Magnetic coupling through flux branching of adjacent type-I and -II superconductors in a neutron star

TL;DR

This paper investigates the magnetic coupling between type-I and type-II superconducting regions in neutron star cores through flux branching, revealing energy requirements and implications for magnetic field decay.

Contribution

It introduces the concept of flux branching as a magnetic coupling mechanism between different superconducting phases in neutron stars, with quantitative energy estimates and implications for pulsar magnetic moments.

Findings

Flux branching couples type-I and type-II superconductors magnetically.

Energy to separate flux tubes is up to ~10^{12} erg for typical core sizes.

Strong coupling can delay magnetic field decay in neutron stars.

Abstract

The inner and outer cores of neutron stars are believed to contain type-I and -II proton superconductors, respectively. The type-I superconductor exists in an intermediate state, comprising macroscopic flux-free and flux-containing regions, while the type-II superconductor is flux-free, except for microscopic, quantized flux tubes. Here, we show that the inner and outer cores are coupled magnetically, when the macroscopic flux tubes subdivide dendritically into quantized flux tubes, a phenomenon called flux branching. An important implication is that up to $\sim 10^{12} (r_1/10^6 \, {\rm cm}) \, {\rm erg}$ of energy are required to separate a quantized flux tube from its progenitor macroscopic flux tube, where $r_1$ is the length of the macroscopic flux tube. Approximating the normal-superconducting boundary as sharp, we calculate the magnetic coupling energy between a quantized and macroscopic flux tube due to flux branching as a function of, $f_1$, the radius of the type-I inner core divided by the radius of the type-II outer core. Strong coupling delays magnetic field decay in the type-II superconductor. For an idealised inner core containing only a type-I proton superconductor and poloidal flux, and in the absence of ambipolar diffusion and diamagnetic screening, the low magnetic moments ($\lesssim 10^{27} \, {\rm G \, cm^3}$) of recycled pulsars imply $f_1 \lesssim 10^{-1.5}$.