$T_{\rm c}(\rho)$ of the proton critical temperature are shown in Fig. \[tcrho\]a; they are phenomenological and can be regarded as illustrative. In the outer core, proton superfluidity is strong and suppresses the modified Urca process. Accordingly, the slow neutrino emission of low-mass stars is determined by weaker neutrino bremsstrahlung in neutron-neutron collisions. This rises the surface temperature of low-mass stars and allows one to explain observations of hottest sources.

In addition, proton superfluidity penetrates into the inner core ($\rho > \rho_{\rm D}$) and suppresses the direct Urca process within the penetration depth. It broadens the direct Urca threshold and realizes a smooth transition from slowly to rapidly cooling neutron stars with the growth of $M$. Superfluidity 2p penetrates deeper than 1p which widens the range of masses of medium-mass stars. One can now explain observations of all the sources, including the coldest ones. Moreover, these scenarios predict the existence of very cold massive neutron stars (which have not been observed so far).

If the EOS and the critical temperature $T_{\rm c}(\rho)$ in the stellar core were known, one would be able to “weigh” neutron stars [@kyg02]. For instance, the mass of the Vela pulsar would be $1.47\, {\mbox{$M_\odot$}}$ for superfluidity 