Rotochemical heating in millisecond pulsars: modified Urca reactions with uniform Cooper pairing gaps

Context. When a neutron star's rotation slows down, its internal density increases, causing deviations from beta equilibrium that induce reactions, heating the stellar interior. This mechanism, named rotochemical heating, has previously been studied for non-superfluid neutron stars. However, the likely presence of superfluid nucleons will affect the thermal evolution of the star by suppressing the specific heat and the usual neutrino-emitting reactions, while at the same time opening new Cooper pairing reactions.
Aims: We describe the thermal effects of Cooper pairing with spatially uniform and isotropic energy gaps of neutrons Δ_n and protons Δ_p, on the rotochemical heating in millisecond pulsars (MSPs) when only modified Urca reactions are allowed. In this way, we are able to determine the amplitude of the superfluid energy gaps for the neutron and protons needed to produce different thermal evolution of MSPs.
Methods: We integrate numerically, and analytically in some approximate cases, the neutrino reactions for the modified Urca processes with superfluid nucleons to include them in the numerical simulation of rotochemical heating.
Results: We find that the chemical imbalances in the star grow up to the threshold value Δ_thr = min (Δ_n + 3Δ_p, 3Δ_n + Δ_p), which is higher than the quasi-steady state achieved in the absence of superfluidity. Therefore, the superfluid MSPs will take longer to reach the quasi-steady state than their nonsuperfluid counterparts, and they will have a higher a luminosity in this state, given by L_γ^∞,qs = (1-4) × 10³²(Δ_thr/MeV) (dot{P}_-20/P_ms^3) erg s^-1, where dot{P}_-20 is the period derivative in units of 10^-20 and P_ms is the period in milliseconds. We are able to explain the UV emission of the PSR J0437-4715 for 0.05 [MeV] ≲ Δ_thr ≲ 0.45 [MeV]. These results are valid if the energy gaps are uniform and isotropic.