Constraining dense-matter superfluidity through thermal emission from millisecond pulsars

As a neutron star spins down, the gradual decrease of the centrifugal force produces a progressive increase of the density of any given fluid element in its interior. Since the chemical'' (or beta’‘) equilibrium state is determined by the local density, this process leads to a chemical imbalance quantified by a chemical potential difference, e.g., \delta\mu=\mu_n-\mu_p-\mu_e, where n, p, and e denote neutrons, protons, and electrons. In the presence of superfluid energy gaps, in this case \Delta_n and \Delta_p, reactions are strongly inhibited as long as both \delta\mu and kT are much smaller than the gaps. Thus, no restoring mechanism is available, and the imbalance will grow unimpeded until \delta\mu=\delta\mu_{thr}=\Delta_n+\Delta_p. At this threshold, the reaction rate increases dramatically, preventing further growth of \delta\mu, and converting the excess chemical energy into heat. The thermal luminosity resulting from this ``rotochemical heating’’ process is $L\sim 2\times 10^{-4}(\delta\mu_{thr}/0.1\MeV)\dot E_{rot}$, similar to the typical x-ray luminosity of pulsars with spin-down power \dot E_{rot}. The threshold imbalance, and therefore the luminous stage, are only reached by millisecond pulsars. A preliminary study of eleven millisecond pulsars with reported ROSAT observations shows that the latter can already be used to start constraining superfluid energy gaps in the theoretically interesting range, ~ 0.1 - 1 MeV.