Magnetic field structures inside magnetars with strong toroidal field

We have analyzed the magnetized equilibrium studies with strong toroidal magnetic fields and found that the negative toroidal current density inside the star is very important for the strong toroidal magnetic fields. The strong toroidal magnetic fields require the strong poloidal current, but the strong poloidal current results in the localized strong toroidal current density in the axisymmetric system. This localized toroidal current changes the magnetic field configuration and makes the size of the toroidal magnetic field region smaller. As a result, the toroidal magnetic field energy can not become large. We need to cancel out the localized toroidal current density in order to obtain the large toroidal fields solutions. We have found and showed that the negative toroidal current cancels out the localized toroidal current density and sustain the large toroidal magnetic field energy inside the star. We can explain the magnetized equilibrium studies with strong toroidal magnetic fields systematically using the negative current density. Physical meaning of the negative current is key to the magnetar interior magnetic fields.