Redshift factor and diffusive equilibrium of unbound neutrons in the single nucleus model of accreting neutron star crust

Abstract

Using a Wigner-Seitz approximation with spherical cells, we re-analyze a widely used single nucleus model of accreting neutron star crust. We calculate beta disequilibrium within the crust, which is sizable, and implies that neutron and baryon chemical potentials, $\mu_n$ and $\mu_{\rm b}$, are not equal. We include also non-equilibrium reactions, driven by matter compression, and proceeding in the reaction layers. The constancy of $e^{\Phi}\mu_n$, where the spacetime metric component $g_{00}=e^{2\Phi}$, in the shells between the reaction layers is not applicable, because single electron captures are blocked, so that the neutron fraction is fixed, and therefore neutrons are not an independent component of the crust matter. The absence of neutron diffusion in the shells between the reaction layers, stems from the constancy of the neutron fraction (concentration) in these shells. In the reaction layers, the outward force resulting from neutron fraction gradient is balanced by the inward gravitational force acting on unbound neutrons. Neglecting the thickness of the reaction layers compared to the shell thickness, we obtain condition $e^{\Phi(r)}f_Q(r)g(r)=$constant, where $g$ is Gibbs energy per nucleon, undergoing discontinuous drops on the reaction surfaces, and $f_Q(r)g(r)={\widetilde{g}}(r)$ is a continuous function, due to the factor $f_Q(r)$ canceling the discontinuities (drops) in $g(r)$. The function $f_Q(r)$ is calculated using the Tolman-Oppenheimer-Volkov equations from $f_Q(P)$ and $g(P)$ obtained from the equation of state (EOS) with discontinuites. The constancy of of $e^{\Phi(r)}{\widetilde{g}}(r)$ is an extension of the standard relation $e^{\Phi(r)}{{\mu}}_{\rm b}(r)=$constant, valid in hydrostatic equilibrium for catalyzed crust.