Modeling the structure of magnetic fields in Neutron Stars: from the interior to the magnetosphere

The phenomenology of the emission of pulsars and magnetars depends dramatically on the structure and properties of their magnetic field. In particular it is believed that the outbursting and flaring activity observed in AXPs and SRGs is strongly related to their internal magnetic field. Recent observations have moreover shown that charges are present in their magnetospheres supporting the idea that their magnetic field is tightly twisted in the vicinity of the star. In principle these objects offer a unique opportunity to investigate physics in a regime beyond what can be obtained in the laboratory. We will discuss the properties of equilibrium models of magnetized neutron stars, and we will show how internal and external currents can be related. These magnetic field configurations will be discussed considering also their stability, relevant for their origin and possibly connected to events like SNe and GRBs. We will also show what kind of deformations they induce in the star, that could lead to emission of gravitational waves. In the case of a twisted magnetosphere we will show how the amount of twist regulates their general topology. A general formalism based on the simultaneous numerical solution of the general relativistic Grad-Shafranov equation and Einstein equations will be presented.