Cooling Theory Faced with Old Warm Neutron Stars: Role of Non-Equilibrium Processes with Proton and Neutron Gaps

Recent observations have found several candidates for old warm neutron stars whose surface temperatures are above the prediction of the standard neutron star cooling scenario, and thus require some heating mechanism. Motivated by these observations, we study the non-equilibrium beta process in the minimal cooling scenario of neutron stars, which inevitably occurs in pulsars. This out-of-equilibrium process yields the late time heating in the core of a neutron star, called the rotochemical heating, and significantly changes the time evolution of the neutron star surface temperature. To perform a realistic analysis of this heating effect, we include the singlet proton and triplet neutron pairing gaps simultaneously in the calculation of the rate and emissivity of this process, where the dependence of these pairing gaps on the nucleon density is also taken into account. We then compare the predicted surface temperature of neutron stars with the latest observational data. We show the simultaneous inclusion of both proton and neutron gaps is advantageous for the explanation of the old warm neutron stars since it enhances the heating effect. It is then found that the observed surface temperatures of the old warm millisecond pulsars, J2124-3358 and J0437-4715, are explained for various choices of nucleon gap models. The same setup is compatible with the observed temperatures of ordinary pulsars including old warm ones, J0108-1431 and B0950+08, by choosing the initial rotational period of each neutron star accordingly. In particular, the upper limit on the surface temperature of J2144-3933 can be satisfied if its initial period is $\gtrsim 10\,\mathrm{ms}$.