Rotochemical heating in millisecond pulsars with Cooper pairing

When a rotating neutron star loses angular momentum, the reduction in the centrifugal force makes it contract. This perturbs each fluid element, raising the local pressure and originating deviations from beta equilibrium that enhance the neutrino emissivity and produce thermal energy. This mechanism is named rotochemical heating and has previously been studied for neutron stars of non-superfluid matter, finding that they reach a quasi-steady state in which the rate that the spin-down modifies the equilibrium concentrations is the same to that of the neutrino reactions restoring the equilibrium. On the other hand, the neutron star interior is believed to contain superfluid nucleons, which affect the thermal evolution of the star by suppressing the neutrino reactions and the specific heat, and opening new Cooper pairing reactions. In this work we describe the thermal effects of Cooper pairing with spatially uniform energy gaps of neutrons and protons on rotochemical heating in millisecond pulsars (MSPs) when only modified Urca reactions are allowed. We find that the chemical imbalances grow up to a value close to the energy gaps, which is higher than the one of the nonsuperfluid case. Therefore, the surface temperatures predicted with Cooper pairing are higher and explain the recent measurement of MSP J0437-4715.Comment: VIII Symposium in Nuclear Physics and Applications: Nuclear and Particle astrophysics. Appearing in the American Institute of Physics (AIP) conference proceeding

Rotochemical Heating in Millisecond Pulsars. Formalism and Non-superfluid case

Rotochemical heating originates in a departure from beta equilibrium due to spin-down compression in a rotating neutron star. The main consequence is that the star eventually arrives at a quasi-equilibrium state, in which the thermal photon luminosity depends only on the current value of the spin-down power, which is directly measurable. Only in millisecond pulsars the spin-down power remains high long enough for this state to be reached with a substantial luminosity. We report an extensive study of the effect of this heating mechanism on the thermal evolution of millisecond pulsars, developing a general formalism in the slow-rotation approximation of general relativity that takes the spatial structure of the star fully into account, and using a sample of realistic equations of state to solve the non-superfluid case numerically. We show that nearly all observed millisecond pulsars are very likely to be in the quasi-equilibrium state. Our predicted quasi-equilibrium temperatures for PSR J0437-4715 are only 20% lower than inferred from observations. Accounting for superfluidity should increase the predicted value.Comment: 34 pages, 8 figures, AASTeX. Accepted for publication in Ap

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

Context: When a rotating neutron star loses angular momentum, the reduction in the centrifugal force makes it contract. This perturbs each fluid element, raising the local pressure and originating deviations from beta equilibrium that enhance the neutrino emissivity and produce thermal energy. This mechanism is named rotochemical heating and has previously been studied for neutron stars of nonsuperfluid matter, finding that they reach a quasi-steady configuration in which the rate at which the spin-down modifies the equilibrium concentrations is the same at which neutrino reactions restore the equilibrium. Aims: We describe the thermal effects of Cooper pairing with spatially uniform energy gaps of neutrons \Delta_n and protons \Delta_p on the rotochemical heating in millisecond pulsars (MSPs) when only modified Urca reactions are allowed. By this, we may determine the amplitude of the superfluid energy gaps for the neutron and protons needed to produce different thermal evolution of MSPs. Results: We find that the chemical imbalances in the star grow up to the threshold value \Delta_{thr}= min(\Delta_n+ 3\Delta_p, 3\Delta_n+\Delta_p), which is higher than the quasi-steady state achieved in 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_\gamma ~ (1-4) 10^{32}\Delta_{thr}/MeV \dot{P}_{-20}/P_{ms}^3 erg s^-1. We can explain the UV emission of the PSR J0437-4715 for 0.05 MeV<\Delta_{thr}<0.45 MeV.Comment: (accepted version to be published in A&A

Order-of-magnitude physics of neutron stars

We use basic physics and simple mathematics accessible to advanced undergraduate students to estimate the main properties of neutron stars. We set the stage and introduce relevant concepts by discussing the properties of "everyday" matter on Earth, degenerate Fermi gases, white dwarfs, and scaling relations of stellar properties with polytropic equations of state. Then, we discuss various physical ingredients relevant for neutron stars and how they can be combined in order to obtain a couple of different simple estimates of their maximum mass, beyond which they would collapse, turning into black holes. Finally, we use the basic structural parameters of neutron stars to briefly discuss their rotational and electromagnetic properties.Comment: 13 pages, 3 figures, accepted for publication in European Physical Journal

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.Comment: 23 pages including 5 figures, uuencoded postscript files, IASSNS-AST 94/5