Deviations from chemical equilibrium due to spin-down as an internal heat source in neutron stars

Abstract

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 uid element increases, changing its chemical equilibrium state. The relaxation towards the new equilibrium takes a nite 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 quanti ed by the chemical potential di erence p + e n . A nite increases the reaction rates and the neutrino emissivity. If large enough (j j > 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 (for non-super uid matter) shows the e ect of this heating mechanism on the thermal evolution of neutron stars. It is particularly noticeable for old, rapidly spinning stars with weak magnetic elds. If the timescale for variations of the rotation rate is much longer than the cooling time, a quasi-equilibrium state is reached in which heating and cooling balance each other and the temperature is completely determined by the current value of _ P=P 3 (or the spin-down power). If only modi ed Urca reactions are allowed, the predicted quasi-equilibrium x-ray luminosity of some millisecond pulsars approaches the upper limits obtained by Danner et al. (1994) from ROSAT data. The predicted x-ray luminosity is much lower if direct Urca or other fast reactions are allowed. In both cases, the luminosity is probably increased if the stellar interior is mostly super uid.

Neutrino emission from neutron stars

“…These estimates show that τ cool is of the same order of magnitude as τ rel for both cases of the slow and fast cooling, with τ rel being several times shorter. Using this fact Reisenegger (1995) concluded that the beta relaxation occurs much faster than the cooling. From our point of view, the difference by a factor of several is not decisive and the non-equilibrium state may persist for some time, i.e, the value of ξ may decrease slowly as the cooling proceeds.…”

“…It is much longer than the typical timescale ∼ 10 −3 s, corresponding to the local compression of matter during the collapse of a neutron star into the black hole (Gourgoulhon and Haensel 1993) or to the radial pulsations of the neutron star. In sufficiently old pulsars, τ rel can also be much longer than the timescale of the local compression of matter, implied by the slowing-down of pulsar rotation (Reisenegger 1995). In all these cases, one has to consider the Urca processes in neutron star matter out of beta equilibrium (Haensel 1992, Reisenegger 1995.…”

“…In sufficiently old pulsars, τ rel can also be much longer than the timescale of the local compression of matter, implied by the slowing-down of pulsar rotation (Reisenegger 1995). In all these cases, one has to consider the Urca processes in neutron star matter out of beta equilibrium (Haensel 1992, Reisenegger 1995.…”

“…In the absence of beta equilibrium, there is a finite difference of the chemical potentials, δµ ≡ µ n − µ p − µ e = 0 (our definition of δµ agrees with that of Haensel 1992, and differs in sign from that of Reisenegger 1995). To be specific, we set δµ > 0, which means an excess of neutrons and deficit of protons and electrons with respect to the fully equilibrium values.…”

“…However, the chemical heating term ∆Γ δµ prevails for ξ > ∼ 5.5, leading to the net heating of the matter. Such a situation is relevant for the neutron star collapse (Gourgoulhon and Haensel 1993), and for the interplay of cooling and spin-down of older pulsars (Reisenegger 1995).…”