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

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]