slow and fast ones.

Accordingly, the density profile of the neutrino emissivity in a neutron star core can look like that plotted in Fig. \[concept\]a; $\rho_1$ and $\rho_2$ mark the density range, where the slow cooling transforms into the fast one. This $Q_\nu(\rho)$ profile translates into a characteristic dependence of the neutrino luminosity $L_\nu$ on the stellar mass shown in Fig. \[concept\]b; the transition between the slow cooling to the fast one takes place in the mass range from $M_1$ to $M_2$.

Thus, at the present stage of investigation, the cooling theory *has potential to test four main parameters* of the neutrino emissivity function $Q_\nu(\rho)$ or the luminosity function $L_\nu(M)$. They specify the lower and upper levels of neutrino emission (either $Q_\mathrm{slow}$ and $Q_\mathrm{fast}$ or $L_\mathrm{slow}$ and $L_\mathrm{fast}$) and the position of the transition zone between the slow and fast emission (either $\rho_1$ and $\rho_2$ or $M_1$ and $M_2$).

Examples of physical models
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Two examples [@kyg02] of successful realization of the above scheme are given in Figs. \[cool4\]c and d. The figures show the cooling curves for neutron stars of $M=(1.1-1.97)\, {\mbox{$M_\odot$}}$ with strong proton superfluidity (1p or 2p) in the core (but with normal neutrons). The density profiles 