Various baryons in neutron star matter can be in superfluid state produced by Cooper pairing of baryons due to an attractive component of baryon-baryon interaction (as already mentioned in § 1.2). Superfluidity of a particular baryon species is switched on when the temperature T falls below some critical temperature Tc. Superfluidity is a Fermi-surface phenomenon; it has almost no effect on the EOS, neutron stars masses and radii.

The theory predicts superfluidity of free neutrons and of nucleons in atomic nuclei in the inner neutron star crust. Neutrons, protons and other baryons in the stellar core can also be superfluid. Superfluidity of charged particles (for instance, protons) means superconductivity.

As outlined in § 1.2, free neutrons in the stellar crust undergo singlet-state (:So) pairing which disappears in the core (Wolf 1966). However, neutrons in the core can be superfluid due to a weaker triplet-state (3P2) pairing. The idea of such pairing in neutron star cores, first estimates and simplified calculations of the superfluid gap have been published by a number of authors (particularly, by Ruderman 1967, Maekawa & Tamagaki 1968, Tamagaki 1969, Hoffberg et al. 1970), mostly in conference proceedings. Usually these results are solely attributed to the first journal publication (Hoffberg et al., 1970). The foundation of the strict relevant theory was laid by Tamagaki (1970). Proton superfluidity in the core is thought to be mainly produced by singlet-state proton pairing. One has also invented superfluidity of hyperons (Balberg & Barnea, 1998) and quarks (Bailin & Love, 1984). Pion and kaon condensates affect superfluidity of nucleons (Takatsuka & Tamagaki, 1995, 1997a,b).

Critical temperatures Tc of various particle species have been calculated by many authors as reviewed by Lombardo & Schulze (2001) (more references can be found in Yakovlev et al. 1999). The results are extremely sensitive to strong interaction models and many-body theories employed. In all the cases mentioned above microscopic calculations give density-dependent critical temperatures Tc < 1010 K and lower. As a rule, superfluidities weaken and disappear at essentially supranuclear densities, where the attractive part of strong interaction becomes inefficient. For example, in the left panel of Fig. 1.4 we present Tc(p) in a neutron star core composed of npe matter with a moderately stiff EOS of Prakash et al. (1988) (after Yakovlev et al. 2002). We plot four purely phenomenological models: models 1p and 2p for single-state proton pairing and models 2nt and 3nt for triplet-state neutron pairing. The curves in the right panel are explained in § 1.3.5.

In addition, Alford et al. (1998) proposed a new type of quark superfluidity associated with color superconductivity (§ 8.8.3). For a typical Fermi energy of quarks ~ 500 MeV, one may expect Tc ~ 50 MeV ~ 5 x 1011 K.

Superfluidity affects the heat capacity and neutrino emission of neutron stars. It induces also a number of macroscopic quantum phenomena. For instance, consider the core of a rotating neutron star composed of neutrons, protons and electrons. A rotation of neutron superfluid is realized in the form of quantized (Feynman-Onsager) vortices parallel to the spin axis (Ginzburg & Kirzhnits 1964, Baym et al. 1969). The total amount of vortices in the star is estimated as ~ 2 x 1016/P, where P is the stellar spin period in seconds. The vortex motion of neutron superfluid, averaged over small macroscopic fluid elements, reproduces a solid-body rotation. The vortices occur also in the inner crust, where free neutrons are superfluid. A pulsar spindown induces the outward drift of vortices and their disappearance at the boundary of the superfluid region.

Below the critical temperature, superconductivity of protons (and other charged baryons) in the neutron-star core is described by the Ginzburg-Landau theory. The proton coherence length (2-6 fm) is typically much smaller than the London screening length (100-300 fm) of electric currents in superconducting medium. This probably means type II superconductivity. If an initially normal hot core contained a quasi-uniform magnetic field B, superconductivity splits the field into fluxoids (Abrikosov vortices), which are thin quantized magnetic flux tubes parallel to the initial field. The total number of fluxoids is ~ 1031 (B/1012 G).

Neutron vortices may pin to atomic nuclei or lattice defects in the crust and to fluxoids in the core. The pinning may be accompanied by vortex creep. These and related phenomena are invoked, for instance, to explain observations of pulsar glitches and to study the evolution of internal magnetic fields (§ 1.4.4).

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