Nuclear impurities in the superfluid crust of neutron stars: Quantum calculation and observable effects on the cooling

We perform a quantum mechanical calculation of the neutron-specific heat in the Wigner-Seitz cells that characterize the inner crust of a neutron star, where a Coulomb lattice of neutron-rich nuclei is permeated by a sea of unbound superfluid neutrons. We then evaluate the observable thermal properties associated with such a microscopic structure and compare them to the standard values obtained in a uniform density model of the crust superfluidity with no nuclear impurities. We first argue that the bottleneck for the diffusion of heat is represented by the outermost layers of the inner crust. We then show that the crust-cooling times obtained for the nonuniform system are significantly different from those obtained in the standard uniform model. We find that the sign and magnitude of this difference depend sensitively on the local matter temperature and the pairing potential. We conclude that in order to give predictive power to observations of the thermal emission from neutron stars, it is crucial to include in any realistic cooling code the effect of nuclear impurities on the neutron superfluid.