3 Theory and observations

Figures 1 and 2 compare the observational data with theoretical cooling curves, $T_{\rm s}^\infty(t)$.

Adopting model 1p of proton superfluidity and model 2nt of neutron superfluidity we obtain (Fig. 1) a family of cooling curves for NSs with different masses M. Actually, superfluidity 2nt is rather weak and has almost no effect on NS cooling. The properties of such cooling models are discussed in KYG and YGKP. One can distinguish NSs of three types:
(I) Low-mass NSs, $M \la M_{\rm I}$, are very slowly cooling NSs where modified or direct Urca processes are strongly suppressed by proton superfluidity; their cooling curves are almost independent of NS mass and EOS.
(II) Medium-mass NSs, $M_{\rm I} \la M \la M_{\rm II}$, undergo moderately fast cooling via direct Urca process partly reduced by proton superfluidity; their cooling is very sensitive to NS mass, EOS, and $T_{\rm cp}(\rho)$model.
(III) Massive NSs, $M \ga M_{\rm II}$, show fast cooling via direct Urca process in the NS centers almost unaffected by proton superfluidity. At $t \sim 10^5$ yr, for our NS models, we have $M_{\rm I} \sim 1.36~M_\odot$and $M_{\rm II} \sim 1.52~M_\odot$. These values are easily varied by choosing other EOSs and proton superfluid models (KGY, YGKP).

The situation would be drastically different if we adopted neutron superfluidity 3nt instead of 2nt. We would get a number of cooling curves plotted in Fig. 2. As long as a NS is hot and its internal temperature is larger than the maximum of $T_{\rm cnt}(\rho)$, the neutron superfluidity is absent and the star cools as shown in Fig. 1. However, the appearance of a moderately strong neutron superfluidity induces powerful neutrino emission due to the Cooper pairing of neutrons, which leads to a very fast cooling. In low-mass NSs ( $M \leq M_{\rm D}$), where direct Urca process is forbidden, this fast cooling has nothing to do with direct Urca process. As seen from Fig. 2, one can easily explain the upper limit of $T_{\rm s}^\infty$ for PSR J0205 by cooling of such a star. Moreover, by changing the maximum of $T_{\rm cn}(\rho)$, one can explain all relatively cool sources in Figs. 1 and 2 (including the coldest ones such as Vela and Geminga) by cooling of low-mass NSs with their own models of neutron superfluidity in the NS cores. In this way, it seems that the current observational data do not require direct Urca process (or similar processes in pion or kaon condensed matter, or in quark matter).

However, the main point is that NSs may have different masses, surface magnetic fields, etc., but they must have the same EOS and superfluid properties of their cores. Thus, all the sources should be explained by one set of models of $T_{\rm cn}(\rho)$ and $T_{\rm cp}(\rho)$. A natural explanation (KYG, YGKP) is to assume a weak neutron superfluidity in the NS cores (e.g., model 2nt, Fig. 1) and the presence of direct Urca process in massive NSs. If this is true, the two hottest sources for their ages, RX J0822 and PSR 1055, can be treated as low-mass NSs of type I, while 1E 1207, RX J002, Vela, PSR 0656, Geminga, and RX J1856 can be treated as medium-mass NSs of type II. This interpretation would be impossible without introducing the direct Urca process. Notice that the revised age of RX J1856 (Walter & Lattimer 2002) changes its status from a type I NS (e.g., KYG) to a type II NS. If PSR J0205 has the surface temperature just below the inferred upper limit, it belongs to the family of type II NSs and requires direct Urca process in its core. The appropriate cooling curve (e.g., the $M=1.42~M_\odot$ curve in Fig. 1) would lie above the cooling curves for Vela and Geminga which means that Vela and Geminga would be colder for their ages than PSR J0205. In other words, the well-known observational data on Vela and Geminga (e.g., Pavlov et al. 2001; Halpern & Wang 1997) provide stronger arguments in favor of direct Urca process than the newly reported data on PSR J0205. Let us recall that our interpretation enables one to measure the masses of type II NSs for a fixed EOS and superfluid properties of NS interiors (see KYG and YGKP). In the above scenario (Fig. 1), the mass of PSR J0205 would be lower than the masses of Vela and Geminga.