Axisymmetric and stationary magnetic field structures in neutron star crusts under various boundary conditions

We have calculated many Hall equilibrium states within the neutron star crust under various boundary conditions in order to investigate the influences of the boundary conditions clearly. We have found two important features of these solutions. First, the magnitude of the core magnetic fields affects the toroidal to total magnetic field energy ratio within the crust (E-t/E). If the core magnetic fields are vanished, the crustal toroidal magnetic fields become weak and the typical energy ratio is only E-t/E similar to 0.1%. If the core magnetic fields are strong, however, the crustal toroidal magnetic fields become strong and the typical ratio reaches E-t/E similar to 15%. Second, the core toroidal magnetic fields and the twisted magnetosphere around the star make the size of the crustal toroidal magnetic field regions large. Therefore if the strong core magnetic fields have strong toroidal component, both strength and size of the crustal toroidal magnetic fields become large. These results show that the Hall MHD evolutions would be deeply affected by both inner and outer boundary conditions.