How different is the magnetic field at the core-crust interface from that at the neutron star surface? -- the range allowed in magnetoelastic equilibrium

This study was focused on the investigation of a magnetic field penetrating from the core of a neutron star to its surface. The range of possible field configurations in the intermediate solid crust is less limited owing to the elastic force acting on the force balance. When the Lorentz force is excessively strong, the magnetoelastic equilibrium does not hold, and thus, the magnetic field becomes constrained. By numerically solving for the magnetoelastic equilibrium in a thin crust, the range of the magnetic field at the core-crust interface was determined, while assuming the exterior to be fixed as a dipole in vacuum. The results revealed that the toroidal component should be smaller than the poloidal component at the core-crust interface for the surface dipole, $B_{0} > 2.1 \times 10^{14}$G. Consequently, a strong toroidal field, for example, $B \sim 10^{16}$G, as suggested by free precession of magnetars should be confined to a deep interior core and should be reduced to $B \sim 10^{14}$G at the bottom of the crust. The findings of this study provide insights into the interior field structure of magnetars. Further investigations on more complicated geometries with higher multipoles and exterior magnetosphere are necessary.