Neutron Stars (NSs) are the most compact objects in the universe endowed with an internal structure. Proposed originally by Baade & Zwicky (1934) in the context of supernova explosions, they where discovered only in 1967 by Hewish et al. (1968) as radio pulsars. Today, NSs are among the most studied objects in high-energy astrophysics because they are known to power many astrophysical source of high energy emission. The extreme conditions characterizing their interior make them also interesting objects from the point of view of nuclear and condense matter physics, and future combined observations of both mass and radius of such compact objects may finally discriminate on the different equations of state (EoS) so far proposed Feroci et al (2012) .

It was immediately evident that NSs can also harbor very high magnetic fields, usually inferred to be in the range 10^8-12G for normal pulsars. It is indeed this very strong magnetic field that is responsible for most of their phenomenology and emission. The amplification of magnetic fields form the initial values prior to collapse to those enhanced values is believed to take place during the formation of the compact object itself: surely due to the compression associated with the collapse of the core of the progenitor star (Spruit 2009), it can be further increased by differential rotation in the core leading to the twisting of fieldlines (Burrows et al 2007), and to possible dynamo effects (Bonanno et al. 2003, Rheinhardt & Geppert 2005). In principle there is a large store of free energy available during and immediately following the collapse of the core and the formation of a proto-NS, such that a magnetic field as high as 10^17-18G could be even reached.

Magnetic fields are a key element in the physics and phenomenology of NS. Virtually nothing of their observed properties can be understood, without considering their effects. In particular the geometry of the magnetic field plays an important role, and even small differences can leads to changes in the physical processes that might be important for NS phenomenology (Harding & Muslimov 2011).

The magnetar model for Anomalous X-Ray Pulsars and Soft Gamma Repeaters (Thompson & Duncan 1996, Mereghetti 2008) suggests that the magnetic field can reach at least values close to 10¹⁶G at the surface of NSs. Accounting also for the effects of dissipative processes (Vigano et al. 2013), given the typical ages of known magnetars (~ 10 ⁴yr), it is not unreasonable to expect that younger magnetars with even higher magnetic fields might exist, and more so immediately after collapse and formation, due to the processes discussed above.

Magnetars could be fundamental also to explain another class of objects typical of high-energy astrophysics, namely Gamma Ray Bursts (GRBs). The combination of a rapid millisecond-like rotation of a compact NS with a magnetic field of typical magnetar strength, can easily drive a relativistic outflow with energetics of the order of ~ 10^49-50 erg s^-1, enough to power a classical Long GRB. Short GRBs have been instead usually associated to merger events, rather than to core collapse of stellar objects, leading to the formation of a rotating Black Hole (BH), similarly to the collapsar scenario for Long GRBs (Woosley 1993, MacFadyen & Woosley 1999). However, the recent discovery, on the one hand of extended emission and flaring activity (pointing to a long-lived engine) (Rowlinson et al. 2010 , Norris & Bonnel 2006), and on the other of a NS of mass 2.1M_sun (Romani et al 2012), suggests that it is not unreasonable to expect a high-mass NS, rather than a BH, to form from the merger of two low-mass NSs. Indeed, these assumptions are in part at the base of the so-called millisecond magnetar models for Long and Short GRBs (Bucciantini et al 2012 , Metzger et al 2011 , Bucciantini et al 2009).

These extremely strong magnetic fields will inevitably introduce deformations of the neutron stars (Haskell et al 2008, Mastrano et al 2013, and references therein). A purely toroidal field is known to make the star prolate, while a poloidal field will tend to make it oblate. Also the distribution of matter in the interior will be affected, depending on the softness or stiffness of the EOS describing the nuclear matter. Deformations could even be revealed: if the system is rotating a natural consequence will be the emission of Gravitational Waves (GWs), and the new generations of detectors could search for the emission by these objects. (Mastrano et al. 2011, Gualtieri et al. 2011, Cutler 2002, DallOsso & Stella 2007) have all estimated the losses of energy due to GWs for newly formed NSs, a process that will compete with the emission of relativistic outflows. More recently an upper limit to the magnetic field inside the Crab Pulsar of 7 x 10¹⁶G has been set from the non-detection of GWs (Mastrano et al 2011).

A newly born proto-NS with magnetic field of the order of 10^15-16G is expected to rapidly settle into an equilibrium configuration, given that the corresponding Alfven crossing time is much smaller than the typical Kelvin-Helmholz timescale (Pons et al 1999). Theoretical models for equilibria of classical magnetized stars have a long tradition, dating back to Chandrasekar & Fermi (1953), up to more recent developments. The main difficulty in solving for magnetized equilibrium models in GR is due to the non-linear nature of Einstein equations for the metric. In particular for Twisted Torus configurations and if rotation is included, many metric terms must be retained and a large set of coupled elliptic partial differential equations has to be solved by means of numerical methods. However, it is well known that non-linear elliptical equations can be numerically unstable, depending on the way the non-linear terms are cast. This might in part explain the discrepancies sometimes present in the literature.

Here are some Wikipedia entries about Neutron Stars

• Neutron Stars
• Pulsars
• Magnetars

A few good books on Neutron Stars

• Black Holes, White Dwarfs and Neutron Stars: The Physics of Compact Objects (by Stuart L. Shapiro, Saul A. Teukolsky )
• Rotation and Accretion Powered Pulsars (by Pranab Ghosh)
• Neutron Stars and Pulsars (ed. Werner Becker)