Born in the furnace of gravitational core-collapse supernova explosions, a newly formed proto-neutron star is extremely hot, with internal temperatures of the order of a few thousand billion degrees . About one minute later, the proto-neutron star becomes transparent to neutrinos that are copiously produced in its interior, and thus rapidly cools down. After several decades, the interior of the star reaches a thermal equilibrium with temperatures of about a hundred million degrees (except for a thin outer heat-blanketing envelope). The last cooling stage takes place after about a hundred thousand years, when heat from the interior diffuses to the surface and is dissipated in the form of radiation. The interior of a mature neutron star is expected to become cold enough for the appearance of superfluids and superconductors, which are frictionless quantum liquids respectively electrically neutral and charged .
Superfluidity and superconductivity are among the most striking macroscopic manifestations of quantum mechanics. Neutrons and protons are fermions, and due to the Pauli Exclusion Principle, they generally tend to avoid themselves. This individualistic behaviour, together with the strong repulsive nucleon-nucleon interaction at short distance, provide the necessary pressure to counterbalance the huge gravitational pull in a neutron star, thereby preventing it from collapsing. However at low enough temperatures, nucleons may form pairs . These pairs are bosons that can behave coherently on a very large scale and the nucleon condensate can flow without any viscosity, analogous to superfluid helium-3. While helium-3 becomes a superfluid only below 1 mK, nuclear superfluidity is sustainable even at a temperature of a billion degrees in a neutron star due to the enormous pressure involved. The interior of a neutron star is expected to contain at least three different kinds of superfluids and superconductors  (see right panel of figure 1): (i) an inhomogeneous neutron superfluid permeating the inner region of the crust, (ii) a homogeneous neutron superfluid in the outer core, and (iii) a homogeneous proton superconductor in the outer core.
The oddity of such quantum liquids is that they are characterised by the coexistence of two different dynamical components: a superfluid that can flow without resistance, and a normal viscous fluid. Despite the absence of viscous drag, superfluids can still be mutually coupled by nondissipative entrainment, whereby momentum and velocity are not aligned. These effects arise from the strong interactions between nucleons. Superfluids can also interact with nonsuperfluid constituents via peculiar friction forces. The most unusual feature of superfluids and superconductors is the quantisation of the momentum circulation in the form of vortex lines and magnetic flux tubes. Such defects have been observed in various laboratory superfluids and superconductors, as illustrated in figure 2. For all these reasons, the hydrodynamics of superfluid-superconducting mixtures can be very different from that of ordinary liquids [5, 6].
Various astrophysical phenomena are believed to be intimately related to the presence of superfluids and superconductors in neutron stars.
Pulsars are neutron stars spinning extremely rapidly with periods ranging from milliseconds to seconds and delays of a few milliseconds per year at most, thus providing the most accurate clocks in the Universe. Nevertheless, irregularities have been detected in long-term pulsar timing observations as shown in figure 2. In particular, some pulsars like the emblematic Vela pulsar exhibit sudden increases in their rotational frequency and spin-down rate, which are followed by a relaxation over days to years [7, 8]. Since these phenomena have not been observed in any other celestial bodies, they have to deal with specific properties of neutron stars, most presumably superfluidity and superconductivity (see, e.g. Section 12.4 of , see also ). The idea is the following. A rotating superfluid is threaded by a regular array of quantised vortex lines, each carrying a quantum of angular momentum. Similarly, neutron stars are expected to contain a huge number of vortices, of order 1017 for a pulsar like Vela. The superfluid is supposed to be weakly coupled to the crust by mutual friction forces and to thus follow its spin-down via the motion of vortices away from the rotation axis unless vortices are pinned to the crust. In this case, the superfluid would rotate more rapidly than the crust. The lag between the superfluid and the crust would induce a Magnus force acting on the vortices thereby producing a crustal stress. At some point, the vortices would suddenly unpin, the superfluid would spin down and, by the conservation of angular momentum the crust would spin up leading to a glitch [11, 12]. However, this vortex-mediated glitch theory has been recently challenged by the realisation that the neutron superfluid in the crust does not contain enough angular momentum due to the previously ignored effects of Bragg scattering [13, 14].
Thermal relaxation of transiently accreting neutron stars during quiescence
As depicted in figure 3, a neutron star accretes matter from a companion star during several years or decades, driving the neutron-star crust out of its thermal equilibrium with the core. After the accretion stops, the heated crust relaxes towards equilibrium (see, e.g., Section 12.7 of , see also ). The thermal relaxation has been already monitored in a few systems (see, e.g.,  and references therein). The thermal relaxation time depends on the properties of the crust, especially the heat capacity. If neutrons were not superfluid, they could store so much heat that the thermal relaxation would last longer than what is observed. On the other hand, the thermal relaxation of these systems is not completely understood. For instance, additional heat sources of unknown origin are needed in order to reproduce the observations [17, 18]. These discrepancies may also originate from a lack of understanding of superfluid properties .
Rapid cooling of Cassiopeia A
Cassiopeia A is the remnant of a star that exploded 330 years ago at a distance of about 11000 light years from us. It owes its name to its location in the constellation Cassiopeia. The neutron star is not only the youngest known, thermally emitting, isolated neutron star in our Galaxy, but it is also the first isolated neutron star for which the cooling has been directly observed (see left panel of figure 1). Ten-year monitoring of this object has revealed that its temperature has dropped by a few percent since its discovery in 1999 . This cooling rate is significantly faster than that expected from nonsuperfluid neutron-star cooling theories. Indeed, the onset of neutron superfluidity opens a new channel for neutrino emission from the continuous breaking and formation of neutron pairs. This process, which is most effective for temperatures slightly below the critical temperature of the superfluid transition, enhances the cooling of the star during several decades. As a consequence, observations of Cassiopeia A put stringent constraints on the critical temperatures of the neutron superfluid and proton superconductor in neutron-star cores [20, 21]. However, this interpretation has been recently questioned and alternative scenarios have been proposed (most of which still requiring superfluidity and/or superconductivity in neutron stars) [22-25].
Rotational evolution of pulsars and quasi-periodic oscillations in soft gamma-ray repeaters
Superfluidity and superconductivity may also leave their imprint on other observed neutron-star phenomena such as long-term free precession , pulsar braking indices [27, 28], or the quasi-periodic oscillations (QPOs) in the giant flares from the soft gamma-ray repeaters SGR 1806-20, SGR 1900+14, and SGR 0526-66 . However, these phenomena warrant further studies.
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Author: Nicolas Chamel (Institute of Astronomy and Astrophysics, Université Libre de Bruxelles, Belgium)