Explaining the thermal emission of old neutron stars with rotochemical heating and magnetized superconducting protons
The detection of likely thermal ultraviolet emission from a few old neutron stars suggests that at least one internal heating mechanism is present in these stars. One proposed mechanism is rotochemical heating, in which the continuous contraction of the neutron star due to its spin-down produces chemical imbalances that induce Urca reactions, and the latter deposit heat in the neutron star core. If the protons in the star are superconducting, their energy gap suppresses the reactions, except in microscopic magnetized regions (such as quantized flux tubes) in which the protons act as if they were normal. Therefore, the strength of the internal magnetic field controls the rate at which reactions proceed and thus affects the thermal evolution of the neutron star. Here, we present the first comprehensive study of the effect of an internal magnetic field in the superconducting interior on rotochemical heating. We simulate the evolution of neutron stars for different internal magnetic field strengths and neutron energy gaps, comparing the results to Hubble Space Telescope observations of old neutron stars. All the observational data can be accounted for if the proton energy gap is large ($\sim 1.5\,\mathrm{MeV}$) and the neutron energy gap is small ($\lesssim 0.1\,\mathrm{MeV}$) or vanishing, while the millisecond pulsar PSR~J0437$-$4715 needs to have a very weak internal magnetic field. Our results suggest that neutron-star cores are characterized by a large proton pairing gap and a small or vanishing neutron gap, and that millisecond pulsars have very weak internal magnetic fields. Under these conditions, rotochemical heating alone can account for the observed thermal emission of old neutron stars.