We construct hydromagnetic neutron star equilibria which allow for a non-zero electric current distribution in the exterior. The novelty of our models is that the neutron stars interior field is in equilibrium with its magnetosphere, thus bridging the gap between previous work in this area which either solves for the interior assuming a vacuum exterior or solves for the magnetosphere without modelling the star itself. We consider only non-rotating stars in this work, so our solutions are most immediately applicable to slowly-rotating systems such as magnetars. Nonetheless, we demonstrate that magnetospheres qualitatively resembling those expected for both magnetars and pulsars are possible within our framework. The inside-out approach taken in this paper should be more generally applicable to rotating neutron stars, where the interior and exterior regions are again not independent but evolve together.
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