The relation between white dwarf mass and orbital period in wide binary radio pulsars

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Abstract
We have re-examined a scenario for the evolution of a binary system, initially comprising a neutron star and a low-mass giant and ending as a wide binary containing a radio pulsar and a white dwarf in a nearly circular orbit. The evolution is driven by the nuclear evolution of the giant, which results in the stable transfer of much or all of the envelope of the giant to the neutron star. The angular momentum associated with the transferred mass may spin the neutron star up to high rotation rates, yielding a ‘recycled’ pulsar; the white dwarf, which had been the core of the giant progenitor, remains as a fossil relic of the giant. This scenario provides a unique test of the theory of advanced stages of stellar evolution, in that it predicts the existence of a testable relationship between observable quantities: the mass, Mwd, of the white dwarf and the orbital period, Porb, of the binary. The relationship arises because (1) stellar evolution theory predicts the existence of a rather tight relationship between the core mass, Mc, of a giant and the radius, Rg, of its envelope; and (2) in the scenario under consideration, the giant envelope is expected to fill its Roche lobe until the termination of mass transfer. The final orbital separation should thus be a well-defined function of Rg at the end of the mass-transfer phase (i.e. at the time when the envelope of the giant is exhausted), while Mwd will be essentially identical to the final value of Mc at the termination of mass transfer. Using refined stellar evolution calculations, we have redetermined the most likely value of Rg as a function of Mc for core masses in the range 0.15<Mc⊙, and we have devised analytic fitting formulae to our results for both the Rg–Mc relation and the concomitant Porb-Mwd relation. We have also, for the first time, obtained a quantitative estimate of the spread in the value of Rg at each value of Mc, resulting both from variations in the initial chemical composition and main-sequence mass of the giant and from the theoretical uncertainty in the value of the convective mixing-length parameter. We find that the maximum spread about the median value of Rg at any given value of Mc is a factor of ∼ 1.8, and that the corresponding maximum spread in Porb at fixed Mwd is a factor of ∼ 2.4 smaller spreads are obtained if one or more of the parameters (e.g. the initial composition of the giant) are assumed to be known. We have compared our results against the observational parameters of 23 radio pulsars in wide, nearly circular, binary orbits with low-mass white dwarf companions. We have also examined the applicability of our results (appropriately modified for non-negligible orbital eccentricity) to two conventional wide-binary systems (Sirius and Procyon) containing more massive white dwarf companions. We find overall good agreement between our theoretical results and the available observational data; however, any comparison between theory and observation is limited by the generally large uncertainties in the masses of the white dwarfs in the binary pulsar systems and by the paucity of known systems containing more massive white dwarfs.

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