The strongest known magnetic fields are found in neutron stars. I briefly discuss how they are inferred from observations, as well as the evidence for their time-evolution. I go on to show how these extremely strong fields are actually weak in terms of their effects on the stellar structure. This is also the case for magnetic stars on the upper main sequence and magnetic white dwarfs, which have similar total magnetic fluxes, perhaps pointing to an evolutionary connection. I suggest that a stable hydromagnetic equilibrium (containing a poloidal and a toroidal field component) could be established soon after the birth of the neutron star, aided by the strong compositional stratification of neutron star matter, and this state is slowly eroded by non-ideal magnetohydrodynamic processes such as beta decays and ambipolar diffusion in the core of the star and Hall drift and breaking of the solid in its crust. Over sufficiently long time scales, the fluid in the neutron star core will behave as if it were barotropic, because, depending on temperature and magnetic field strength, beta decays will keep adjusting the composition to the chemical equilibrium state, or ambipolar diffusion will decouple the charged component from the neutrons. Therefore, the still open question regarding stable hydromagnetic equilibria in barotropic fluids will become relevant for the evolution, at least for magnetar fields, which are likely too strong to be stabilized by the solid crust.

The crust of Neutron Stars can be approximated by a highly conducting solid crystal lattice. The evolution of the magnetic field in the crust is mediated through Hall effect, namely the electric current is carried by the free electrons of the lattice and the magnetic field lines are advected by the electron fluid. Here, we present the results of a time-dependent evolution code which shows the effect Hall drift has in the large-scale evolution of the magnetic field. In particular we link analytical predictions with simulation results. We find that there are two basic evolutionary paths, depending on the initial conditions compared to Hall equilibrium. We also show the effect axial symmetry combined with density gradient have on suppressing turbulent cascade.

Although barotropic matter does not constitute a realistic model for magnetic stars on short timescales, it would be interesting to confirm a recent conjecture that states that magnetized stars with a barotropic equation of state would be dynamically unstable (Reisenegger 2009). In this work we construct a set of barotropic equilibria, which can eventually be tested using a stability criterion. A general description of the ideal MHD equations governing these equilibria is summarized, allowing for both poloidal and toroidal magnetic field components. A new finite-difference numerical code is developed in order to solve the so-called Grad-Shafranov equation describing the equilibrium of these configurations, and some properties of the equilibria obtained are briefly discussed.

We have analyzed the magnetized equilibrium studies with strong toroidal magnetic fields and found that the negative toroidal current density inside the star is very important for the strong toroidal magnetic fields. The strong toroidal magnetic fields require the strong poloidal current, but the strong poloidal current results in the localized strong toroidal current density in the axisymmetric system. This localized toroidal current changes the magnetic field configuration and makes the size of the toroidal magnetic field region smaller. As a result, the toroidal magnetic field energy can not become large. We need to cancel out the localized toroidal current density in order to obtain the large toroidal fields solutions. We have found and showed that the negative toroidal current cancels out the localized toroidal current density and sustain the large toroidal magnetic field energy inside the star. We can explain the magnetized equilibrium studies with strong toroidal magnetic fields systematically using the negative current density. Physical meaning of the negative current is key to the magnetar interior magnetic fields.

We have calculated many Hall equilibrium states within the neutron star crust under various boundary conditions in order to investigate the influences of the boundary conditions clearly. We have found two important features of these solutions. First, the magnitude of the core magnetic fields affects the toroidal to total magnetic field energy ratio within the crust (Et/E). If the core magnetic fields are vanished, the crustal toroidal magnetic fields become weak and the typical energy ratio is only Et / E ~ 0.1%. If the core magnetic fields are strong, however, the crustal toroidal magnetic fields become strong and the typical ratio reaches Et / E ~ 15%. Second, the core toroidal magnetic fields and the twisted magnetosphere around the star make the size of the crustal toroidal magnetic field regions large. Therefore if the strong core magnetic fields have strong toroidal component, both strength and size of the crustal toroidal magnetic fields become large. These results show that the Hall MHD evolutions would be deeply affected by both inner and outer boundary conditions.

I will review our current knowledge on the most magnetic objects in the Universe, a small sample of neutron stars called magnetars. The powerful persistent high energy emission and the flares from these strongly magnetized (1015 Gauss) neutron stars are providing crucial information about the physics involved at these extremes conditions, reserving us many unexpected surprises.

The excellent sensitivity of X-ray telescopes, such as Chandra and XMM-Newton, is ideal for the study of cooling neutron stars, which can emit at these energies. In order to exploit the wealth of information contained in the high quality data, a thorough knowledge of the radiative properties of neutron star atmospheres is necessary. A key factor affecting photon emission is magnetic fields, and neutron stars are known to have strong surface magnetic fields. Here I briefly describe our latest work on constructing magnetic (B ≥ 1010 G) atmosphere models of neutron stars and the NSMAXG implementation of these models in XSPEC. Our results allow for more robust extractions of neutron star parameters from observations.

We study the effects of strong magnetic fields in dense stellar matter within an effective relativistic equation of state with the inclusion of hyperons and Δ(1232)-isobar degrees of freedom. The effects of high magnetic field interactions significantly affect the nuclear equation of state and the macroscopic properties of the star. In this framework we investigate the role of the presence of the Δ-isobars degrees of freedom in structure and in the bulk properties of the compact star.

It is now believed that magnetohydrodynamic equilibria can exist in stably stratified stars due to the seminal works of Braithwaite & Spruit (2004) and Braithwaite & Nordlund (2006). What is still not known is whether magnetohydrodynamic equilibria can exist in a barotropic star, in which stable stratification is not present. It has been conjectured by Reisenegger (2009) that there will likely not exist any magnetohydrodynamical equilibria in barotropic stars. We aim to test this claim by presenting preliminary MHD simulations of barotropic stars using the three dimensional stagger code of Nordlund & Galsgaard (1995).