Nucleon superfluidity vs. observations of cooling neutron stars

Cooling simulations of neutron stars (NSs) are performed assuming that stellar cores consist of neutrons, protons and electrons and using realistic density profiles of superfluid critical temperatures $T_{\rm cn}(\rho)$ and $T_{\rm cp}(\rho)$ of neutrons and protons. Taking a suitable profile of $T_{\rm cp}(\rho)$ with maximum ~$5 \times 10^9$ K one can obtain smooth transition from slow to rapid cooling with increasing stellar mass. Adopting the same profile one can explain the majority of observations of thermal emission from isolated middle-aged NSs by cooling of NSs with different masses either with no neutron superfluidity in the cores or with a weak superfluidity, $T_{\rm cn} < 10^8$ K. The required masses range from ~$1.2 {M}_\odot$ for (young and hot) RX J0822-43 and (old and warm) PSR 1055-52 and RX J1856-3754 to ≈ $1.45 {M}_\odot$ for the (rather cold) Geminga and Vela pulsars. Observations constrain the $T_{\rm cn}(\rho)$ and $T_{\rm cp}(\rho)$ profiles with respect to the threshold density of direct Urca process and maximum central density of NSs