2 Cooling theory

We confront the observational data with our simulations of NS cooling, using the recent cooling theory summarized in KYG and YGKP. For simplicity, we consider the models of NSs with the cores composed of neutrons, protons, and electrons (npe matter). We use the equation of state (EOS) in the NS core proposed by Prakash et al. (1988) (version I of the symmetry energy, with the compression modulus K=240 MeV of the saturated nuclear matter; it is denoted as EOS A in KYG and YGKP). The maximum NS mass for this EOS is $M_{\rm max}= 1.977~M_\odot$ (with the central density $\rho_{\rm c}^{\rm max}= 2.575 \times 10^{15}$ g cm^-3). The adopted EOS opens direct Urca process in the NSs with masses $M > M_{\rm D} =1.358~M_\odot$ and central densities above $\rho_{\rm D} =7.851 \times 10^{14}$ g cm^-3.

In our simulations, we take into account superfluidity of nucleons in the NS interiors. Superfluidity suppresses many neutrino emission processes (e.g., direct and modified Urca processes, nucleon-nucleon bremsstrahlung) but opens a specific powerful mechanism of neutrino emission due to the Cooper pairing of nucleons (proposed by Flowers et al. 1976; see, e.g., Yakovlev et al. 2001a for details), and also affects the heat capacity of matter. We include the singlet-state pairing of protons and the triplet-state pairing of neutrons in the NS cores but, for simplicity, we neglect the singlet-state pairing of neutrons in the NS crusts. The core superfluids are characterized by the density-dependent critical temperatures  $T_{\rm cp}(\rho)$and  $T_{\rm cnt}(\rho)$ (Fig. 3). We use one model of strong proton superfluidity (model 1p described, e.g., in KYG), and two models of triplet-state neutron superfluidity (model 2nt of weak superfluidity and model 3nt of moderately strong superfluidity). The critical temperatures  $T_{\rm c}(\rho)$ are parameterized by Eq. (1) in KYG. The parameters of model 1p are given in KYG; the parameters of models 2nt and 3nt are the same as for model 1nt in KYG, but the parameter T₀ is now equal to $2 \times 10^9$ K and $1.5 \times 10^{10}$ K, respectively. Our phenomenological superfluid models are consistent with the current microscopic models of nucleon superfluidity in NS cores (e.g., Lombardo & Schulze 2001).