Axisymmetric and stationary structures of magnetized barotropic stars with extremely strong magnetic fields deep inside

We have succeeded in obtaining magnetized star models that have extremely strong magnetic fields in the interior of the star. In our formulation, arbitrary functions of the magnetic flux function appear in the expression for the current density. By appropriately choosing the functional form for one of the arbitrary functions that corresponds to the distribution of the toroidal current density, we have obtained configurations with magnetic field distributions that are highly localized within the central part and near the magnetic axis region. The absolute values of the central magnetic fields are stronger than those of the surface region by two orders of magnitude. By applying our results to magnetars, the internal magnetic poloidal fields could be 10 ¹⁷G, although the surface magnetic fields are about 10 ¹⁵G in the case of magnetars. For white dwarfs, the internal magnetic poloidal fields could be 10 ¹²G, when the surface magnetic fields are 10 ⁹-10 ¹⁰G.