Deviations from Chemical Equilibrium due to Spin-Down as an Internal Heat Source in Neutron Stars

The core of a neutron star contains several species of particles, whose relative equilibrium concentrations are determined by the local density. As the star spins down, its centrifugal force decreases continuously, and the star contracts. The density of any given fluid element increases, changing its chemical equilibrium state. The relaxation towards the new equilibrium takes a finite time, so the matter is not quite in chemical equilibrium, and energy is stored that can be released by reactions. For a neutron star core composed of neutrons (n), protons (p), and electrons (e), the departure from chemical equilibrium is quantified by the chemical potential difference $\delta\mu\equiv \mu_{\rm p}+\mu_{\rm e} -\mu_{\rm n}$. A finite $\delta\mu$ increases the reaction rates and the neutrino emissivity. If large enough ($|\delta\mu|\gta 5kT$), it reduces the net cooling rate because some of the stored chemical energy is converted into thermal energy, and can even lead to net heating. A simple model shows the effect of this heating mechanism on the thermal evolution of neutron stars. It is particularly noticeable for old, rapidly spinning stars with weak magnetic fields. Observational consequences for millisecond pulsars are discussed.