Non-equilibrium beta processes in superfluid neutron star cores

The beta relaxation time of the degenerate neutron star cores, composed of neutrons, protons and electrons, is evaluated taking into account the superfluidity of nucleons. We calculate the implied reduction factors for both direct and modified Urca reactions, with isotropic pairing of the protons or anisotropic pairing of neutrons. We find that due to the non-zero value of the temperature and/or to the vanishing of anisotropic gaps in some directions of the phase-space, superfluidity does not always completely inhibit beta relaxation, allowing for some reactions if the superfluid gap amplitude is not too large in respect to both the typical thermal energy and the chemical potential mismatch. However, we also observe that if the ratio between the critical temperature and the actual temperature is very small, a suprathermal regime is reached for which superfluidity is almost irrelevant. On the contrary, if the gap is large enough, the composition of the nuclear matter can stay frozen for very long durations, unless the departure from beta equilibrium is at least as important as the gap amplitude. These results are crucial for precise estimation of the superfluidity effect on the cooling/slowingdown of pulsars, which we will study elsewhere.