Eccentricities of millisecond pulsars with intermediate-mass progenitors

One channel to form millisecond pulsars with CO white dwarf companions is through the stable Roche lobe overflow of intermediate-mass ((Formula presented)) stars at the end of the main sequence (Case A) or the beginning of the hydrogen shell burning phase (Case B). We reproduce previous numerical calculations of this channel and supplement them with a simple analytical model that relates the final orbital period (Formula presented) to the white dwarf’s mass and to its progenitor’s initial mass M. We also theoretically calculate for the first time the eccentricity e in this process, which is set by the fluctuating gravitational quadrupole moment of the progenitor’s convective envelope during Roche lobe detachment. Intermediate-mass progenitors detach when their non-degenerate cores ignite helium, in contrast to low-mass ((Formula presented)) stars with degenerate cores that detach when their envelopes become too light to support a burning shell. Despite the order of magnitude higher envelope mass at detachment (Formula presented) in our case, the eccentricity is barely affected because (Formula presented), explaining why intermediate-mass ((Formula presented) CO white dwarfs have similar eccentricities to lower mass helium white dwarfs. Massive CO and ONe white dwarfs ((Formula presented), on the other hand, probably formed through a different channel of unstable Roche lobe overflow during helium shell burning (Case C), followed by common envelope inspiral. The measured eccentricities of these massive white dwarfs remain to be explained.