General relativistic models for rotating magnetized neutron stars in conformally flat spacetime

The extraordinary energetic activity of magnetars is usually explained in terms of dissipation of a huge internal magnetic field of the order of $10^{15-16}$G. How such a strong magnetic field can originate during the formation of a neutron star is still subject of active research. An important role can be played by fast rotation: if magnetars are born as millisecond rotators dynamo mechanisms may efficiently amplify the magnetic field inherited from the progenitor star during the collapse. In this case, the combination of rapid rotation and strong magnetic field determine the right physical condition not only for the development of a powerful jet driven explosion, manifesting as a gamma ray burst, but also for a copious gravitational waves emission. Strong magnetic fields are indeed able to induce substantial quadrupolar deformations in the star. In this paper we analyze the joint effect of rotation and magnetization on the structure of a polytropic and axisymmetric neutron star, within the ideal magneto-hydrodynamic regime. We will consider either purely toroidal or purely poloidal magnetic field geometries. Through the sampling of a large parameter space, we generalize previous results in literature, inferring new quantitative relations that allow for a parametrization of the induced deformation, that takes into account also the effects due to the stellar compactness and the current distribution. Finally, in the case of purely poloidal field, we also discuss how different prescriptions on the surface charge distribution (a gauge freedom) modify the properties of the surrounding electrosphere and its physical implications.