Appearance of the prolate and the toroidal magnetic field dominated stars: Analytic approach

We have analyzed magnetized equilibrium states and shown a condition for the appearance of the prolate and the toroidal magnetic field-dominated stars using analytic approaches. Both observations and numerical stability analysis support that the magnetized star would have prolate and large internal toroidal magnetic fields. In this context, many investigations concerning magnetized equilibrium states have been tried to obtain the prolate and the toroidal dominant solutions, but many of them have failed to obtain such configurations. Since the Lorentz force is a cross-product of current density and magnetic field, the prolate-shaped configurations and the large toroidal magnetic fields in stars require a special relation between current density and the Lorentz force. We have analyzed simple analytical solutions and found that the prolate and the toroidal-dominant configuration require non-force-free toroidal current density that flows in the opposite direction with respect to the bulk current within the star. Such current density results in the Lorentz force which makes the stellar shape prolate. Satisfying this special relation between the current density and the Lorentz force is a key to the appearance of the prolate and the toroidal magnetic field-dominated magnetized star.