$^1$S$_0$ Neutron Pairing vs Observations of Cooling Neutron Stars

As shown recently by Kaminker et al. (2001), current observations of thermal emission of isolated middle-aged neutron stars (NSs) can be explained by cooling of NSs of different masses with the cores composed of neutrons, protons and electrons, assuming rather strong superfluidity (SF) of protons, weak triplet-state SF of neutrons and neglecting singlet-state SF of neutrons. We show that this explanation remains correct in the presence of singlet-state SF of neutrons in the NS crust and outermost core but under stringent constraints on the density profile of the SF critical temperature $T_{\rm cns}(\rho)$. In order to explain observations of (young and hot) RX J0822--43 and (old and warm) PSR 1055--52 and RX J1856--3754 as cooling not too massive NSs, the maximum $T_{\rm cns}^{\rm max}$ should be rather high ($\ga 5 \times 10^9$ K) and the decrease of $T_{\rm cns}(\rho)$ outside the maximum should be sharp. These results place important constraints on the models of nucleon SF in dense matter.