For the sources 13 and 16, $T_{\rm s}^\infty$ is determined [@hoetal07; @motchetal03] using models of hydrogen atmospheres of finite depth (mHA$^*$ and HA$^*$, respectively). One important source, 1E 1207.4–5209, is not included in the table because of the problems to interpret its spectrum (although modeling of such spectra is progressing [@mori_ho]).

Basic cooling curves
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For illustration, we will mainly use models of neutron stars whose cores contain neutrons, protons and electrons and have a stiff phenomenological EOS proposed in Ref. [@pal88] (model I for the symmetry energy and the bulk energy model that gives the compression modulus of saturated nuclear matter $K=240$ MeV). The most massive stable neutron star, for this EOS, has the gravitational mass $M_{\rm max}=1.977\,{\mbox{$M_\odot$}}$ and the central density $\rho_{\rm c}=2.578 \times 10^{15}$ g cm$^{-3}$; the direct Urca process is allowed at $\rho \geq \rho_{\rm D}=7.851 \times 10^{14}$ g cm$^{-3}$ (in stars with $M \geq M_{\rm D}=1.358\,{\mbox{$M_\odot$}}$).

![ (a) Superfluid transition temperature versus density for two models (1p and 2p) of proton superfluidity in the neutron star core. (b) Neutrino emissivity profiles in the core at $T=3\times10^8$ K for nonsuperfluid matter (noSF) and for matter with superfluid protons (models 1p or 2p).[]{data-label="tcrho"}](yakfig1.ps){width="7.8cm"}

![Observational limits on surface 