in the chemical composition influence the initial orbital period (Pi) versus final orbital period (Pf) relation. The initial chemical composition has a more significant effect on this relation for shorter Pi than for longer Pi. We have found systematic differences for the Pf versus white dwarf mass relation for various chemical compositions. For converging systems, we have found that there is a boundary orbital period (Pb) such that if Pi < Pb, a system will evolve through the period gap (there are no low‐mass X‐ray binaries with orbital periods between one and three hours) with a Roche lobe overflowing secondary, but the accretion on to the neutron star is forbidden owing to the ‘propeller effect’. The systems will end their evolution as ultra‐short period and very bright, low‐mass X‐ray binaries. If Pb < Pi < Pbif, then short orbital period millisecond binary pulsar systems will be formed.

Abstract.We investigate the evolution of a binary system which initially has an orbital period of 2 h 17 m and contains a 0.5 M helium star with a white dwarf companion of 0.97 M , similar to the suggested SN Ia candidate progenitor KPD 1930+2752. We show that the helium star completes core helium burning and becomes a white dwarf before the components merge. The most probable outcome of the merger of the components is the formation of a massive white dwarf, despite the fact that initially the total mass of the system is above the Chandrasekhar mass.