Twisted magnetar magnetospheres

Magnetar magnetospheres are strongly twisted, and are able to power sudden energetic events through the rapid release of stored electromagnetic energy. In this paper, we investigate twisted relativistic force-free axisymmetric magnetospheres of rotating neutron stars. We obtain numerical solutions of such configurations using the method of simultaneous relaxation for the magnetic field inside and outside the light-cylinder. We introduce a toroidal magnetic field in the region of closed field-lines that is associated with a poloidal electric current distribution in that region, and explore various mathematical expressions for that distribution. We find that, by increasing the twist, a larger fraction of magnetic field-lines crosses the light-cylinder and opens up to infinity, thus increasing the size of the polar caps and enhancing the spin-down rate. We also find that, for moderately to strongly twisted magnetospheres, the region of closed field-lines ends at some distance inside the light-cylinder. We discuss the implications of these solutions on the variation of magnetar spin-down rates, moding and nulling of pulsars, the relation between the angular shear and the twist, and the overall shape of the magnetosphere.