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Magnetized stars with differential rotation and a differential toroidal field

We have succeeded in obtaining magnetized equilibrium states with differential rotation and differential toroidal magnetic fields. If an internal toroidal field of a proto-neutron star is wound up from the initial poloidal magnetic field by differential rotation, the distribution of the toroidal magnetic field is determined by the profile of this differential rotation. However, the distributions of the toroidal fields in all previous magnetized equilibrium studies do not represent the magnetic winding by the differential rotation of the star. In this paper, we investigate a formulation of a differential toroidal magnetic field that represents the magnetic field wound up by differential rotation. We have developed two functional forms of differential toroidal fields which correspond to a v-constant and a j-constant field in analogy to differential rotations. As the degree of the differential becomes very high, the toroidal magnetic field becomes highly localized and concentrated near the rotational axis. Such a differential toroidal magnetic field would suppress the low-T/|W| instability more efficiently even if the total magnetic field energy is much smaller than that of a non-differential toroidal magnetic field.