Allard Valentin

Neutrons stars are leftovers of supernovae. Predicted decades before their discovery by Bell and Hewish, their masses are of the order of the Sun for a radius of the order of a city like Paris classifying them, with their densities exceeding that of the atomic nuclei (~10^17 kg/m³), as the most compact objects in the Universe (behind black holes). Such compact stars are supposed to contain nuclear superfluids (whose existence was predicted years before the discovery of neutron stars) in their interiors. In particular, the outer core of a neutron star is thought to contain a neutron-proton superfluid mixture. Because a superfluid can flow without resistance and carries no heat, the dynamics of a neutron star is a multifluid dynamics : it must be described by three distinct components, at least ; a neutron superfluid, a proton superconductor, and a normal fluid.

Despite the absence of a viscous drag, neutrons and protons cannot flow independently and are, due to strong nuclear interactions, mutually entrained : the mass current of one nucleon species is thus given as a combination of the superfluid velocities of both species and of the normal velocity Vex of thermal excitations. Such effect is also encountered, with the name "Andreev-Bashkin effect" or "Entrainment effects", in laboratory superfluid mixtures (like 4He-3He mixtures) and it is described using an entrainment matrix whose elements, the entrainment coefficients, characterize the coupling between the various (superfluid) species constituting the mixture. Andreev-Bashkin effects are supposed to play a role in the global dynamics of rotating neutron stars : as known in terrestrial experiments, a rotating superfluid is pierced by vortices so rotating neutron stars are supposed to contain neutron vortices which will, due to entrainment, induce proton circulation and, consequently, a magnetic field with which electrons and muons will interact giving birth to a frictional coupling between the neutron superfluid and the normal charged particles.

Entrainment matrix can be studied microscopically using the tools of condensed matter, in particular, the energy-density functional which postulates the energy of the neutron-proton system can be written as a functional of density matrices characterizing protons and neutrons. Minimizing this energy at fixed temperature and nucleon number will give the equations of evolution for the density matrices called the time-dependent Hartree-Fock Bogoliubov (TDHFB) equations which are highly non-linear equations due to their self-consistency : the solution define the equations of evolution which, in their turn, give the solution, and so on (see the consistency loop). Working with TDHFB equations, one can find a continuity equation giving the expression of the mass currents. To find the entrainment matrix, one needs to chose the energy-density functional : if the functional was known, one could have solved exactly the N-body problem but, in practice, the exact form of the functional is unknown so we use phenomenological functionals which are functionals whose parameters are fitted to nuclear experiments, astrophysical observations and N-body computations.

Specifying the functional, one can find the form of the entrainment matrix for every temperature T giving consistent entrainment coefficients : for temperatures T above the critical proton (resp. neutron) temperature (temperature above which the proton (resp. neutron) superfluid is destroyed) the proton (resp. neutron) superfluid do not contribute to the mixture anymore so the associated entrainment coefficients rho_np = rho_pn and rho_pp (resp rho_nn) vanish. The results are also consistent with previous theories using low temperature approximation (Fermi-Liquid theory) and can be used to investigate the induced neutron vortex magnetic field or the NS oscillations. These results can also be used to improve the nuclear energy-density functionals and giving a better understanding of matter at huge densities.