The mean-field theory of superfluid-superconducting vortex states in the outer core of neutron stars

TL;DR

This paper extends mean-field theory to characterize superfluid-superconducting vortex states in neutron star cores, considering pressure and temperature effects, and reveals vortex core sizes and limitations of London's approximation.

Contribution

The work introduces a generalized mean-field model for vortex states in neutron star cores, accounting for differing critical temperatures and pressure dependence of pairing gaps.

Findings

Vortex core size exceeds magnetic penetration depth in the outer core.

Entrainment increases vortex core size and decreases magnetic penetration depth.

London's approximation is invalid for neutron vortices in the outer core.

Abstract

Purpose: Characterize superfluid-superconducting vortex states at arbitrary pressures with $T_{cp}\neq T_{cn}$, assuming both proton and neutron mean-fields are formed by spin-0 Cooper pairs. Method: The existing mean-field theory is extended to account for $T_{cp}\neq T_{cn}$. The pressure dependence of the pairing gap energy $\Delta_{\alpha0}$ is quantitatively established on the basis of the effective chiral field theory. To link $T_{c\alpha}$ with $\Delta_{\alpha0}$, I use the weak-coupling result $T_{c\alpha}\approx0.57\Delta_{\alpha0}$. A quadratic scaled-temperature ($T/T_{cp}$) dependence of the thermodynamic magnetic field is postulated in analogy with pure superconductors. The $T/T_{c\alpha}$-dependence of the gap $\Delta_{\alpha T}$ is inferred from the many-body approximations for the pure neutron matter. Results: An empirical $T/T_{c\alpha}$-dependence for the mean-field is constructed to account for the interplay between the condensation and the magnetic energies. The superfluid entrainment is found to increase the size of the vortex core and to decrease the effective magnetic penetration depth. The size of the neutron vortex core is found to be larger than the magnetic penetration depth in the outer core. Conclusions: The usual approximation of infinitely thin vortex line (the London's approximation) for the neutron vortex is found to be irrelevant in the entire outer core and for the proton vortex is found to be limited to vicinity of the crust-core transition. The developed mean-field theory paves the way to study the vortex microscopic structure, the angular momentum, the magnetization and the vortex-fluxtube interaction energy.