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Astron. Astrophys. 343, 650-660 (1999)

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1. Introduction

Neutron star interiors contain matter of nuclear and supranuclear density. Various microscopic theoretical models predict different compositions of this matter (neutrons, protons and electrons; hyperons; kaon or pion condensates; quarks), different equations of state (from very soft to very stiff) and superfluid properties of strongly interacting baryonic components (nucleons, hyperons, quarks).

Neutron stars are born very hot in supernova explosions but cool gradually in time. Their cooling depends on properties of stellar matter. Comparison of theoretical cooling models with observational data on the surface temperatures of isolated neutron stars yields a potentially powerful method to constrain models of superdense matter.

Young and middle-age ([FORMULA]-[FORMULA] yr) neutron stars cool mainly via neutrino emission from their interiors. It is important, thus, to know the main neutrino production mechanisms. For simplicity, we restrict ourselves by consideration of neutron stars whose cores are composed of neutrons (n), with some admixture of protons (p) and electrons (e). The neutrino emission mechanisms in the stellar cores may be divided into two groups, leading to standard or rapid cooling (e.g., Pethick 1992).

The standard cooling lowers the surface temperature to about [FORMULA] K in [FORMULA] yr. It goes mainly via the modified Urca process (e.g., Friman & Maxwell 1979, Yakovlev & Levenfish 1995)

[EQUATION]

and the nucleon-nucleon bremsstrahlung

[EQUATION]

where N is a nucleon (n or p).

Rapid cooling is strongly enhanced by the direct Urca reaction

[EQUATION]

which can operate (Lattimer et al. 1991) only in the central regions of rather massive neutron stars with not too soft equations of state. If the direct Urca reaction is allowed, the neutrino emissivity is typically 4-5 orders of magnitude higher than in the standard reactions (1) and (2), and the surface temperature decreases to several times of [FORMULA] K in [FORMULA] yr.

Cooling of neutron stars can be strongly affected by superfluidity of neutrons and protons in the stellar cores. The superfluidity is generally thought to be of BCS type produced under nuclear attraction of nucleons. At subnuclear densities [FORMULA] (where [FORMULA] g cm-3 is the standard nuclear-matter density) the neutron pairing occurs due to nn attraction in the 1S0 state. The superfluid gaps depend sensitively on nn interaction model. Various microscopic theories (e.g., Tamagaki 1970, Amundsen & Ostgaard 1985, Baldo et al. 1992, Takatsuka & Tamagaki 1993 , 1970) predict these gaps to vary in the range from some ten keV to some MeV. However, the singlet-state interaction of neutrons becomes repulsive at [FORMULA], and, therefore, the singlet-state neutron superfluidity vanishes near the boundary between the neutron star core and the crust. Deeper in the core ([FORMULA]), the triplet-state (3P2) nn interaction can be attractive to produce the superfluid with an anisotropic gap. Since the number density of protons is much smaller than that of neutrons, the singlet-state pp interaction is thought to be attractive in the stellar core leading to proton superfluidity. The superfluidity of n and p affects the main neutrino generation mechanisms (1) - (3) in the neutron star cores, and hence cooling of neutron stars with superfluid interiors. Superfluidity always suppresses these reactions decreasing the neutrino luminosity of neutron stars.

However, the appearance of neutron or proton superfluid in a cooling neutron star initiates an additional specific neutrino production mechanism associated with the direct interband transition of a nucleon,

[EQUATION]

The mechanism is allowed due to existence of a superfluid gap in the nucleon dispersion relation. The process may be called the neutrino emission due to Cooper pair formation . It was proposed and calculated for singlet-state neutron superfluidity by Flowers et al. (1976). It was rediscovered later by Voskresensky & Senatorov (1986 , 1987). Until recently, the process has been `forgotten', and has not been used in the studies of the neutron star cooling. It has been included into recent cooling simulations by Schaab et al. (1997), Page (1998), Yakovlev et al. (1998) and Levenfish et al. (1998) although its effect has not been analysed in details.

In Sect. 2 we present the derivation of the neutrino energy generation rate due to singlet and triplet Cooper pairing of nucleons. In the particular case of singlet-state nn pairing we reproduce the result by Flowers et al. (1976). The cases of triplet pairing of neutrons and singlet paring of protons appear to be different. In Sect. 3 we compare the Cooper-pair neutrino emission with other neutrino processes in the neutron star cores, and in Sect. 4 we analyse the effect of the Cooper-pair neutrinos on cooling of neutron stars with superfluid cores.

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© European Southern Observatory (ESO) 1999

Online publication: March 1, 1999
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