Hall drift of axisymmetric magnetic fields in solid neutron-star matter

Abstract

Context. The Hall drift, namely, the transport of magnetic flux by the moving electrons giving rise to the electrical current, may be the dominant effect causing the evolution of the magnetic field in the solid crust of neutron stars. It is a nonlinear process that, despite a number of theoretical efforts, is still not fully understood. Aims. Through mostly analytic arguments and solutions, we intend to help understand this highly nonlinear process. Methods. We use the Hall induction equation in axial symmetry to obtain some general properties of nonevolving fields, as well as analyzing the evolution of purely toroidal fields, their poloidal perturbations, and current-free, purely poloidal fields. We also analyze energy conservation in Hall instabilities and write down a variational principle for Hall equilibria. Results. We show that the evolution of any toroidal magnetic field can be described by Burgers' equation, as previously found by Vainshtein and collaborators in a plane-parallel geometry. This evolution leads to sharp current sheets, which dissipate on the Hall time scale, yielding a stationary field configuration that depends on a single, suitably defined coordinate. This field, however, is unstable to poloidal perturbations, which grow as their field lines are stretched by the background electron flow, as in the instabilities found numerically by Rheinhardt and Geppert. On the other hand, current-free poloidal configurations are stable and could represent a long-lived crustal field supported by currents in the fluid stellar core. There may be additional, stable configurations, corresponding to restricted local minima or maxima of the magnetic energy. Conclusions. Hall equilibria can be described by a simple variational principle. Long-lived, toroidal fields are not expected in neutron star crusts or other regions where Hall drift is the dominant evolution mechanism. However, other stable configurations do exist, such as current-free poloidal fields and possibly others.

Stable magnetic equilibria and their evolution in the upper main sequence, white dwarfs, and neutron stars

“…In principle, the Hall drift might also play a role in the field evolution in neutron stars (Jones 1988;Urpin & Shalybkov 1991;Reisenegger et al 2005Reisenegger et al , 2007. However, its time scale in neutron star cores tends to be somewhat longer than those of the other processes considered here .…”

“…This decomposition makes it explicit that the field depends only on two scalar functions, B and A, and explicitly satisfies the condition of zero divergence independently for both components. As in Reisenegger et al (2007), we choose to write it in terms of the gradient of the azimuthal angle, ∇φ =φ/ , instead of the unit vector,φ, in order to make easy use of the identity ∇ × ∇φ = 0. For reference, we also write the toroidal and poloidal components of the current,…”

“…However, its time scale in neutron star cores tends to be somewhat longer than those of the other processes considered here . Moreover, in an axially symmetric, equilibrium magnetic field configuration, the effect of the Hall drift is exactly cancelled by bulk fluid motions (Reisenegger & Thompson 2009), so we do not take it into account. For simplicity, we also refrain from discussing the role of the solid crust of the neutron star, as well as the effects of superconductivity and superfluidity, which alter the magnetic stresses (Easson & Pethick 1977;Akgün & Wasserman 2008) as well as the dissipative processes.…”

Model X‐Ray Spectra of Magnetic Neutron Stars with Hydrogen Atmospheres

“…2006; Ho 2007; see also Geppert et al 2006;Potekhin et al 2007;Reisenegger et al 2007;Aguilera et al 2008, and references therein, for recent work on surface B and T distributions). Therefore, we provide results from two sets of models: one set with a single B and T eff , as has been done previously for NS atmospheres with fully ionized hydrogen (see NSA; Pavlov et al 1995), and a set which is constructed with B and T eff varying across the surface.…”

Multi-Fluid Simulation of the Magnetic Field Evolution in Neutron Stars

“…The processes of [2] occur in thousands of years, compared to dynamical time scales of milliseconds to seconds, thus we use a slow-motion approximation in which we neglect the inertial terms in the equations of motion for the particles. The plasma species are described as three fluids coupled by collisions and electromagnetic forces, satisfying the equations of motion [2,6]:…”