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Entrainment in Superfluid Neutron-Star Crusts: Hydrodynamic Description and Microscopic Origin

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Abstract

In spite of the absence of viscous drag, the neutron superfluid permeating the inner crust of a neutron star cannot flow freely and is entrained by the nuclear lattice similarly to laboratory superfluid atomic gases in optical lattices. The role of entrainment on the neutron superfluid dynamics is reviewed. For this purpose, a minimal hydrodynamical model of superfluidity in neutron-star crusts is presented. This model relies on a fully 4-dimensionally covariant action principle. The equivalence of this formulation with the more traditional approach is demonstrated. In addition, the different treatments of entrainment in terms of dynamical effective masses or superfluid density are clarified. The nuclear energy density functional theory employed for the calculations of all the necessary microscopic inputs is also reviewed, focusing on superfluid properties. In particular, the microscopic origin of entrainment and the different methods to estimate its importance are discussed.

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Notes

  1. Since gravity is neglected here, the total momentum covectors \(\pi ^{_{\mathrm{X}}}_{\, \nu }\) reduce to the material momentum covectors \(\mu ^{_{\mathrm{X}}}_{\, \nu }\), as can be seen from Eq. (152) of Ref. [28] after setting the Newtonian gravitational potential \(\phi =0\).

  2. Because of the local electric charge neutrality condition \(n_{ p}=n_{ e}\), where \(n_{ e}\) is the electron number density, the electron chemical potential is included in \(\mu ^p\).

  3. Let us recall that the chemical potential \(\mu ^{p}\) includes the contribution of electrons.

  4. These equations are also called Bogoliubov–de Gennes equations in condensed matter physics.

  5. The pairing contributions to \(h_q\) are typically very small, and therefore often neglected.

  6. In principle, one should also solve the density functional theory equations for electrons. But in the extreme environment of neutron stars it is usually a very good approximation to treat electrons as an ideal relativistic Fermi gas.

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Acknowledgements

This work was supported by the Fonds de la Recherche Scientifique—FNRS (Belgium) under Grant No. CDR J.0187.16, and the European Cooperation in Science and Technology (COST) action MP1304 NewCompStar. This work was completed at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611.

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Chamel, N. Entrainment in Superfluid Neutron-Star Crusts: Hydrodynamic Description and Microscopic Origin. J Low Temp Phys 189, 328–360 (2017). https://doi.org/10.1007/s10909-017-1815-x

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