Relativistic dynamics of superfluid-superconducting mixtures in the presence of topological defects and an electromagnetic field with application to neutron stars

TL;DR

This paper derives relativistic equations describing superfluid-superconducting mixtures in neutron stars, accounting for electromagnetic interactions, vortices, and temperature effects, providing a foundation for realistic neutron star modeling.

Contribution

The paper presents a comprehensive relativistic framework for superfluid-superconducting mixtures in neutron stars, including magnetic fields and vortices, extending previous nonrelativistic models.

Findings

Derived relativistic dynamic equations for neutron star matter.

Identified discrepancies with previous nonrelativistic formulations.

Simplified equations applicable to typical neutron star conditions.

Abstract

The relativistic dynamic equations are derived for a superfluid-superconducting mixture coupled to the electromagnetic field. For definiteness, and bearing in mind possible applications of our results to neutron stars, it is assumed that the mixture is composed of superfluid neutrons, superconducting protons, and normal electrons. Proton superconductivity of both I and II types is analysed, and possible presence of neutron and proton vortices (or magnetic domains in the case of type-I proton superconductivity) is allowed for. The derived equations neglect all dissipative effects except for the mutual friction dissipation and are valid for arbitrary temperatures (i.e. they do not imply that all nucleons are paired), which is especially important for magnetar conditions. It is demonstrated that these general equations can be substantially simplified for typical neutron stars, for which a kind of magnetohydrodynamic approximation is justified. Our results are compared to the nonrelativistic formulations existing in the literature and a number of discrepancies are found. In particular, it is shown that, generally, the electric displacement ${\pmb D}$ does not coincide with the electric field ${\pmb E}$, contrary to what is stated in the previous works. The relativistic framework developed here is easily extendable to account for more sophisticated microphysics models and it provides the necessary basis for realistic modelling of neutron stars.