Effect of BCS pairing on entrainment in neutron superfluid current in neutron star crust

The relative current density ni of “conduction” neutrons in a neutron star crust beyond the neutron drip threshold can be expected to be related to the corresponding particle momentum covector pi by a linear relation of the form ni=Kijpj in terms of a physically well-defined mobility tensor Kij. This result is describable as an “entrainment” whose effect-wherever the crust lattice is isotropic-will simply be to change the ordinary neutron mass m to a “macroscopic” effective mass m⋆ such that in terms of the relevant number density n of unconfined neutrons we shall have Kij=(n/m⋆)γij. In a preceding work based on a independent particle treatment beyond the Wigner-Seitz approximation, using Bloch type boundary conditions to obtain the distribution of energy Ek and associated group velocity vki=∂Ek/∂ℏki as a function of wave vector ki, it was shown that the mobility tensor would be proportional to a phase space volume integral Kij∝∫d3kvkivkjδ{Ek−μ}, where μ is the Fermi energy. Using the approach due to Bogoliubov, it is shown here that the effect of BCS pairing with a superfluid energy gap ΔF and corresponding quasiparticle energy function €k=(Ek−μ)2+ΔF2 will just be to replace the Dirac distributional integrand by the smoother distribution in the formula Kij∝∫d3kvkivkjΔF2/€k3. It is also shown how the pairing condensation gives rise to superfluidity in the technical sense of providing (meta) stability against resistive perturbations for a current that is not too strong (its momentum pi must be small enough to give 2|pivki|<€k2/|Ek−μ| for all modes). It is concluded that the prediction of a very large effective mass enhancement in the middle layers of the star crust will not be significantly effected by the pairing mechanism.