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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 idealized 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 |$f_1 \lesssim 10^{-1.5}$|⁠. [ABSTRACT FROM AUTHOR]

The coupled, time-dependent Gross–Pitaevskii and Ginzburg–Landau equations are solved simultaneously in three dimensions to investigate the equilibrium state and far-from-equilibrium, spin-down dynamics of an interpenetrating neutron superfluid and proton type-II superconductor, as an idealized description of the outer core of a neutron star. The simulations generalize previous calculations without the time-dependent Ginzburg–Landau equation, where proton feedback is absent. If the angle θ between the rotation and magnetic axes does not equal zero, the equilibrium state consists of geometrically complicated neutron vortex and proton flux-tube tangles, as the topological defects pin to one another locally but align with different axes globally. During spin-down, new types of motion are observed. For θ = 0, entire vortices pair rectilinearly with flux tubes and move together while pinned. For θ ≠ 0, vortex segments pair with segments from one or more flux tubes, and the paired segments move together while pinned. The degree to which proton feedback impedes the deceleration of the crust is evaluated as a function of θ and the pinning strength, η. Key geometric properties of vortex-flux-tube tangles, such as filament length, mean curvature, and polarity are analysed. It is found that proton feedback smooths the deceleration of the crust, reduces the rotational glitch sizes, and stabilizes the vortex tangle dynamics. The dimensionless control parameters in the simulations are mutually ordered to match what is expected in a real neutron star, but their central values and dynamic ranges differ from reality by many orders of magnitude due to computational limitations. [ABSTRACT FROM AUTHOR]